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An abstract hyperbolic population model for the transdermal transport of ethanol in humans: estimating the distribution of random parameters and the deconvolution of breath alcohol concentration
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An abstract hyperbolic population model for the transdermal transport of ethanol in humans: estimating the distribution of random parameters and the deconvolution of breath alcohol concentration
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An Abstract Hyperbolic Population Model for the Transdermal Transport of Ethanol in Humans: Estimating the Distribution of Random Parameters and the Deconvolution of Breath Alcohol Concentration By Zheng Dai A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of Requirements for the Degree DOCTOR OF PHILOSOPHY Applied Mathematics December 2020 Copyright 2020 Zheng Dai Contents List of Tables iv List of Figures v Abstract vii 1 Introduction 1 1.1 The Alcohol Biosensor Problem . . . . . . . . . . . . . . . . . . . . . 1 1.2 Need for Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Population Model Approach . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4.1 Earlier works of Rosen and Luczak (deterministic, with cali- bration) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4.2 Replacing the alcohol challenge with a drinking diary . . . . 7 1.4.3 Sirlanci et al.'s work on population models based on random parabolic PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 Summary and Outline of the Remainder of the Thesis . . . . . . . . 8 2 Preliminaries 10 3 A Hyperbolic Model for Diusion and the Transdermal Trans- port of Ethanol 14 3.1 Parabolic Model for Diusion . . . . . . . . . . . . . . . . . . . . . . 14 3.2 The Innite Speed of Propagation . . . . . . . . . . . . . . . . . . . 15 3.3 The Telegraph Equation . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3.1 The Telegraph Equation Model for Diusion . . . . . . . . . 17 4 Abstract Formulation of the Deterministic Hyperbolic Model 19 4.1 Abstract Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2 Parabolic Regularization for Hyperbolic Model . . . . . . . . . . . . 21 4.3 Discrete-Time Formulation of Telegraph Equation with Zero-Order Hold Input on the Boundary of the Domain . . . . . . . . . . . . . . 23 4.4 Finite-Dimensional Approximation and Convergence . . . . . . . . . 27 ii 4.4.1 Telegraph Equation with Parabolic Regularization . . . . . . 27 4.4.2 Telegraph Equation with Zero-Order Hold Input on the Bound- ary of the Domain . . . . . . . . . . . . . . . . . . . . . . . . 30 5 Abstract Formulation of the Hyperbolic Model with Random Parameters 35 5.1 Random Regularly Dissipative Operators and Associated Semigroups 35 5.2 Parabolic Regularization for Hyperbolic Model with Random Param- eters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.3 Discrete-Time Formulation with Parabolic regularization . . . . . . . 42 5.4 Discrete-Time Formulation of Telegraph Equation with Zero-Order Hold Input on the Boundary of the Domain . . . . . . . . . . . . . . 43 5.5 Finite-Dimensional Approximation and Convergence . . . . . . . . . 46 6 Estimating Distribution of Random Parameter 51 6.1 Training, Calibrating, or Fitting the Model to Data . . . . . . . . . . 51 6.2 Deconvolution Problem: Obtaining BrAC from TAC . . . . . . . . . 53 7 Application to the Alcohol Biosensor Problem 57 8 Numerical studies 66 8.1 Computational Considerations . . . . . . . . . . . . . . . . . . . . . 66 8.2 Numerical Results with Simulated Data . . . . . . . . . . . . . . . . 72 8.3 Numerical Results with Alcohol Biosensor Data . . . . . . . . . . . . 77 8.3.1 Dataset 1 - Single Subject with 11 Drinking Episodes . . . . 77 8.3.2 Dataset 2 - Multiple Subjects, each with a Single Drinking Episode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 8.4 Comparison between the results of the hyperbolic model and parabolic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 9 Discussion and concluding remarks 94 iii List of Tables 8.1 Optimal distribution for each episode . . . . . . . . . . . . . . . . . . 79 8.2 Dataset 1:Estimation of the peak BrAC value . . . . . . . . . . . . . 88 8.3 Dataset 1:Estimation of the peak BrAC time . . . . . . . . . . . . . 88 8.4 Dataset 2:Estimation of the peak BrAC value . . . . . . . . . . . . . 90 8.5 Dataset 2:Estimation of the peak BrAC time . . . . . . . . . . . . . 90 8.6 Comparison between the hyperbolic and parabolic models- Estima- tion of the peak BrAC value . . . . . . . . . . . . . . . . . . . . . . . 92 8.7 Comparison between the hyperbolic and parabolic models-Estimation of the peak BrAC time . . . . . . . . . . . . . . . . . . . . . . . . . . 93 8.8 Comparison between the hyperbolic and parabolic models-Estimation of the peak BrAC value . . . . . . . . . . . . . . . . . . . . . . . . . 93 iv List of Figures 1.1 Two examples of BrAC and its corresponding TAC, with TAC ex- hibiting markedly dierent attenuation and latency properties. . . . 2 8.1 Input function u(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 8.2 Simulated TAC corresponding to u(t) for training . . . . . . . . . . . 73 8.3 Simulated TAC corresponding to u(t) for testing . . . . . . . . . . . 74 8.4 Training on 100 simulated episodes . . . . . . . . . . . . . . . . . . . 75 8.5 Simulated Test Episode 1: Estimating the input function u(t) with TAC from Test Episode 1 and the optimal distribution of the param- eters from the averaged training episodes . . . . . . . . . . . . . . . 76 8.6 Simulated Test Episode 2: Estimating the input function u(t) with TAC from Test Episode 2 and the optimal distribution of the param- eters from the averaged training episodes . . . . . . . . . . . . . . . 76 8.7 Simulated Test Episode 3:Estimating the input function u(t) with TAC from Test Episode 3 and the optimal distribution of the param- eters from the averaged training episodes . . . . . . . . . . . . . . . 77 8.8 BrAC and TAC measurements for Dataset 1 . . . . . . . . . . . . . . 78 8.9 Numerical results for each episode computed by a truncated multi- variate normal distribution with each their own optimal parameters 80 8.10 Numerical results for each episode computed by a truncated multi- variate normal distribution with the median of 11 sets of optimal distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8.11 The density function of the optimal distribution from 11 episodes . . 83 8.12 The density function of the optimal distribution from Group 1 . . . 84 8.13 The density function of the optimal distribution from Group 2 . . . 85 8.14 Estimated BrAC using the TAC from Episode 3 and the optimal distribution of the parameters from a) all 11 episodes (top left), b) Group 1 (top right), c) Group 2 (bottom ) . . . . . . . . . . . . . . . 86 8.15 Estimated BrAC using the TAC from Episode 9 and the optimal distribution of the parameters from a) all 11 episodes (top left), b) Group 1 (top right), c) Group 2 (bottom ) . . . . . . . . . . . . . . . 86 v 8.16 Estimated BrAC using the TAC from Episode 10 and the optimal distribution of the parameters from a) all 11 episodes (top left), b) Group 1 (top right), c) Group 2 (bottom ) . . . . . . . . . . . . . . . 87 8.17 The density function of the optimal distribution from Dataset 2 . . . 89 8.18 Numerical results for the three test episode computed by a truncated multivariate normal distribution with optimal distribution . . . . . . 90 8.19 The density function of the optimal distribution from Group 1 in Dataset 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 vi Abstract This research is motivated by a blind deconvolution problem [25, 26, 30, 31, 33] involving the data from a transdermal alcohol biosensor,tracking the num- ber of ethanol molecules in perspiration. To make these devices practicable, we developed a hyperbolic population model to convert biosensor-measured trans- dermal alcohol concentration (TAC) into blood alcohol concentration (BAC) or breath alcohol concentration (BrAC) consistently and reliably. Two steps are required to solving this problem: one is a parameter estimation prob- lem, and the other is a deconvolution problem. All un-modeled sources of uncertainty observed in individual data were attributed to random eects by dening the parameters in the system as random variables with joint density function. The distribution was then estimated by estimating the joint density function. An abstract approximation framework and convergence theory was developed to estimate the distribution of random parameters in an innite- dimensional discrete-time system. By taking expectation on both sides of the hyperbolic PDEs, the randomness of the parameters of a hyperbolic system was embedded in sesquilinear form or equivalently in random regularly dis- sipative operators. Then, each random variable could be treated as another space-like independent variable. In this way, the parameter estimation prob- lem became an optimization problem over the space of all feasible distributions for the random parameters. In this optimization problem, aggregate popula- tion data,instead of individual data, were used to t the model describing the population's mean behavior. This process allowed us to estimate the distri- bution of the parameters instead of individual parameters. After solving the optimization problem and obtaining the optimal distribution, the model was inverted to produce credible bands for BAC/BrAC corresponding to TAC by solving a deconvolution problem. We demonstrated that the alcohol biosensor model satises all the assumptions we made in developing the approximation framework and convergence theory. Last, numerical results were obtained and presented when using simulated data, multiple episodes from the same subject, and a single episode from multiple subjects. vii 1 Introduction 1.1 The Alcohol Biosensor Problem This research is motivated by a blind deconvolution problem involving the data from a transdermal alcohol biosensor, tracking the number of ethanol molecules in perspiration. The consumption of alcoholic beverages plays an important role when people are socializing, celebrating, and relaxing. However, excessive alcohol con- sumption can take a severe toll on human health, especially on the heart, liver, and pancreas. The denition of \too much" drinking and how it can be measured are crucial to achieving a balance between enjoying alcoholic beverages and avoiding severe health risks. Blood alcohol concentration (BAC) is widely accepted among researchers and law enforcement to measure the alcohol level in the human body. However, a BAC measurement is inconvenient as it requires an analysis of one's blood sample. To solve that, in 1967, British scientists Bill Ducie and Tom Parry Jones developed the rst electronic breathalyzer. It measures the amount of alcohol in one's breath instead of measuring BAC directly. This breakthrough, has led to breath alcohol concentration (BrAC) becoming a popular measurement standard when determining the legal alcohol limit for driving in almost all countries. Despite the portability of a breathalyzer, many factors can in uence the accuracy of its re- sults. Similarly, many alcohol studies rely on self-reporting data from participants regarding the volume and frequency of their alcohol consumption, which can be un- reliable. Aiming for convenience in the procedure and accuracy in subjects' alcohol data, researchers introduced wearable alcohol biosensors. Such devices measure the transdermal alcohol concentration (TAC), which is the amount of ethanol released through the epidermis via perspiration after ingesting alcohol. A biosensor can be easily worn on one's arm or leg (limb), and oers continuous and non-invasive monitoring of one's TAC. However, TAC is still dierent from BAC and BrAC. In the recent forty year, BAC and BrAC have been commonly accepted as the mea- surement standards of alcohol level in humans. In addition, the two measurements have provided better interpretable information about alcohol consumption. There- 1 fore, a reliable and valid calibration process is needed to convert TAC into accurate estimates of BAC/BrAC. With this calibration process, biosensors will oer con- tinuous and non-invasive monitoring of BAC/BrAC, bringing countless benets to researchers and people suering from alcohol-related issues. Figure 1.1: Two examples of BrAC and its corresponding TAC, with TAC exhibiting markedly dierent attenuation and latency properties. 1.2 Need for Calibration Some diculties exist in developing the aforementioned calibration process between BAC/BrAC and TAC. Figure 1.1 illustrates two plots of contemporaneous BrAC and TAC, collected from two dierent subjects, with the two subjects using dier- ent biosensor devices measuring TAC. First, as the gure depicts, the relationship between BrAC and TAC is not linear, and the BrAC measurement cannot be eas- ily converted from TAC through a fundamental linear transformation. Second, the TAC measurements dier signicantly between the two plots, while the measure- ments for both BrAC share similar patterns across subjects. As the two subjects used dierent biosensor devices to measure TAC, this indicates that the relation- ship between BrAC and TAC varies across subjects and devices. Furthermore, the TAC results are not consistent when measured at dierent times, even for the same subject measured by the same device. This inconsistency arises because numerous factors in uence the TAC measurement. Since the biosensor device measures the amount of alcohol in perspiration through the epidermis, the thickness, porosity, and tortuosity of a subject's skin will aect the TAC measurements. Even for the 2 same person with the same level of alcohol, the body and ambient temperature, skin hydration, and vasodilation at dierent times may also contribute to the vari- ance of TAC measurements. Therefore, even if the devices produce accurate TAC measurements, the map between the TAC data and BrAC will be inconsistent. The above issues would also hold true when converting TAC to BAC. As a result, al- cohol research and clinical communities have not widely accepted these devices. The challenge is thus to make these devices practicable by developing a system to convert biosensor-measured TAC into BAC or BrAC consistently and reliably. 1.3 Population Model Approach In works to date on developing a system for estimating BAC/BrAC from TAC [29], Rosen et al. took a deterministic approach to convert TAC into BAC/BrAC. First- principles, physics-based models were t to individual calibration data to capture the transporting process (the transport of ethanol molecules from the blood to the sensor). After tting, the system expressed TAC as a convolution of BAC or BrAC with a kernel or lter. Last, the system estimates the BAC or BrAC from the biosensor measurements of TAC via deconvolution. Furthermore, Sirlanci and Rosen [33, 31, 30] introduced "a rst-principles, physics- based parabolic model to describe the dynamics that are common to the entire population (all individuals, devices, and environmental conditions)." They also at- tributed all un-modeled sources of uncertainty observed in individual data to ran- dom eects. They rst redened the parameters in Rosen's deterministic model as random variables. As a result, the model became a random, parabolic, partial dif- ferential equation (PDE) wherein the parameters are random variables. Aggregate population data, instead of individual data, were then used to t the model de- scribing the population's mean behavior. This process allowed them to estimate the distribution of the parameters instead of individual parameters. The model could ultimately be used to produce means and associated credible bands. More speci- 3 cally, Gittelson', Andreev', and Schwab's theory [10] allowed Sirlanci and Rosen to embed the randomness of parameters of a parabolic system in the sesquilinear form or equivalently in the random regularly dissipative operators; Then, all random variables could be treated as another space-like independent variable. Moreover, the nite-dimensional estimation was treated in the same way as in Banks and Ito's [2] deterministic approach. Then, they formulated an optimization problem (see [31, 30, 33]) over the space of all feasible distributions for the random pa- rameters. For the optimization problem, the goal was to minimize the estimation error, which is the dierence between the expectation of the model's output and the observed TAC. After solving the optimization problem and obtaining the opti- mal distribution, the model was inverted to produce credible bands for BAC/BrAC corresponding to TAC. The partial dierential operators in Sirlanci's [33] ethanol diusion model are ab- stract parabolic operators. Though Sirlanci's model oered some excellent results, questions still remain that the parabolic PDE model cannot answer. In parabolic PDEs, the speed of propagation in the transport process is innity. A speed of innity means that ethanol takes no time to reach the biosensor through the skin. However, ethanol takes time to get through the skin and change the readings of biosensors (see an example in Section 2.2). To improve Sirlanci's model, we use a hyperbolic PDE model with weak damping (a type of telegraph equation) to replace the parabolic PDE model. The hyperbolic PDE model has been used to describe transport processes in mathematical biology [12] because it does not exhibit the unwanted eect of innitely fast propagation. In addition, observation of the alcohol concentration on the boundary is no longer an unbounded output when using this second-order hyperbolic PDE model (see details in Section 4.4). This greatly simplies some mathematics. For example, in the abstract formulation, the output can now be formulated in space H instead of being restricted in H's subspace V . Another benet is that the hyperbolic PDE 4 model yields not only the alcohol concentration but also the rate of the change of concentration. This rate is useful, especially when forecasting BAC/BrAC in the future; however,it will lead us back to the unbounded output when the observation is on the boundary. We can solve this problem by formulating it in the V space, just as Sirlanci did with the parabolic model. Furthermore, there are some challenges to improve Sirlanci's approach. Sir- lanci rst formulated her model abstractly in a Gelfand triple setting [28], before introducing a regular, dissipative operator that generates holomorphic semigroups [11, 13, 21]. With the help of the Trotter-Kato theorem and Gittelson's work [10], she developed an abstract approximation framework and convergence theory to estimate random parameters in these systems. In this thesis, we are solving the following three challenges after replacing parabolic PDEs with hyperbolic PDEs. First, the partial dierential operators in the hyperbolic system are not regularly dissipative, which means that some adjustments are required instead of a direct application of semigroup theorems. Thanks to Banks and Ito's work [2], we can solve this problem by adding parabolic regularization in the form of Kelvin-Voight damping to the hyperbolic system. Second, Gittelson's work is the bridge between random PDEs and well-studied deterministic PDEs. However, Gittelson's work has focused on parabolic PDEs, and we must extend his work to include hyperbolic PDEs. Last, we must demonstrate the convergence of the solutions to a sequence of nite-dimensional estimating problems to the solutions of the original innite- dimensional problem (see futher details in Section 6). 1.4 Literature Review A common approach to solve the alcohol biosensor calibration problem is to treat it as a parameter estimation problem. Parameter estimation problems are often related to the mathematical modeling of natural phenomena in many areas, such as physics, chemistry, ecology, and pharmacokinetics. Researchers rst choose a 5 suitable mathematical model to describe the system's dynamics and then charac- terize it by estimating the parameters. Depending on the problem, the available observations, inputs, and eects in the model, many approaches can be utilized to formulate a parameter estimation problem. Based on the parameters' properties, we can categorize parameter estimation problems as either deterministic, stochastic, or probabilistic. 1.4.1 Earlier works of Rosen and Luczak (deterministic, with calibra- tion) In the case of a deterministic approach, we assume that the parameters are xed con- stants. To solve this calibration problem, Rosen, Luczak, et al. [8, 17, 18, 25, 24, 30] introduced an approach involving the development and identication of physical models containing parameters. First, the authors tted the rst-principles, physics- based models in term of a distributed parameter system with unbounded inputs and outputs. Then, they identied the parameters in the model with individual cali- bration data. The model captured the dynamics of the transportation of alcohol molecules moving from the blood, through the skin, and nally reaching the biosen- sor. In the second step, the inverted model provides the estimated BAC/BrAC (eBAC/eBrAC) based on the TAC. To illustrate the process more accurately, they rst used two-parameter, parabolic diusion equations with appropriate boundary conditions, containing BAC/BrAC as the input and TAC as the output. Then, the parameterized equations could become an impulse-response function or a con- volution lter with TAC as the output and BAC/BrAC as the input. With the help of individual calibration data, they identied the parameters by solving a non-negatively constrained quadratic programming problem and they deconvolved the BAC or BrAC from the convolution lter to obtain eBAC/eBrAC from TAC thereafter. The entire process is a blind deconvolution scheme, which produces eBAC/eBrAC from TAC. 6 1.4.2 Replacing the alcohol challenge with a drinking diary To signicantly increase the utility of devices and reduce the burden on researchers and patients in the calibration process, Rosen, Luczak, and Dai explored a method that does not require the collection of BAC/BrAC. To obtain accurate BAC/BrAC data, researchers must keep volunteers in a lab for hours and measure their alco- hol concentration levels every 5 minutes, which is time-consuming and inecient. Therefore, there is a need to replace BAC/BrAC with easily measured data, and two approaches are available to achieve this. The rst is to employ a drinking diary to estimate BAC/BrAC instead of collecting these measurements in a lab. Many types of research, such as the Widmark's formula, focus on converting a drink- ing diary into BAC/BrAC. This formula employs a drinking diary and a subject's gender, weight, and drinking periods in hours to estimate the peak BAC. In this research, Widmark uses a nonlinear pharmacokinetics model to capture the pro- cess of digesting alcohol after a drink. This model converts a drinking diary into an estimated BAC/BrAC. The estimated BAC/BrAC and corresponding TAC are then used in the calibration process instead of the \actual" BAC/BrAC during that episode. This approach saves researchers and patients from having to stay in a lab and take measurements for hours. The second approach uses a machine learning technique. Assuming that a large number of results are available from Rosen's calibration process, our biosensors when worn, can also detect information other than TAC, for example temperature, a subject's skin conductivity, humidity, and heart rate. With the available informa- tion, we can construct a regression model, whose output consists of the parameters of the calibration process, while the input includes the factors that aect the value of TAC. After identifying all coecients in the regression, we can easily calculate the parameters without any calibration. In other words, collecting BAC/BrAC data in a lab is no longer necessary. To obtain a relatively accurate eBAC/eBrAC, one would only need to wear the device and enter one's personal information. However, 7 these two approaches also have some disadvantages. While reducing the burden of collecting actual BAC/BrAC, the reliability of our eBAC/eBrAC is now in question. 1.4.3 Sirlanci et al.'s work on population models based on random parabolic PDEs In the case of a probabilistic approach, Sirlanci, Rosen, et al. began with the rst-principles, physics-based model, and assumed that the parameters are random variables with unknown distribution. Then, un-modeled sources of uncertainty from all individual data become part of the random parameters. They rst built their model based on random PDEs wherein the parameters in the model are random variables. Thereafter, they estimated the distribution of the parameters by tting aggregate population data. The t model then describes the mean behavior of the population. Finally, they solved an optimization problem to nd the distribution of the parameters. In this case, the model produced means and associated condence or credible bands. 1.5 Summary and Outline of the Remainder of the Thesis The remainder of this thesis is structured as follows. Chapter 2 lists the primary results, including all denitions and theorems from the functional analysis used in the thesis. Chapter 3 then introduces the mathematical model in the form of an ab- stract hyperbolic PDE for the alcohol biosensor problem. In Chapter 4, we present the abstract formulation of our mathematical model when we treat the parameters as constants, and we state and prove related theorems. We subsequently allow the model's parameters to be random variables and derive the abstract formulation in Chapter 5.Moreover, the theorems related to the model's well-posedness and nite- dimensional approximation and convergence are stated and proved in this chapter. Then, in Chapter 6, we formulate the input estimation problem and present the results for the deconvolution problem. In addition, the corresponding theorems are 8 stated and proved. Chapter 7 demonstrates that our alcohol biosensor model satis- es all the assumptions we made in previous chapters, and Chapter 8 contains the numerical results. We also presented the numerical results when using simulated data, multiple episodes from the same subject, and a single episode from multiple subjects. Finally, Chapter 9 provides some analysis, discussions, and concluding remarks. 9 2 Preliminaries In this chapter, we present some primary results from the function analysis that we will use in the remainder of this thesis. The theorems and their proofs in this section can be found in [11] and [13]. Let X be a real or complex Banach space, andfT (t) : 0t1g be a family of operators inL(X), the bounded linear operators from X to X. Denition 2.1. The familyfT (t) : 0 t 1g is called a semigroup in X. C 0 semigroup or strongly continuous in X if 1. T (0) =I, 2. T (t +s) =T (t)T (s) for every t;s> 0. Note: the family T (t) : 0t inf is called a C 0 semigroup or is strongly contin- uous in X if T (t) is a semigroup, and for every x2 X, the mapping t! T (t)x is continuous on [0;1). Denition 2.2. The family T (t) is called an analytic semigroup in X if T (t) is a C 0 semigroup, and for every x2X, the mapping t!T (t)x is a real analytic on (0;1). Denition 2.3. OperatorA is called the innitesimal generator ofC 0 semigroup T (t) if dom(A) =fx2X : lim t!0 T (t)xx t existsg and Ax = lim t!0 T (t)xx t for x2 dom(A) Theorem 2.4. For each C 0 semigroup T (t), real numbers M 1 and w2 R exist such thatjT (t)Me wt j for t 0. If a linear operator is the innitesimal generator of a C 0 semigroup T (t) and satises such exponential growth, then we write it as A2G(M;w). 10 Denition 2.5. A2G(1; 0) is the innitesimal generator of the contraction semi- group wherejT (t)xT (t)yjjxyj Theorem 2.6. Let A(q) be the innitesimal generator of C 0 semigroup T (t). Then, T (t)x = lim n!1 (I t n A) n x for each x2X, and the limit is uniform in t on compact subsets of [0;1]. Theorem 2.7. Let A(q) be the innitesimal generator ofC 0 semigroupT (t). Then for each x2 domA, the mapping t! T (t)x2 domA is continuously dierentiable on (0;1) and d dt T (t)x =AT (t)X =T (t)Ax: Denition 2.8. LetX be a normed space, andA : domA!X is a linear operator such that domAX. The set of all satisfying the following conditions is called a resolvent set, denoted by (A): 1. The range of operator IA : domA!X is dense; 2. IA has a continuous inverse on this set. Denote the resolvent operator by R (A) = (IA) 1 . Theorem 2.9. Let A2G(M;w) be the innitesimal generator for T (t). Then R (A)x = Z 1 0 e t T (t)xdt; for any satisfying Re >w and all x2X. Theorem 2.10. (Hille-Yosida) For Mleq1;w2 R, we have A2 G(M;w) if and only if 1. A is closed and densely dened, dom(A) =X; 2. for real >w, we have 2(A), and R (A) satises jR (A) n j M (w) n 11 n = 1; 2;::: . Denition 2.11. LetA be a linear operator inX;F denotes the duality mapping in X. If f2 Fu H exists satisfying Re (f;Au) 0 for any u2 D(A), then A is called a dissipative operator, and ifA is a dissipative operator, thenA is called an accretive operator. Denition 2.12. If A is a dissipative operator and R(A) = X for some satisfying Re > 0, then A is maximal dissipative. Theorem 2.13. Every dissipative operator has a maximal dissipative extension Theorem 2.14. Let A be a closed dissipative operator. A is maximal dissipative if and only if A is a dissipative operator. In this case, A is also a maximal dissipative operator. Theorem 2.15. Let A be a closed operator with its domain D(A) dense. Both A andA are dissipative if and only if half-planef :Re> 0g is contained in (A), andj (A) 1 j V 1 Re holds in the half-plane. Theorem 2.16. (Lumer-Philips) Let A be a linear operator in a Hilbert space X. 1. IfA is densely dened, thenAwI is dissipative for some realw andR( 0 I A) =A for some 0 with Re 0 >w, then A2G(1;w). 2. If A2G(1;w), then A is densely dened, and AwI is dissipative R( 0 I A) =A for all 0 with Re 0 >w. Denition 2.17. Let a(u;v) be a sesquilinear form dened on VV . That is, each u;v2 V corresponds to a complex number a(u;v), which is linear in u and antilinear in v: a(u 1 +u 2 ;v) =a(u 1 ;v) +a(u 2 ;v) a(u;v 1 +v 2 ) =a(u;v 1 ) +a(u;v 2 ) a(u;v) =a(u;v);a(u;v) = a(u;v) 12 Denition 2.18. (Gelfand triple) Let V and H be Hilbert spaces with V ,! H (V is continuously and densely embedded in H).We obtain the Gelfand triple V ,! H ,! V , where V is the dual space of V . Let <; > H denote the H inner product andjj H ;jj V denote the norms on H and V . Denition 2.19. (Lax-Milgram) LetV ,!H ,!V be a Gelfand triple anda(u;v) be a sesquilinear form dened on VV satisfying the following conditions: 1. ja(u;v)j juj V jvj V for all u;v2V 2. a constant > 0 exists such thatja(u;u)jjuj 2 V . Then, the operator A : V ! V dened by a(u;v) =< Au;v > V ;V is a linear isomorphism between V and V . Moreover, A 1 is continuous from V to V with jA 1 j L(V ;V ) 1 . Theorem 2.20. (Trotter-Kato) Let X and X N be Hilbert spaces such that X N X. Dene orthogonal projection P N :X!X N . Assume P N x!x as N!1 for all x2 X. Let A N and A be an innitesimal generator of C 0 semigroup S N (t) and S(t) on X N and X, satisfying the following conditions: 1. M;w exists such thatjS N (t)jMe wt for each N; 2. dense subset D exists in X such that for some , (IA)D is dense in X and A N P N x!Ax for all x2D; 3. (replacement of the second condition)2(A)\ 1 N=1 (A N ) exists withRe()> w so that R (A N )P N x!R (A)x for each X2X. Then, for each x2X, S N (t)x!S(t)x uniformly in t on compact intervals [0,T]. 13 3 A Hyperbolic Model for Diusion and the Trans- dermal Transport of Ethanol 3.1 Parabolic Model for Diusion Rosen and Sirlanci [31] derived the diusion model from Fick's law and conservation of mass. Let '(t;) denote ethanol's concentration in moles/cm 2 in the skin at t seconds and depth cm. The following equation is the general form for the heat or diusion equation in one dimension: @' @t (t;) =D @ 2 ' @ 2 (t;); 0<<L;t> 0 (3.1) where D represents the diusivity coecient ( in cm 2 per second), and L is the thickness of the epidermal layer (in cm). They assumed the ux of ethanol passing through the epidermis (which does not have a blood supply) is proportional to the concentration of ethanol in the blood. They also assumed that the ux of ethanol evaporating from the skin's surface is proportional to the alcohol concentration at the boundary of the epidermal layer. The following are boundary conditions: @' @ (t; 0)'(t; 0) = 0; @' @ (t;L) =(t);t> 0 (3.2) where > 0 and > 0 are the proportionality constants (in cm / seconds and cm seconds BAC/BrAC's units). Moreover, they assumed that initially there is no ethanol in the epidermal layer, thus leading to the following initial conditions: '(0;) = 0; 0<<L (3.3) After appropriate changes to variables or conversion to dimensionless quantities, the following diusion system is the one treated by Sirlanci et al. in [31] (see further details in Section 3.4): 14 @' @t (t;) =q 1 @ 2 ' @ 2 (t;); 0<< 1;t> 0 (3.4) q 1 @' @ (t; 0)'(t; 0) = 0;q 1 @' @ (t; 1) =q 2 (t);t> 0 (3.5) '(0;) = 0; 0<< 1 (3.6) y(t) ='(t; 0);t> 0 (3.7) where q 1 and q 2 are two unknown dimensionless and, in general, subject-, environment-, and device-dependent parameters. Their underlying modeling as- sumptions are that a) the ethanol ux through the upper surface of the skin ( = 0) is proportional to the ethanol concentration at the surface, and b) the ethanol ux through the lower surface of the epidermal layer ( = 1) is proportional to the level of alcohol in the blood (or breath as a surrogate). In [31],they assumed that the alcohol concentration is zero in the epidermal layer att = 0. They also believed that the TAC sensor measures alcohol concentration (i.e., TAC) at the upper surface of the epidermal layer ( = 0). 3.2 The Innite Speed of Propagation As mentioned in the introduction, the disadvantage of using a parabolic PDE model is the assumption of the innite speed of propagation. The following is a simple example: @' @t (t;) =D @ 2 ' @ 2 (t;);t> 0 (3.8) '(0;) = 0 () (3.9) 15 where 0 () is a Dirac delta function. Then, the solution to this heat equation can be written as x(t;) = Z 1 1 1 p 4Ddt exp (y) 2 4t 0 (y)dy = 1 p 4Dt exp 2 4Dt : For a short time t = and any x, we have x(;) = 1 p 4D exp 2 4D > 0. Therefore, the parabolic model suggests that the biosensor will have TAC readings immediately after the alcohol concentration in the blood becomes nonzero, which one would not expect to be true. 3.3 The Telegraph Equation Apart form parabolic PDEs, the telegraph equation also sometimes describes diu- sion (see [7]). In this thesis, we replace Rosen' and Sirlanci's parabolic model with a hyperbolic tTelegraph equation, which is a hyperbolic PDE with weak damping. The following is the general form of the telegraph equation, a damped wave equation ([19]): @ 2 ' @t 2 + @' @t =D @ 2 ' @ 2 '(t;) is the fundamental solution to the equation above with a point source at x = 0 and t = 0: '(t;) = 8 > > < > > : 1 N exp[ t 2 ]I 0 [ 1 N p ]; jj< q D t; 0 otherwise: '(t;) converges to the solution '(t;) = 1 p 4Dt exp 2 4Dt of the diusion as ! 0. Here I 0 is the modied Bessel function, = Dt 2 =x 2 and N = p D. The solution demonstrates that perturbations governed by the telegraph equation spread with a nite speed q D , as expected for a damped wave equation. 16 3.3.1 The Telegraph Equation Model for Diusion In earlier eorts, researchers used a one-dimensional rst-order Fick's Law-based diusion (or heat) equation to model ethanol's transport through the skin (see, for example, [6, 9, 26]). In this thesis, we use a second-order hyperbolic system, namely, the telegraph equation [20], because we want the time rate of change of concentration to be one of the state variables. This transport model, which takes the form of a damped wave equation, also has an advantage over the standard diusion model: it does not exhibit innite speed of propagation. After converting to essentially dimensionless quantities (see [26]), we obtain the input/output model ' tt (t;) +q 2 ' t (t;) =q 1 ' (t;); 0<< 1; t> 0; (3.10) q 1 ' (t; 0)'(t; 0) = 0; t> 0; (3.11) q 1 ' (t; 1) =q 3 u(t); t> 0; (3.12) '(0;) =' 0 (); ' t (0;) =' 1 (); 0<< 1; (3.13) y(t) = ['(t; 0);' t (t; 0)] T t> 0; (3.14) in the form of a one-dimensional telegraph equation with input and output on the boundary and three unknown parameters, q = (q 1 ;q 2 ;q 3 ). In the system (3.10)- (3.14), '(t;) is essentially the concentration of ethanol in the interstitial uid in the epidermis of the skin at depth and time t. Moreover, u is the concentration of alcohol in the blood (BAC) measured by a breath analyzer (BrAC), and y is the TAC. The boundary condition (3.11) models the evaporation of ethanol from the skin's surface. The condition (3.12) captures ethanol molecules' exchange between the dermal and epidermal layers of the skin, and the output equation (3.14) models the biosensor measured TAC at the skin's surface and the time rate of the TAC change. We typically assume that there is no alcohol in the skin initially; therefore, in general, ' 0 =' 1 = 0 in (3.13). We use linear semigroup theory [14, 22, 32], and tools from functional analysis 17 to reformulate (3.10)-(3.14) as a discrete-time single input system in an innite- dimensional Hilbert space. In (3.10)-(3.14), the input and output are on the boundary, (3.12) and (3.13). The resulting input and output operators are conse- quently unbounded for the standard state-space,H 1 (0; 1)L 2 (0; 1). By introducing parabolic regularization [16] and restricting the system to zero-order hold inputs, we can obtain a discrete or sampled time formulation wherein the input and output operators are bounded. 18 4 Abstract Formulation of the Deterministic Hy- perbolic Model 4.1 Abstract Formulation To formulate the abstract form of our model, we start with some denitions. Let V and H be Hilbert spaces with the embeddings V ,! H ,! V being dense and continuous (Gelfand triple setting); V denotes the space of continuous linear func- tionals on V . Leth;i andjj denote the H inner product and norm, respectively, and lethh;ii andjjjj denote the inner product and norm on V . Furthermore, letfQ;d Q g denote the compact subset of R 3 of feasible parameters. Then for q2fQ;d Q g, a(q;;) : VV ! R is known as a V -bounded, Q-continuous, and V -coercive bilinear form if it satises the following conditons: (i) (Boundedness)ja(q; 1 ; 2 )j 0 jj 1 jjjj 2 jj, 1 ; 2 2V , q2Q; (ii) (V-Coercivity) a(q; ; ) + 0 j j 2 0 jj jj 2 , 2V , q2Q; (ii)' (H-Semi-coercivity) a(q; ; ) 1 j j 2 , 2V , q2Q; (iii) (Continuity) For 1 ; 2 2 V , the map q7! a(q; 1 ; 2 ) is continuous in the sense thatja(q ; 1 ; 2 )a(q ; 1 ; 2 )jd Q (q ;q )jj 1 jjjj 2 jj, for allq ;q 2 Q. For q2fQ;d Q g, a(q;;) : HH! R is an H-bounded, Q-continuous, and H- coercive bilinear form if it satises (i)-(iii) with the V norm,jjjj, replaced by the H norm,jj. In light of conditions (i)-(iii) above, we deneV norm in the following way: jj jj =fa(q; ; ) + 0 j j 2 g 1 2 ; 2V; 19 with the corresponding inner product << 1 ; 2 >>=a(q; 1 ; 2 ) + 0 < 1 ; 2 >; 1 ; 2 2V: Then, forq2fQ;d Q g, leta 1 (q;;) :VV !R be aV -bounded,Q-continuous, and V -coercive bilinear form and a 2 (q;;) : HH ! R be an H-bounded, Q- continuous, and H-coercive bilinear form. Furthermore, b(q),c 1 (q), and c 2 (q) are elements in V with the maps q7!b(q), q7!c 1 (q), and q7!c 2 (q) continuous from Q into V , and they consider an input/output system in weak form as given by h '; i V ;V +a 2 (q; _ '; ) +a 1 (q;'; ) =hb(q); i V ;V u; y = [hc 1 (q);'i V ;V ;hc 2 (q); _ 'i V ;V ] T 2V; (4.1) where '(0) = ' 0 2 V and _ '(0) = ' 1 2 H. h;i V ;V denote the natural extension of theH inner product to the duality pairing betweenV andV . If we assume that b(q);c 2 (q); 1 2H,u2L 2 (0;T ), then for someT > 0, one can demonstrate (see, for example, [16]) that the system (4.1) admits a unique solution 2 C[0;T ;V ) with _ 2C[0;T ;H),y2C[0;T ;R 2 ) and the mapsfu; 0 ; 1 g7!f; _ g andfu; 0 ; 1 g7! y continuous fromL 2 (0;T )VH intoL 2 (0;T ;V )L 2 (0;T ;H) andL 2 (0;T ;R 2 ), respectively. Forq2Q, theq-dependent bilinear form onVV ,a 1 (q;;) :VV !R, denes a bounded linear operator A 1 (q)2L(V;V ) byhA 1 (q) 1 ; 2 i V ;V =a 1 (q; 1 ; 2 ), for 1 ; 2 2 V . Then, if we let ~ H denote any of the spaces V;H or V , we can consider the linear operator A 1 (q) to be the unbounded operator, A 1 (q) : D q ~ H! ~ H where D q = V in the case ~ H = V , and D q =f 2 V : A 1 (q) 2 ~ Hg in the case ~ H = H or ~ H = V . We can also show (see, for example, [3, 4, 32]) that A 1 (q) is a closed, densely dened unbounded linear operator on ~ H. In the Gelfand triple setting, the operatorA 1 (q) is an example of a regularly dissipative operator that generates a holomorphic semigroup (see [14, 22, 32]). Analogously for q2 Q, the q-dependent bilinear form on HH, a 2 (q;;) : HH!R, denes a bounded linear operatorA 2 (q)2L(H;H) byhA 2 (q) 1 ; 2 i = 20 ~ a 2 (q; 1 ; 2 ), for 1 ; 2 2 H. If b(q);c 2 (q) 2 H, then dene the q-dependent bounded linear operatorsB 1 (q) fromR intoH,C 1 (q) fromV intoR, andC 2 (q) from H into R by B 1 (q)w = b(q)w, C 1 (q) =hc 1 (q); i V ;V and C 2 (q) =hc 2 (q); i. Then let H = VH and V = VV and dene operators A(q) : Dom(A(q)) H! H, B(q)2 L(R;H), and C(q)2 L(H;R 2 ) by A(q)(; ) = ( ;A 1 (q) + A 2 (q) ), (; ) 2 Dom(A) = f(;) 2 H : 2 V;A 1 (q) +A 2 (q) 2 Hg, B(q)w = (0;B 1 (q)w), w2 R, and C(q)(;) = (C 1 (q);C 2 (q), (;)2 H, re- spectively. With these denitions, if we assume further that 1 2 H so that x 0 = ( 0 ; 1 )2 H, then the second-order system that is given in (4.1) can be rewritten as a rst-order abstract evolution system in H as _ x(t) =A(q)x(t) +B(q)u(t); t> 0; (4.2) x(0) =x 0 2H; (4.3) y(t) =C(q)x(t); (4.4) where x(t) = ('(t); _ '(t)), t 0. [3, 4, 16] have demonstrated that A(q) is the in- nitesimal generator of aC 0 -semigroupfT(t;q) :t 0g of bounded linear operators on H. The solution to the system (4.2)-(4.4) is given by x(t;q) =T(t;q)x 0 + Z t 0 T(ts;q)B(q)u(s)ds; t 0; (4.5) with x2C(0;T ;H) and y(t;q) =C(q)T(t;q)x 0 + Z t 0 C(q)T(ts;q)B(q)u(s)ds: t 0; (4.6) 4.2 Parabolic Regularization for Hyperbolic Model Given "> 0 and for q2fQ;d Q g use (4.1) to dene the V -bounded, Q-continuous, and V -coercive bilinear form a 2;" (q;;) :VV !R by a 2;" (q; 1 ; 2 ) =a 2 (q; 1 ; 2 ) +"a 1 (q; 1 ; 2 ); 1 ; 2 2V: (4.7) 21 Then instead of the system (4.1), we consider the system with viscosity given by h '; i V ;V +a 2;" (q; _ '; ) +a 1 (q;'; ) =hb(q); i V ;V u; y = [hc 1 (q);'i V ;V ;hc 2 (q); _ 'i V ;V ] T 2V: (4.8) We now have that for q2Q, the q-dependent bilinear form on VV , a 2;" (q;;) : VV !R given by (4.7), denes a bounded linear operator A 2;" (q)2 L(V;V ) byhA 2;" (q) 1 ; 2 i V ;V = ~ a 2;" (q; 1 ; 2 ), for 1 ; 2 2V . We represent the Gelfand triple V ,! H ,! V and note that in this case, V = V V . Then, dene the operator A " (q) : Dom(A " (q)) H! H, by A " (q)(; ) = ( ;A 1 (q) + A 2;" (q) ), (; )2 Dom(A " ) =f(;)2 V : 2 V;A 1 (q) +A 2;" (q)2 Hg. [3, 4, 32] have demonstrated that A " (q) is now an innitesimal generator of a holomorphic or analytic semigroup [22]fT " (t;q) :t 0g of bounded linear operators onH,V, andV . We now assume 1 2V ,b(q);c 2 (q)2V and consider the viscosity evolution system (4.8) rewritten in the rst order form given by _ x " (t) =A " (q)x " (t) +B(q)u(t); t> 0; (4.9) x " (0) =x 0 2H; (4.10) y " (t) =C(q)x " (t): (4.11) It has been shown in [3, 4, 16] that A(q) is the innitesimal generator of a C 0 - semigroupfT(t;q) :t 0g of bounded linear operators on H. The following is the solution to the system (4.9)-(4.11): x " (t;q) =T " (t;q)x 0 + Z t 0 T " (ts;q)B(q)u(s)ds; t 0; (4.12) with x " 2C(0;T ;H) and y " (t;q) =C(q)T " (t;q)x 0 + Z t 0 C(q)T " (ts;q)B(q)u(s)ds: t 0; (4.13) 22 Now withc 2 (q)2V andx " (t) = (' " (t); _ ' " (t))2C(0;T ;H),t 0, and _ ' " (t)2H, (4.13) is consequently still undened. 4.3 Discrete-Time Formulation of Telegraph Equation with Zero-Order Hold Input on the Boundary of the Domain Let be the sampling interval's length and consider the zero-order hold input given by u(t) = u k = [u k 1 ;u k 2 ] T , t2 [k; (k + 1)), k = 0; 1; 2;:::. Then, for k = 1; 2;::: consider the linked set of initial-boundary value problems for a damped wave equation with state ' k (t;) for t2 [(k 1);k] given by ' k tt (t;) +q 2 ' k t (t;) =q 1 ' k (t;); 0<< 1; t2 ((k 1);k); (4.14) q 3 ' k (t; 0)q 4 ' k (t; 0) =q 5 u k1 1 ; t2 [(k 1);k]; (4.15) q 6 ' k (t; 1) +q 7 ' k (t; 1) =q 8 u k1 2 ; t2 [(k 1);k]; (4.16) ' k ((k 1);) =' k1 ((k 1);); 0<< 1; (4.17) ' k t ((k 1);) =' k1 t ((k 1);); 0<< 1; (4.18) where q 1 > 0, q 2 ;q 3 ;q 4 ;q 5 ;q 6 ;q 7 ;q 8 0, q 3 (q 6 +q 7 ) +q 4 q 7 6= 0, ' 0 (0;) = ' 0 (), and ' 0 t (0;) =' 1 (), 0 1. Dene' : [0;1) [0; 1]!R by'(t;) =' k (t;),t2 [k; (k + 1)), 0 1. Then for k = 0; 1; 2;:::, let x k = [x k 1 ;x k 2 ] T = lim t!k ['(t;);' t (t;)] T . For j2 f0; 1; 2;:::g; we dene v(t) = [v 1 (t);v 2 (t)] T 2L 2 (0; 1)L 2 (0; 1) by v 1 (t) =' j+1 (t;) 0 (q;)u j 1 1 (q;)u j 2 ; t2 (j; (j + 1)); (4.19) v 2 (t) =' j+1 t (t;); t2 (j; (j + 1)); (4.20) v 1 (j) =x j 1 0 u j1 1 1 u j1 2 ; (4.21) v 2 (j) =x j 2 ; (4.22) 23 where 0 (q;); 1 (q;)2H 1 (0; 1) are the ramp functions given respectively by 0 (q;) = q 5 q 7 q 3 (q 6 +q 7 ) +q 4 q 7 + q 5 (q 6 +q 7 ) q 3 (q 6 +q 7 ) +q 4 q 7 and 1 (q;) = q 3 q 8 q 3 (q 6 +q 7 ) +q 4 q 7 + q 4 q 8 q 3 (q 6 +q 7 ) +q 4 q 7 : It then follows that q 3 q 5 @ @ 0 (q; 0) q 4 q 5 0 (q; 0) = 1; (4.23) q 6 q 8 @ @ 0 (q; 1) q 6 q 8 0 (q; 1) = 0; (4.24) q 3 q 5 @ @ 1 (q; 0) q 4 q 5 1 (q; 0) = 0; (4.25) q 6 q 8 @ @ 1 (q; 1) q 6 q 8 1 (q; 1) = 1: (4.26) The derivative of v with respect to t for t2 (j; (j + 1)) then yields @ @t v 1 (t) = @ @t ' j+1 (t;) @ @t 0 (q;)u j 1 @ @t 1 (q;)u j 2 = @ @t ' j+1 (t;) (4.27) @ @t v 2 (t) = @ @t ' j+1 t (t;); t2 (j; (j + 1)); (4.28) (4.29) A straightforward calculation reveals thatv 1 (t) =v 1 (t;);v 2 (t) =v 2 (t;) satises v 1 (t) +q 2 _ v 1 (t) =q 1 D 2 v(t); t2 (j; (j + 1)); (4.30) _ v 1 (t) =v 2 (t) (4.31) q 3 q 5 Dv 1 (t)j =0 q 4 q 5 v 1 (t)j =0 = 0; (4.32) q 6 q 8 Dv 1 (t)j =1 + q 7 q 8 v 1 (t)j =1 = 0; (4.33) v 1 (j) =x j 1 0 u j1 1 1 u j1 2 ; (4.34) v 2 (j) =x j 2 (4.35) 24 where D denotes the dierentiation operator on H 1 (0; 1): Consider the weak-form system (4.30)-(4.35); for any 2V , we have < v 1 (t); > +q 2 < _ v 1 (t); > =q 1 <D 2 v(t); >; t2 (j; (j + 1)); (4.36) =q 1 (Dv(t)) ()j =1 =0 q 1 <Dv(t);D >; (4.37) Next, we apply the boundary condition Dv 1 (t)j =0 = q5 q3 q4 q5 v 1 (t)j =0 = q4 q3 v 1 (t)j =0 andDv 1 (t)j =1 = q8 q6 q7 q8 v 1 (t)j =1 = q7 q6 v 1 (t)j =1 to the right-hand side of the weak form (4.37), RHS =q 1 (Dv(t)) ()j =1 =0 q 1 <Dv(t);D >; = q 1 q 7 q 6 v 1 (t) ()j =1 q 1 q 4 q 3 v 1 (t) ()j =0 q 1 <Dv(t);D >: (4.38) LetH =L 2 (0; 1) andV =H 1 (0; 1) with their regular inner products and norms. We hence have the Gelfand triple H 1 (0; 1),!L 2 (0; 1),!H 1 (0; 1). We dene the sesquilinear forms a 1 (q;;) :VV !R, a 2 (q;;) :HH!R as a 1 (q; 1 ; 2 ) = q 1 q 7 q 6 1 (1) 2 (1) + q 1 q 4 q 3 1 (0) 2 (0) +q 1 Z 1 0 0 1 () 0 2 ()dx; 1 ; 2 2V; (4.39) and a 2 (q; 1 ; 2 ) =q 2 Z 1 0 1 () 2 ()dx; 1 ; 2 2V: (4.40) 25 Therefore, we can rewrite the system (4.30)-(4.35) in a weak form h v; i V ;V +a 2 (q; _ v; ) +a 1 (q;v; ) = 0 (4.41) Dene bounded linear operators A 1 (q)2L(V;V ), A 2 (q)2L(H;H) by hA 1 (q) 1 ; 2 i V ;V =a 1 (q; 1 ; 2 ); 1 ; 2 2V hA 2 (q) 1 ; 2 i =a 2 (q; 1 ; 2 ); 1 ; 2 2H and A(q)( ;) = (;A 1 (q) A 2 (q)) (4.42) where ( ;)2 Dom(A) =f( ;)2 VH : 2 V;A 1 (q) +A 2 (q)2 Hg. With these denitions, the second-order system given in (4.41) can be re-written as a rst-order abstract system in VH as _ w(t;q) =A(q)w(t;q) t2 (k; (k + 1)) (4.43) w(k;q) = [x k 1 0 (q)u k1 1 1 (q)u k1 2 ;x k 2 ] T (4.44) where w(t;q) = (v(t);v t (t)) T . Since A(q) is the innitesimal generator of a C 0 -semigroup of contractionsfT(t;q) :t 0g on VV , we have w(t;q) =T(tk;q)w(k;q) t2 [k; (k + 1)): The continuity ofw, and the denitions of' k andw then imply that [' k ;' k t ] T = 26 w(k;q) + [ 0 (q)u k1 1 + 1 (q)u k1 2 ; ~ 0] T , and hence that for k = 0; 1; 2; 3;::: x k+1 = [' k+1 ;' k+1 t ] T =w((k + 1);q) + [ 0 (q)u k 1 + 1 (q)u k 2 ; ~ 0] T (4.45) =T(;q)f[' k ;' k t ] T [ 0 (q)u k 1 + 1 (q)u k 2 ; ~ 0] T g + [ 0 (q)u k 1 + 1 (q)u k 2 ; ~ 0] T (4.46) =T(;q)[' k ;' k t ] T +fIT(;q)g[ 0 (q)u k 1 + 1 (q)u k 2 ; ~ 0] T (4.47) =fT(;q)g k+1 [' 0 ;' 0 t ] T + k X i=0 fT(;q)g ki fIT(;q)g[ 0 (q)u i 1 + 1 (q)u i 2 ; ~ 0] T (4.48) x 0 = [' 0 ;' 0 t ] T = [' 0 ;' 1 ] T (4.49) where 0 (q;) = q 5 q 7 q 3 (q 6 +q 7 ) +q 4 q 7 + q 5 (q 6 +q 7 ) q 3 (q 6 +q 7 ) +q 4 q 7 ; (4.50) 1 (q;) = q 3 q 8 q 3 (q 6 +q 7 ) +q 4 q 7 + q 4 q 8 q 3 (q 6 +q 7 ) +q 4 q 7 : (4.51) That is, we have x k+1 =T(;q)x k +B(q)u k ;k = 0; 1; 2;:::; x 0 = [' 0 ;' 1 ] T ; (4.52) whereB(q)2L(R 2 ;VH) is given byB(q)u = [fIT(;q)g[ 0 (q)u 1 + 1 (q)u 2 ; ~ 0] T . 4.4 Finite-Dimensional Approximation and Convergence 4.4.1 Telegraph Equation with Parabolic Regularization For n = 1; 2;::: , let H n = spanf n j g n j=0 V , and let P n : H! H n denote the orthogonal projection of H onto H n with respect to the H inner product. We require the following assumption on the approximating subspaces H n . 27 Assumption 4.1 For each 2V , a sequencef n g exists with n 2H n , such that jj n jj! 0; as n!1: (4.53) It follows from Assumption 4.1 and the Gelfand triple setting that lim n!1 P n = in H for 2H and if the basis of the spacef n j g n j=0 are chosen as linear spline functions, using the Schmidt inequality (see, for example, [27]), in V as well for 2 V . For n = 1; 2;::: , q 2 Q, and k = 1; 2, dene A n k (q)2 L(H n ;H n ) to be the nite-dimensional linear operator whose matrix representation is given by bA n k (q)c i;j = [h n i ; n j i] 1 [a k (q; n i ; n j )], for i;j = 0; 1; 2;:::;n. Under certain con- ditions, particularly whenA 1 (q) 1 exists, and 0 in (ii) in Section 4.1 is non-positive, it is not dicult to show thatA n 1 (q) = (P n a A 1 (q) 1 ) 1 , whereP n a is the orthogonal projection ofV ontoH n with respect to the inner producth;i a1 =a 1 (q;;) onV . Let H n =H n H n , let "> 0, and dene the operator A n " (q)2L(H n ;H n ) by A n " (q) = 2 6 4 0 I A n 1 (q) A n 2 (q)"A n 1 (q) 3 7 5 (4.54) We then set ^ A" n (q) = e A n " (q) . Recalling that H is dense in V , let ^ P n denotes the standard bounded extension of P n to V ; set B n (q) = 2 6 4 0 ^ P n B 1 (q); 3 7 5 (4.55) and set ^ B n " (q) = R 0 e A n " (q)s B n (q)ds2 L(R;H n ). As in the innite-dimensional case, if A n " (q)2 L(H n ;H n ) is invertible, then it follows that ^ B n " (q) = ( ^ A n " (q) I)(A n " ) 1 (q)B n (q)2L(R;H n ). Finally, letting ^ C n (q) =C(q)2L(H n ;R), we ob- tain the nite-dimensional approximating discrete-time input/output system given 28 by x n ";j+1 = ^ A n " (q)x n ";j + ^ B n " (q)u j ; j = 0; 1; 2;:::; (4.56) y n ";j = ^ C n (q)x n ";j ; j = 0; 1; 2;:::; (4.57) with x n ";0 = x n 0 = (P n 0 ;P n 1 )2 H n . Once again if we assume that the alcohol concentration is zero in the epidermal layer at t = 0, then x 0 =x n ";0 =x n 0 = 0, and it follows from (4.56) and (4.57) that y n ";i (q) = i1 X j=0 C(q)( ^ A n " (q)) ij1 B n " (q)u j = i1 X j=0 C(q)( ^ A n " (q)) ij1 ( ^ A n " (q)I)(A n " ) 1 (q)B n (q)u j = i1 X j=0 ^ h n ";ij (q)u j ; i = 0; 1; 2;::: (4.58) wherein (4.58) ^ h n ";i (q) =C(q) ^ A n " (q) i1 ( ^ A n " (q)I)(A n " ) 1 (q)B n (q); i = 1; 2;::: . If the nite dimensional subspaces H n are constructed as follows using linear spline functions, a convergence result can be obtained. For each n = 1; 2;:::, let f n j g n j=0 denote the set of standard linear B-splines on the interval [0; 1] dened for the usual uniform mesh,fj=ng n j=0 , and set H n = spanf n j g n j=0 V =H 1 (0; 1) (note the n j are the usual "pup tent" or "chapeau" functions with height one and support of width 2=n, j1 n ; j+1 n \ [0; 1]). If P n :H!H n denotes the orthogonal projection ofH =L 2 (0; 1) ontoH n , it is well known (see for example, (Schultz, [27]) that lim n!1 P n = inH for 2H, and inV for 2V . Then, once again using the Trotter-Kato semigroup (see, for example, [4, 14, 22]) approximation theorem, it is not dicult to argue (see [9, 26]) that for each"> 0, lim n!1 e A n " (q)t P n =T " (t;q), strongly inH and inV, uniformly inq forq2fQ;dg, whereP n :H!H n denotes the orthogonal projection of H onto H n given by P n ( 0 ; 1 ) = (P n 0 ;P n 1 ). It then follows that for each " > 0, x n ";j (q)! x ";j (q) in V , y n ";j (q)! y ";j (q), and ^ h n i (q)! ^ h i (q) as n!1, uniformly in q for q2fQ;dg, and uniformly in i for i in 29 bounded subsets ofZ. 4.4.2 Telegraph Equation with Zero-Order Hold Input on the Boundary of the Domain In the previous discussion, the H-semicoercivity condition on a 2 (q;;) is strength- ened to V-coercivity by adding the parabolic regularization term, in order to prove that a(q;;) is V-coercive. However, the same results can be proved without the help of parabolic regularization. We start with the weak form system (4.1): h '; i V ;V +a 2 (q; _ '; ) +a 1 (q;'; ) =hb(q); i V ;V u; y = [hc 1 (q);'i V ;V ;hc 2 (q); _ 'i V ;V ] T 2V; where a 1 (q;;) satises conditions (i),(ii) with 0 = 0 and (iii), while a 2 (q;;) satises conditions (i),(ii') and (iii). In [3], Bank and Ito applied a resolvent con- vergence form of the Trotter-Kato theorem to obtain the same results as those in Section 4.5.1. We will present their approach in the following. First, we consider the resolvent of the operatorA(q) given in (4.42): A(q)( ;) = (;A 1 (q) A 2 (q)) and its resolvent R (A(q)). In H = V H, dene = R (A(q)) for = ( ;), = (; ). It can also be viewed as solving for 2 dom(A(q)) in the equation (IA(q)) =, which may be written as follows: = +A 1 (q) +A 2 (q) = (4.59) By Substituting the rst equation =+ into the second equation, we arrived at 2 +A 1 (q) +A 2 (q) = + +A 2 (q); (4.60) 30 which must be solved for 2V . Next, for > 0, we dene the sesquilinear form a (q) :VV !R given by a (q)( ;&) = 2 < ;& > H +a 1 (q)( ;&) +a 2 (q)( ;&): (4.61) Since a 1 is V-coercive and a 2 is H-semicoercive, for any 2V , we have a (q)( ; ) = 2 j j 2 H +a 1 (q)( ; ) +a 2 (q)( ; ) 2 j j 2 H + 0 k k 2 V + 1 j j 2 H 0 k k 2 V + ^ j j 2 H ( ^ = 2 + 1 > 0) > 0 k k 2 V (4.62) Therefore, the form a (q)(;) is V-coercive. The equation (4.60) is solvable in H, which means that for any dened in (4.59) and any = (; ), we can nd = ( ;) in domA(q) that solves (4.59). In other words, for > 0, R (A(q)) exists in L(H). Next, we consider the Galerkin type approximation scheme described in Section 4.5.1 for the following equation: _ w(t) =A(q)w(t) +F (t) (4.63) where w(t;q) = (v(t);v t (t)) T , and A(q)( ;) = (;A 1 (q) A 2 (q)) (4.64) where ( ;)2 Dom(A) =f( ;)2 V H : 2 V;A 1 (q) +A 2 (q)2 Hg. Moreover, we assume the approximation subspacesH N V satisfying Assumption 4.1. Then for any w2 V = V V and each N, there exists ^ w N 2 H N H N satisfying ^ w N w V ! 0 asN!1. We then restricteda(q) toH N =H N H N and obtained the operators A N (q) :H N !H N . Since we have demonstrated that 31 (4.62) holds, we have R (A N (q)), which exists in L(H N ) for > 0. Let P N be the orthogonal projection ofH ontoH N . Then, we prove the following convergence results. Theorem 4.1. Assume that a 1 (q;;) satises conditions (i),(ii) with 0 = 0 and (iii); a 2 (q;;) satises conditions (i),(ii'), and (iii); and let q N ! q in Q. Then, for > 0, we have R (A N (q N ))P N &!R (A(q))& in the V norm for any &2H. Proof. Let2H be arbitrary; then, w =w(q) =R (A(q)) andw N =w N (q N ) = R (A N (q N ))P N . Let w = ( ;) and = (; ), such that = , + A 1 (q) +A 2 (q) = . Similarly, let w N = ( N =; N =) and P N = ( N ; N ). When q =q N , then N N = N , N +A 1 (q) N +A 2 (q) N = N . According to (4.60) and (4.61), for &2V , we have a (q)( ;&) =< ;& > H +<;& > H +a 2 (q)(;&) a (q N )( N ;&) =< N ;& > H +< N ;& > H +a 2 (q N )( N ;&): (4.65) We then choose ^ w N = ( ^ N ; ^ N ) in H N such that ^ w N w V ! 0 as N !1. Next, choosing & =& N N ^ N in H N V , we obtain the following equation: a (q)( ;& N )a (q N )( N ;& N ) =< N ;& N > H +< N ;& N > H +a 2 (q)(;& N )a 2 (q N )( N ; N ): (4.66) Since a (q)( ; ) 0 k k 2 V + ^ j j 2 H > 0 k k 2 V , then for & = N ^ N we have 0 k&k 2 V + ^ j&j 2 H a (q)(& N ;& N ) =a (q N )( N ;& N )a (q)( ;& N ) +a (q)( ;& N )a (q N )( ^ N ;& N ) =< N ;& N > H +< N ;& N > H +a 2 (q)(;& N ) +a 2 (q N )( N ;& N ) +a (q)( ;& N )a (q N )( ^ N ;& N ) (4.67) Applying the boundedness and continuity in q a 1 and a 2 (conditions (i) and (iii) 32 from Section 4.1) along withjj H kkk V to the equation above, we get a 2 (q)(;& N ) +a 2 (q N )( N ;& N ) +a (q)( ;& N )a (q N )( ^ N ;& N ) =< N ;& N > H +< N ;& N > H +a 2 (q)( N ;& N )a 2 (q)( N ;& N ) +a 2 (q N )( N ;& N ) +a (q)( ^ N ;& N ) +a (q)( ^ N ;& N )a (q N )( ^ N ;& N ) 0 N V & N V +d(q;q N ) N V & N + ( 2 k 2 + 0 + 0 ) ^ N V & N V + (1 +)d Q (q;q N ) ^ N V & N V : (4.68) Since P N is the orthogonal projection of H onto H N , Assumption 4.1 implies that ( N ; N ) = P N (; )! (; )2 H. Therefore, eta N K for some constant K. Since ^ N ! 2V , we can also assume that ^ N K. Combine (4.66),(4.67) and (4.68) together, we have 0 k&k 2 V kj N j H + (k + 0 ) N V +K(2 +)d Q (q;q N ) +C ^ N V (4.69) where C = ( 2 k 2 + 0 + 0 ). Thus, & N = N ^ N ! 0 in V is proved, which implies that N ! in V . Furthermore, since N = N N and = , then we nd N ! in V. Therefore, w N = ( N ; N )! ( ;) in the V norm. LetT N (t;q N ) be theC 0 semigroup generated byA N (q N ) inH N ; then, applying the Trotter-Kato approximation theorem (Theorem 2.20), the following convergence results are obtained immediately. Theorem 4.2. Under the same assumption of Theorem 4.1, for each2H andt> 0, we have T N (t;q N )P N ! T(t;q) in H. Moreover, T N (t;q N )P N uniformly converges to T(t;q) in t on compact subintervals. In particular, for solutions to (4.63) and their approximation, we have v N (t;q N )!v(t;q) in V norm 33 v N t (t;q N )!v t (t;q)in H norm: 34 5 Abstract Formulation of the Hyperbolic Model with Random Parameters In this chapter, we follow the same denitions in Chapter 4;however, we assume the parameters are random variables. We treat the parameter vector q as a p dimensional random vector that has compact support Q p i=1 [a i ;b i ], where1< a< a i <b i < b<1 for all i = 1; 2;:::;p. Let ! a = [a i ] p i=1 , ! b = [b i ] p i=1 , and R r for some r is closed and bounded. A measure = ( ! a; ! b; ! ) can represent the distribution ofq , where ! 2 . In the following section, we will prove thata(q;;) is a sesquilinear form satisfying assumptions (i)-(iv) below. (i) (Boundedness)ja(q; 1 ; 2 )j 0 jj 1 jjjj 2 jj, 1 ; 2 2V , q2Q, (ii) (V-Coercivity) a(q; ; ) + 0 j j 2 0 jj jj 2 , 2V , q2Q, (ii)' (H-Semi-coercivity) a(q; ; ) 1 j j 2 , 2V , q2Q, (iii) (Continuity) For 1 ; 2 2 V , the map q7! a(q; 1 ; 2 ) is continuous in the sense thatja(q ; 1 ; 2 )a(q ; 1 ; 2 )jd Q (q ;q )jj 1 jjjj 2 jj, for allq ;q 2 Q. (iv) (Measurability) For all 1 ; 2 2 V , the map q! a(q; 1 ; 2 ) is measurable on Q with respect to all measures dened in term of the densities in F (Q), where R r is the set of feasible parameters. 5.1 Random Regularly Dissipative Operators and Associated Semigroups Dene ~ Q = R p R p R r = R 3p+r and the Bochner spaces ~ V = L 2 ( ~ Q;V ) and ~ H = L 2 ( ~ Q;H), where V and H are the same spaces dened in Section 4.1. Since V , H, and V form the Gelfand triple, ~ V , ~ H and ~ V also form the Gelfand triple 35 ~ V ,! ~ H ,! ~ V , where ~ V =L 2 ( ~ Q;V ), ~ H, and its dual ~ H =L 2 ( ~ Q;H ). For Q p i=1 [a i ;b i ] satisfying1 < a < a i < b i < b <1 for all i = 1; 2;:::;p , ! 2 and = ( ! a; ! b; ! ), we can dene ()-averaged norms: ~ H normjj ~ H and ~ V normkk ~ V by j j ~ H =f Z Q j j 2 d(q;)g 1 2 ; k k ~ V =f Z Q k k 2 d(q;)g 1 2 =f Z Q (a(q; ; ) + 0 j j 2 )d(q;)g 1 2 with the corresponding inner product << 1 ; 2 >> ~ V; ~ V = Z Q fa(q; 1 ; 2 ) + 0 < 1 ; 2 >gd(q;); 1 ; 2 2 ~ V: (Note, H normjjand V normkk are dened in Section 4.1.) Next, we dene ()-averaged sesquilinear forms ~ a 1 (;;) : ~ V ~ V ! R, ~ a 2 (;;) : ~ H ~ H!R by the following equations ~ a 1 (; 1 ; 2 ) = Z Q a 1 (q; 1 (q); 2 (q))d(q;) =E () [a 1 (q; 1 (q); 2 (q))]; ~ a 2 (; 1 ; 2 ) = Z Q a 2 (q; 1 (q); 2 (q))d(q;) =E () [a 2 (q; 1 (q); 2 (q))]; where 1 ; 2 2 ~ V and = ( ! a; ! b; ! ). From the denitions in section 4.1, we know a 1 (q;;) is a V -bounded and V -coercive bilinear form, and a 2 (q;;) is H-bounded and H-coercive. Then, we can demonstrate that ~ a 1 (;;) is also a ~ V - bounded and ~ V -coercive bilinear form, and ~ a 2 (;;) is ~ H-bounded and ~ H-coercive. Theorem 5.1. If a 1 (q;;) is a V -bounded and V -coercive bilinear form, then 36 ~ a 1 (;;) is ~ V -bounded and ~ V -coercive Proof. j ~ a 1 (; 1 ; 2 )j Z Q ja 1 (q; 1 (q); 2 (q))jd(q;) 0 Z Q k 1 (q)kk 2 (q)kd(q;) (Vbounded) 0 f Z Q k 1 (q)k 2 d(q;)g 1 2 f Z Q k 2 (q)k 2 d(q;)g 1 2 (CauchySchwarz inequality) = 0 k 1 k ~ V k 2 k ~ V (5.1) ~ a 1 (; ; ) + 0 j j 2 ~ H = Z Q a 1 (q; (q); (q))d(q;) + 0 Z Q j j 2 d(q;) = Z Q (a 1 (q; (q); (q))d + 0 j j 2 )d(q;) 0 Z Q k (q)k 2 d(q;) (Since a 1 (q;;) is Vcoercive) = 0 k k 2 ~ V (5.2) Theorem 5.2. ~ a 2 (;;) is ~ H-bounded and ~ H-coercive whena 2 (q;;) isH-bounded and H-coercive. Proof. The proof is similar to the previous proof. j ~ a 2 (; 1 ; 2 )j Z Q ja 2 (q; 1 (q); 2 (q))jd(q;) 0 Z Q j 1 (q)jj 2 (q)jd(q;) 0 f Z Q j 1 (q)j 2 d(q;)g 1 2 f Z Q j 2 (q)j 2 d(q;)g 1 2 = 0 j 1 j ~ H j 2 j ~ H (5.3) 37 ~ a 2 (; ; ) + 0 j j 2 ~ H = Z Q a 2 (q; (q); (q))d(q;) + 0 Z Q j j 2 d(q;) = Z Q (a 2 (q; (q); (q))d + 0 j j 2 )d(q;) 0 Z Q j (q)j 2 d(q;) = 0 j j 2 ~ H (5.4) In Chapter 4, b(q), c 1 (q), and c 2 (q) are elements in V with the maps q 7! b(q), q 7! c 1 (q), and q 7! c 2 (q) continuous from Q into V . Moreover, q 7!< b(q); > V ;V , q7!< c 1 (q);' > V ;V ,q7!< c 2 (q); _ ' > V ;V are ()-measurable for any 2 ~ V and '; _ '2 L 2 ([0;T ]; ~ V ). We then dene ~ B() : R ! ~ V , ~ C 1 () : L 2 ([0;T ]; ~ V )!R v , ~ C 2 () :L 2 ([0;T ]; ~ V )!R v by ~ B()u =f Z Q <b(q); > V ;V d(q;)gu; ~ C 1 ()' = Z Q Z T 0 <c 1 (q);'> V ;V d(q;); ~ C 2 () _ ' = Z Q Z T 0 <c 2 (q); _ '> V ;V d(q;); The input/output system stated in (4.1) is also ()-measurable; Therefore, we can rewrite this system when q is a random variable instead of a constant.The following equations provide the new input/output system: h '; i ~ V ; ~ V + ~ a 2 (; _ '; ) + ~ a 1 (;'; ) =h ~ b(); i ~ V ; ~ V u; y = [h~ c 1 ();'i ~ V ; ~ V ;h~ c 2 (); _ 'i ~ V ; ~ V ] T 2 ~ V; (5.5) where '(0) = ' 0 2 ~ V , _ '(0) = ' 1 2 ~ H, andh;i ~ V ; ~ V denotes the natural extension of the ~ H inner product to the duality pairing between ~ V and ~ V . If 38 we assume ~ b(); ~ c 2 (); 1 2 ~ H, u2 L 2 ([0;T ]; ~ V ), for some T > 0, then we can demonstrate (see, for example, [16]) that the system (5.5) admits a unique so- lution 2 C([0;T ]; ~ V ) with _ 2 C([0;T ]; ~ H), y 2 C([0;T ];R 2 ) and the maps fu; 0 ; 1 g7!f; _ g andfu; 0 ; 1 g7! y continuous from L 2 ([0;T ]; ~ V ) ~ V ~ H into L 2 ([0;T ]; ~ V )L 2 ([0;T ]; ~ H) and L 2 ([0;T ];R 2 ), respectively. Next, we demon- strate that the solution to (4.1) and the solution to (5.5) agrees for a.e. and all q2 Q. Theorem 5.3. The solution ' to (4.1) is the unique element of ~ V satisfying (5.5) for all 2 ~ V . Proof. Let'2 ~ V be the unique solution to (4.1) and ~ '2 ~ V be the unique solution to (5.5), then we have h '; i V ;V +a 2 (q; _ '; ) +a 1 (q;'; ) =hb(q); i V ;V u; (5.6) Z Q fh ~ '; i V ;V +a 2 (q; _ ~ '; ) +a 1 (q; ~ '; )gd(q;) = Z Q fhb(q); i V ;V ugd(q;) (5.7) Since is arbitrary in ~ V , we can set = 0 1 E() , where 0 2 V and E2 B( ~ Q); therefore, we have Z Q fh ~ '; 0 1 E() i V ;V +a 2 (q; _ ~ '; 0 1 E() ) +a 1 (q; ~ '; 0 1 E() )gd(q;) = Z Q fhb(q); 0 1 E() i V ;V ugd(q;) Z E fh ~ '; 0 i V ;V +a 2 (q; _ ~ '; 0 ) +a 1 (q; ~ '; 0 )gd(q;) = Z E fhb(q); 0 i V ;V ugd(q;) Z E fh ~ '; 0 i V ;V +a 2 (q; _ ~ '; 0 ) +a 1 (q; ~ '; 0 )hb(q); 0 i V ;V ugd(q;) = 0 Since for any 0 2V and measurable setsE, the equation above holds, the integrand is 0 a.e. in Q. Hence ~ ' satises (4.1) for a.e., which implies that ~ ' = ' in ~ V . 39 For random variable q, the -dependent bilinear form on ~ V ~ V , ~ a 1 (;;) : ~ V ~ V !R denes a bounded linear operator ~ A 1 ()2L( ~ V; ~ V ) byh ~ A 1 () 1 ; 2 i ~ V ; ~ V = ~ a 1 (; 1 ; 2 ), for 1 ; 2 2V . Analogously forq2Q, the-dependent bilinear form on ~ H ~ H, ~ A 2 (;;) : ~ H ~ H ! R denes a bounded linear operator ~ A 2 ()2 L( ~ H; ~ V ) byh ~ A 2 () 1 ; 2 i = ~ a 2 (; 1 ; 2 ), for 1 ; 2 2 ~ H. If ~ b(); ~ c 2 ()2 ~ H, then dene the -dependent bounded linear operators ~ B 1 () fromR into ~ H, ~ C 1 () from ~ V into R, and ~ C 2 () from ~ H into R by ~ B 1 ()w = ~ b()w, ~ C 1 () =h~ c 1 (); i ~ V ; ~ V , and ~ C 2 () = h~ c 2 (); i. Then let ~ H = ~ H ~ H, ~ V = ~ V ~ V , and dene the operators ~ A() : Dom( ~ A()) ~ H! ~ H, ~ B()2 L(R; ~ H), and ~ C()2 L( ~ H;R 2 ) by ~ A()(; ) = ( ; ~ A 1 () + ~ A 2 () ), (; ) 2 Dom( ~ A) = f(;) 2 ~ H : 2 ~ V; ~ A 1 () + ~ A 2 ()2 ~ Hg, ~ B()w = (0; ~ B 1 ()w), w2 R, and ~ C()(;) = ( ~ C 1 (); ~ C 2 (), (;) 2 ~ H, respectively. With these denitions, if we assume further that 1 2 ~ H so that x 0 = ( 0 ; 1 )2 ~ H, then we rewrite the second-order system that is given in (5.5) as a rst-order abstract evolution system in H as _ x(t) = ~ A()x(t) + ~ B()u(t); t> 0; (5.8) x(0) =x 0 2 ~ H; (5.9) y(t) = ~ C()x(t); (5.10) where x(t) = ('(t); _ '(t)), t 0. We can demonstrate (see, for example, [3, 4, 16]) that ~ A() is an innitesimal generator of a ~ C 0 -semigroupf ~ T(t;) : t 0g of bounded linear operators on ~ H. The solution to the system (5.8)-(5.10) is thus as follows: x(t;) = ~ T(t;)x 0 + Z t 0 ~ T(ts;q) ~ B()u(s)ds; t 0; (5.11) with x2C(0;T ; ~ H) and y(t;) = ~ C() ~ T(t;)x 0 + Z t 0 ~ C() ~ T(ts;) ~ B()u(s)ds: t 0; (5.12) 40 5.2 Parabolic Regularization for Hyperbolic Model with Ran- dom Parameters However, we must still face the case that our abstract evolution system (5.8)-(5.10) has unbounded input and output operators. Therefore, as in Section 4.2, here we in- troduce parabolic regularization in the form of Kelvin-Voight visco-elastic damping into the system (5.5) (see, for example [3, 16]). Let " > 0 be given and use the denitions introduced in Section 5.1 to dene the ~ V -bounded and ~ V -coercive bilinear form ~ a 2;" (;;) : ~ V ~ V !R by ~ a 2;" (; 1 ; 2 ) = ~ a 2 (; 1 ; 2 ) +"~ a 1 (; 1 ; 2 ); 1 ; 2 2 ~ V: (5.13) Then instead of the system (5.5), we consider the weak form with viscosity given by h '; i ~ V ; ~ V + ~ a 2;" (; _ '; ) + ~ a 1 (;'; ) =h ~ b(); i ~ V ; ~ V u; y = [h~ c 1 ();'i ~ V ; ~ V ;h~ c 2 (); _ 'i ~ V ; ~ V ] T 2 ~ V: (5.14) Forq2Q, theq-dependent bilinear form on ~ V ~ V , ~ a 2;" (;;) : ~ V ~ V !R given by (5.13), denes a bounded linear operator ~ A 2;" ()2L( ~ V; ~ V ) byh ~ A 2;" () 1 ; 2 i ~ V ; ~ V = ~ a 2;" (; 1 ; 2 ), for 1 ; 2 2 ~ V . We now dene the Gelfand triple ~ V ,! ~ H ,! ~ V and note that in this case ~ V = ~ V ~ V . Then, we dene the operator ~ A " () : Dom( ~ A " ()) ~ H! ~ H, by ~ A " ()(; ) = ( ; ~ A 1 () + ~ A 2;" () ), (; )2 Dom( ~ A " ) =f(;)2 ~ V : 2 ~ V; ~ A 1 () + ~ A 2;" ()2 ~ Hg. According to [3, 4, 32], ~ A " () is now an innitesimal generator of a holomorphic or analytic semigroup [22]f ~ T " (t;) : t 0g of bounded linear operators on ~ H, ~ V, and ~ V . We subsequently assume that 1 2 ~ V , ~ b(); ~ c 2 ()2 ~ V and consider the viscosity 41 evolution system (5.5) rewritten in the rst-order form given by _ x " (t) = ~ A " ()x " (t) + ~ B()u(t); t> 0; (5.15) x " (0) =x 0 2 ~ H; (5.16) y " (t) = ~ C()x " (t): (5.17) It can be demonstrated [3, 4, 16] that ~ A() is the innitesimal generator of a ~ C 0 - semigroupf ~ T(t;) :t 0g of bounded linear operators on ~ H. The solution to the system (5.15)-(5.17) is given by x " (t;) = ~ T " (t;)x 0 + Z t 0 ~ T " (ts;) ~ B()u(s)ds; t 0; (5.18) with x " 2C([0;T ]; ~ H) and y " (t;) = ~ C() ~ T " (t;)x 0 + Z t 0 ~ C() ~ T " (ts;) ~ B()u(s)ds: t 0; (5.19) Note that ~ c 2 ()2 V and x " (t) = (' " (t); _ ' " (t))2 C(0;T ; ~ H), t 0 and conse- quently _ ' " (t)2H. 5.3 Discrete-Time Formulation with Parabolic regularization In the previous section, (5.19) is still undened. To address this, we allow only zero-order hold inputs of the form u(t) = u j , t2 [j; (j + 1)), j = 0; 1;:::, where > 0 denotes the length of the sampling interval, and consider the discrete-time formulation. Using the variation of constants formula (5.18), we obtain x ";j+1 = ^ ~ A " ()x ";j + ^ ~ B " ()u j ; j = 0; 1; 2;:::; (5.20) y ";j = ^ ~ C()x ";j ; j = 0; 1; 2;:::; (5.21) with x ";0 = x 0 2 ~ V where x ";j = x " (j), y ";j = y " (j), x " and y " given by (5.18) and (5.19), respectively, ^ ~ A " () = ~ T " (;)2 L( ~ V; ~ V), ^ ~ B " () = R 0 ~ T " (s;) ~ B()ds2 42 L(R; ~ V), and ^ ~ C() = ~ C()2L( ~ V;R). If we assume that the alcohol concentration is zero in the epidermal layer at t = 0, then x 0 = 0. By the analyticity of the semigroup, n ~ T " (t;) :t 0 o , the operators ^ ~ A " () and ^ ~ B " () are bounded on ~ V (see [3, 4, 5, 16, 23, 32]). With the assumption that ' 1 2 ~ V , it now follows that x ";j , j = 0; 1; 2;::: given by (5.20) is in ~ V, and hence (5.21) is well dened. It is simple to argue [3, 4] (using the Trotter-Kato theorem [14, 22]) that 7! x ";j ()2 C(; ~ V), j = 0; 1; 2;:::. Moreover, the ~ V -coercivity assumption (ii) implies that, following a change of variables, we may assume that ~ A " () : Dom( ~ A " ()) ~ V ! ~ V is invertible with a bounded inverse; it follows that ^ ~ B " () = ( ^ ~ A " ()I) ~ A 1 " () ~ B()2 L(R; ~ V) and from (5.20) and (5.21)that y ";i () = i1 X j=0 ~ C()( ^ ~ A " ()) ij1 ~ B " ()u j = i1 X j=0 ~ C()( ^ ~ A " ()) ij1 ( ^ ~ A " ()I) ~ A 1 " () ~ B()u j = i1 X j=0 ^ ~ h ";ij ()u j ; i = 0; 1; 2;::: (5.22) where ^ ~ h ";i () = ~ C() ^ ~ A " () i1 ( ^ ~ A " ()I) ~ A 1 " () ~ B(); i = 1; 2;::: . Finally, we note that if' 1 ; ~ b(); ~ c 2 ()2 ~ H, then it can be argued either directly (see [16]) or once again using the Trotter Kato Theorem that lim "!0 x " (;) =x(;) in C([0;T ]; ~ H) uniformly in q for 2 R p R p , where x " (;) and x(;) are from (5.18) and (5.11), respectively. Similar convergence results are held fory " (;) and y(;), x ";j () and x(j;), and y ";j () and y(j;), the latter two on nite or bounded subsets of j2Z + . 5.4 Discrete-Time Formulation of Telegraph Equation with Zero-Order Hold Input on the Boundary of the Domain We consider the general discrete-time formulation described in Section 4.4. Then, we have H = L 2 (0; 1) and V = H 1 (0; 1) with their regular inner products and 43 norms and the Gelfand triple H 1 (0; 1) ,! L 2 (0; 1) ,! H 1 (0; 1). The sesquilinear forms a 1 (q;;) :VV !R, a 2 (q;;) :HH!R are a 1 (q; 1 ; 2 ) = q 1 q 7 q 6 1 (1) 2 (1) + q 1 q 4 q 3 1 (0) 2 (0) +q 1 Z 1 0 0 1 () 0 2 ()dx; 1 ; 2 2V; (5.23) and a 2 (q; 1 ; 2 ) =q 2 Z 1 0 1 () 2 ()dx; 1 ; 2 2V: (5.24) Therefore, we can rewrite the system (4.30)-(4.35) in a weak form h v; i V ;V +a 2 (q; _ v; ) +a 1 (q;v; ) = 0 (5.25) Based on the weak form above, we can dene ~ Q =R 8p+r , and the Bochner spaces ~ V = L 2 ( ~ Q;V ) and ~ H = L 2 ( ~ Q;H). Since V , H, and V form the Gelfand triple, ~ V , ~ H, and ~ V also form the Gelfand triple ~ V ,! ~ H ,! ~ V , where ~ V =L 2 ( ~ Q;V ), ~ H =L 2 ( ~ Q;V ), and ~ H =L 2 ( ~ Q;H ). Next, we dene ()-averaged sesquilinear forms ~ a 1 (;;) : ~ V ~ V ! R, ~ a 2 (;;) : ~ H ~ H!R by the following equations ~ a 1 (; 1 ; 2 ) = Z Q a 1 (q; 1 (q); 2 (q))d(q;) =E () [a 1 (q; 1 (q); 2 (q))]; ~ a 2 (; 1 ; 2 ) = Z Q a 2 (q; 1 (q); 2 (q))d(q;) =E () [a 2 (q; 1 (q); 2 (q))]; 44 where 1 ; 2 2 ~ V , and = ( ! a; ! b; ! ). The weak form of the system becomes h ~ v; i ~ V ; ~ V + ~ a 2 (; _ ~ v; ) + ~ a 1 (; ~ v; ) = 0 (5.26) From the denitions in Section 4.1, we know that a 1 (q;;) is a V -bounded and V -coercive bilinear form, anda 2 (q;;) isH-bounded andH-coercive. Then, we can demonstrate that ~ a 1 (;;) is also a ~ V -bounded and ~ V -coercive bilinear form, and ~ a 2 (;;) is ~ H-bounded and ~ H-coercive. Furthermore, the bounded linear operators ~ A 1 ()2L( ~ V; ~ V ), ~ A 2 ()2L( ~ H; ~ H) are dened by by h ~ A 1 () 1 ; 2 i V ;V = ~ a 1 (; 1 ; 2 ); 1 ; 2 2 ~ V h ~ A 2 () 1 ; 2 i = ~ a 2 (; 1 ; 2 ); 1 ; 2 2 ~ H and ~ A()( ;) = (; ~ A 1 () ~ A 2 ()) (5.27) where ( ;)2 Dom( ~ A) =f( ;)2 ~ V ~ H : 2 ~ V; ~ A 1 () + ~ A 2 ()2 ~ Hg. With these denitions, the second-order system given in (5.26) can be rewritten as a rst-order abstract system in ~ V ~ H as follows: _ w(t;) = ~ A()w(t;) t2 (k; (k + 1)) (5.28) w(k;) = [x k 1 0 ()u k1 1 1 ()u k1 2 ;x k 2 ] T (5.29) where w(t;) = (~ v(t); ~ v t (t)) T . Since ~ A() is the innitesimal generator of an C 0 - semigroup of contractionsf ~ T(t;) :t 0g on ~ V ~ V , we have w(t;) = ~ T(tk;)w(k;) t2 [k; (k + 1)): The continuity of w, as well as the denitions of ' k and w, then imply that 45 [' k ;' k t ] T =w(k;)+[ 0 ()u k1 1 + 1 ()u k1 2 ; ~ 0] T , and hence that fork = 0; 1; 2; 3;::: x k+1 = [' k+1 ;' k+1 t ] T =w((k + 1);) + [ 0 ()u k 1 + 1 ()u k 2 ; ~ 0] T (5.30) = ~ T(;)f[' k ;' k t ] T [ 0 ()u k 1 + 1 ()u k 2 ; ~ 0] T g + [ 0 ()u k 1 + 1 ()u k 2 ; ~ 0] T (5.31) = ~ T(;)[' k ;' k t ] T +fI ~ T(;)g[ 0 ()u k 1 + 1 ()u k 2 ; ~ 0] T (5.32) =f ~ T(;)g k+1 [' 0 ;' 0 t ] T + k X i=0 f ~ T(;)g ki fI ~ T(;)g[ 0 ()u i 1 + 1 ()u i 2 ; ~ 0] T (5.33) x 0 = [' 0 ;' 0 t ] T = [' 0 ;' 1 ] T (5.34) where 0 (;) = Z Q f q 5 q 7 q 3 (q 6 +q 7 ) +q 4 q 7 + q 5 (q 6 +q 7 ) q 3 (q 6 +q 7 ) +q 4 q 7 gd(q;); (5.35) 1 (;) = Z Q f q 3 q 8 q 3 (q 6 +q 7 ) +q 4 q 7 + q 4 q 8 q 3 (q 6 +q 7 ) +q 4 q 7 gd(q;): (5.36) That is, x k+1 = ~ T(;)x k +B()u k ;k = 0; 1; 2;:::; x 0 = [' 0 ;' 1 ] T ; (5.37) whereB()2L(R 2 ;VH) is given byB()u = [fI ~ T(;)g[ 0 ()u 1 + 1 ()u 2 ; ~ 0] T . 5.5 Finite-Dimensional Approximation and Convergence As stated at the beginning of this chapter, for i2f1; 2; 3;:::;pg, where1 < a < a i < b i < b <1, dene Q = Q p i=1 [ a; b], = ([ a] p i=1 ; [ b] p i=1 ; )2 , H = L 2 ( ) ( Q;H), and V =L 2 ( ) ( Q;V ). Then, for N = 1; 2;:::; denote ! a N = [a N i ] p i=1 , ! b = [b N i ] p i=1 and N = ( ! a N ; ! b N ; ! N )2 , where1< a<a N i <b N i < b<1. Set Q N = Q p i=1 [a N i ;b N i ], H N =L 2 ( N ) (Q N ;H),V N =L 2 ( N ) (Q N ;V ), and let W N 46 be a nite-dimensional subspace of V N . Dene a linear map F N from H H to H N H N (or V V !V N V N ) byF N ( ) = j Q N for any = ( 1 ; 2 )2 H H (or 2 V V ). Let P N :H N H N !W N W N be the orthogonal projection from H N H N onto W N W N . Then we dene F N : H H ! W N W N by F N = P N F N . Moreover, let H = H H, H N = H N H N , V = V V , V N =V N V N , and W N =W N W N . We assume that the probability measures (;) are continuous and (;) f(;), wheref(;) is a joint density for the random vector q. Then, we dene the operators A N 1 (),A N 2 () and A N 2; () on W N to be the restriction of A 1 (),A 2 () and A 2; () to W N by following equations: <A N 1 () N 1 ; N 2 >= ~ a 1 (; N 1 ; N 2 ) = Z Q N ~ a 1 (q; N 1 (q); N 2 (q))d(q;) = Z Q N ~ a 1 (q; N 1 (q); N 2 (q))df(q;); <A N 2 () N 1 ; N 2 >= ~ a 2 (; N 1 ; N 2 ) = Z Q N ~ a 2 (q; N 1 (q); N 2 (q))d(q;) = Z Q N ~ a 2 (q; N 1 (q); N 2 (q))f(q;)dq; <A N 2; () N 1 ; N 2 >= ~ a 2; (; N 1 ; N 2 ) = Z Q N ~ a 2; (q; N 1 (q); N 2 (q))d(q;) = Z Q N ~ a 2; (q; N 1 (q); N 2 (q))f(q;)dq; where N 1 ; N 2 2W N . In addition, we denote ~ A N ()( N 1 ; N 2 ) = ( N 2 ;A N 1 () N 1 A N 2 () N 2 ); where ( N 1 ; N 2 ) 2 Dom( ~ A N ()) = f(;) 2 V N V N : 2 V N ;A N 1 () + 47 A N 2; ())2H N g, and ~ A N ()( N 1 ; N 2 ) = ( N 2 ;A N 1 () N 1 A N 2; () N 2 ); where ( N 1 ; N 2 ) 2 Dom( ~ A N ()) = f(;) 2 V N V N : 2 V N ;A N 1 () + A N 2; ())2H N g. ~ B N ()u =f Z Q N <b(q); N (q)> V ;V f(q;)dqgu; ~ C N 1 ()' N = Z Q N Z T 0 <c 1 (q);' N > V ;V f(q;)dq; ~ C N 2 () _ ' N = Z Q N Z T 0 <c 2 (q); _ ' N > V ;V f(q;)dq; where ' N 2W N and u2R. When using parabolic regularization, we replace the operator A 2 () by A 2; (), which leads to the following discrete-time formulation: x N ";j+1 = ^ ~ A N " ()x N ";j + ^ ~ B N " ()u j ; j = 0; 1; 2;:::; (5.38) y N ";j = ^ ~ C N ()x N ";j ; j = 0; 1; 2;:::; (5.39) where ^ ~ A n " () = ~ T N " (;) = e ~ A N " () 2 L(W N W N ;W N W N ), ^ ~ B N " () = R 0 ~ T N " (s;) ~ B N ()ds2L(R;W N W N ),and ^ ~ C N () = ~ C N ()2L(W N W N ;R). When there is no parabolic regularization term in our system, we changeA 2; () back to A 2 (). However, regardless whether we use parabolic regularization, we must prove that there is a subsequence of solutions to the nite-dimensional approximating problems. This subsequence converges to the solution of the innite-dimensional estimation problem. 48 First, we start with two additional assumptions: Assumption 5.1 Positive real numbers and exist such that for any 2 , we have 0< f(q;)<1 for ()-a.e. q2 Q. Assumption 5.2 For allw2 V, there existsu N 2W N such that u N F N w V N ! 0 as N!1. Under conditions (i)-(iv) at the beginning of Chapter 5 and Assumption 5.1-5.2, we can prove the following theorem. Theorem 5.4. Dene a sequencef N g such thatf(q; N )!f(q;) for a.e. q2 Q. With the denitions provided in this chapter, the conditions of the Trotter-Kato theorem are satised. We have F N (t; N )P N zP N F(t;)z H N ! 0; as N!1 (5.40) for every z2H, uniformly in t on compact intervals where F N =fF N (t; N :t 0)g is the semigroup on H N generated by ~ A N ( N ), and F =fF(t; : t 0)g is the semigroup on H generated by ~ A() This theorem can be proved by merely choosing q = , H = L 2 (Q;H), V = L 2 (Q;V ),H N = L 2 (Q;H N ), V N = L 2 (Q;V N ),a 1 (q) = ~ a 1 (),a 2 (q) = ~ a 2 (), and a 2; (q) = ~ a 2; (). Then,the results of theorem 4.1 and 4.2 presented in Section 4.5, can be applied to this case. Theorem 5.5. Under the same assumptions of the previous theorem, we have x N j;i ( N )Px j;i () V N ! 0; as N!1; (5.41) y N j;i ( N )y j;i () R v ! 0; as N!1; (5.42) for i = 1; 2; 3;:::;m; j = 0; 1; 2;:::;n i ; where x N j;i ( N ),y N j;i ( N ),x j;i () and y j;i () are given in (5.38) (5.39) (5.20) and (5.21) Since the feasible parameter set is closed and bounded in R 3p+r , together with the two theorems above, we have the following result, 49 Theorem 5.6. If the maps ! f(q;) from to R are continuous for ()- a.e. q2 Q, then each of the approximating estimation problems admits a solution N . Moreover, the sequencef N g has a convergent subsequencef N k g such that N k ! as N k !1, and is the solution to the original estimation problem. 50 6 Estimating Distribution of Random Parameter 6.1 Training, Calibrating, or Fitting the Model to Data Before the model given in (3.10)-(3.14) can be used to estimate BrAC from TAC measurements, calibration to the individual subject is required. That is, the param- eters q2 Q must be estimated based on contemporaneous BrAC and TAC data. We do this via a regularized nonlinear least-squares t data. Given BrAC and TAC training data,f~ u i ; ~ y i g; i = 0; 1; 2;:::;N, respectively, we seek 2fQ;d g, which minimizes the cost functional J() = N X i=0 jy ";i () ~ y i j 2 +r(): (6.1) In (6.1), r() denotes a (Tychonov) regularization term and gives y ";i () by either (5.12) or (5.19) and (5.22) and, in those expressions, with u i given by u i = ~ u i ; i = 0; 1; 2;:::;N. Regularization is required to guard against over-tting and enhance the optimization problem (i.e., making J more convex). To compute a solution to the above optimization problem, we replace y ";i (q) in (6.1) with its nite-dimensional approximationy n ";i (q) given by either (5.27) or (5.20) and (5.21) and in those expressions, with u i given byu i = ~ u i ; i = 0; 1; 2;:::;N. In actual practice, we then seek n 2f;dg, which minimizes the cost functional J n () = N X 0 jy n ";i () ~ y i j 2 +r() (6.2) where u i = ~ u i ; i = 0; 1; 2;:::;N. We can establish the following convergence result (see, for example, [3, 9, 26, 31]) under the assumptions on the construction of the nite-dimensional subspaces stated in Section 5.5. The sequencef n g of solutions to the optimization problems given in (6.2) has a convergent subsequence,f nj g, with lim j!1 nj = , and is the solution to the optimization problem for (6.1). Given these approximating optimization problems, the cost functional (6.2), 51 J n , is solved iteratively using a gradient-based descent method. For a given value of q2 Q, we can now compute the value of J n () with the expressions given in (5.22) or (5.20) and (5.21). The gradient of J n () can be computed accurately (in fact, with zero truncation error) and eciently using the adjoint method [15]. For i = 0; 1; 2;:::;N, set v n ";i = 2[ ^ C() n ] T (y n ";i () ~ y i )2R n+1 z n ";i1 = [ ^ A" n ] T z n ";i +v n ";i1 ;z n ";N =v N ";N ;i =N;N 1;:::; 2; 1: (6.3) where ^ A" n () = exp(A n " ()) with the operatorsA n " () andB n () given by (4.54) and (4.55), respectively. We then compute the gradient of J n at 2 by ~ rJ n () = N X i=1 [z n ";i ] T @ ^ A n " () @ x n ";i1 (A n " ()) 1 @A n " () @ (A n " ()) 1 ( ^ A n " ()I)B n ()~ u i1 @ ^ A n " () @ B n ()~ u i1 ( ^ A n " ()I) @B n () @ ~ u i1 + N X i=0 y n ";i ~ y i T@ ^ C n () @ x n ";i + @r() @ ; (6.4) where z n ";i , i = 0; 1; 2;:::;N is given by (6.3). We compute the tensor @A n " () @ appearing in (6.4) simultaneously when using the sensitivity equations to calculate A n " (). For t 0 and q2 Q, set n (;t) = e A n " ()t . Then n (;) is the unique principal fundamental matrix solution to the initial value problem _ n (;) =A n " () n (;); n (; 0) =I (6.5) We set n (;t) =@ n (;t)=@, dierentiate both sides with respect to q, inter- change the order of dierentiation, and use the product rule, and we obtain _ n (;) =A n " () n (;) + @A n " () @ n (;); n (; 0) = 0 (6.6) 52 By combining the two initial value problems (6.5) and (6.6), we can then solvethe expression given in (6.7) below, 2 6 4 @ ^ A n " () @ ^ A n " ()() 3 7 5 = 2 6 4 n (;) n (;) 3 7 5 =e 2 6 6 6 4 A n " () (@A n " ()=@) 0 A n " () 3 7 7 7 5 2 6 4 0 I 3 7 5: (6.7) 6.2 Deconvolution Problem: Obtaining BrAC from TAC Estimating BrAC from TAC takes the form of a non-negatively constrained, regu- larized deconvolution problem. It can also be viewed as a quadratic programming (see, for example, [16]), though evaluating a performance indicator involves solving an innite-dimensional state equation. We denote the t parameters from Section 5.1, as q and assume that the input isfu j g with u j = u(j), where > 0 is the length of the sampling interval andu2U, whereU is a compact subset ofH 1 (0;T ). Letf^ y k g K k=0 be a set of biosensor measured TAC data. The input estimation problem is then given by u " = arg min U J(u) = arg min U K X k=0 jy ";k (u) ^ y k j 2 +jjujj R ; (6.8) wherejjujj R is a regularization term withjjjj R an appropriately weighted norm on H 1 (0;T ), and y ";k (u) = k1 X j=0 < ^ ~ h ";kj ( );u j > L 2 (Q ) ; k = 0; 1; 2;:::;K; (6.9) with u j =u(j), j = 0; 1;:::;K, and ^ ~ h ";k ( ) is dened in (5.22) with = . To solve the optimization or deconvolution problem given by (6.9), we use nite- dimensional approximation. Let N = (n;m) be an index and note that we mean both componentsN go to innity whenever we use the notation N!1. For each m, letU m be a closed subset ofUH 1 (0;T ). Assume the following approximation assumption holds on the compact subsets U m . 53 Assumption 6.1 For each u2 U, a sequencefu m g exists with u m 2 U m , such that jju m ujj R ! 0; as m!1: (6.10) Piecewise constant or spline-based subspaces (see [27]) typically satisfy (6.10). Given the following approximating nite deconvolution problems : u N " = arg min U m J N (u) = arg min U m K X k=0 y n ";k (u) ^ y k 2 +jjujj R ; (6.11) where y ";k (u) = k1 X j=0 < ^ ~ h ";kj ( );u j > L 2 (Q ) ; k = 0; 1; 2;:::;K; (6.12) with u j =u(j), j = 0; 1;:::;K, and ^ h n ";k ( ) in (6.12) is as it was dened in (5.22) with q =q . Theorem 6.1. For each N, the nite-dimensional approximating optimization problem given in (6.11) has a solution u N " . Under Assumptions 3.1 and 3.2, a subsequence offu N " g,fu Nj " gfu N " g exists with u Nj " !u " as j!1, where u " is a solution to the innite-dimensional estimation problem given in (6.8). Proof. By the continuity of J N , and the compactness of U m , each of the nite- dimensional approximating optimization problems given in (6.11) has a solution. Letfu m g be a convergent sequence in U H 1 (0;T ) with u m 2 U m andjju m ujj R ! 0 as m ! 1, u 2 U. Then if u m;j = u m (j), and u j = u(j), j = 0; 1; 2;:::;K, it follows thatju m;j u j j! 0 as m!1, j = 0; 1; 2;:::;K. Follows that, for each k, j ^ ~ h n ";kj ( )u j ^ ~ h ";kj ( )u j j! 0; as N = (n;m)!1; j;l = 1; 2;:::;K: (6.13) 54 It follows from (6.13) that J N (u m )!J(u) as N!1. Letfu N g U m U be a sequence of solutions to the nite-dimensional approximating optimization problems given in (6.11). By the compactness of U, a subsequencefu Nj g offu N g exists, with u Nj ! u 2 U as j!1. For any u2U, we have J(u ) =J( lim j!1 u Nj ) = lim j!1 J Nj (u Nj ) lim j!1 J Nj (u mj ) =J( lim j!1 u mj ) =J(u); (6.14) where N j = (n j ;m j ). The sequencefu mj gU with u mj 2U mj in (6.14) denotes the approximations to u that exists because of Assumption 6.1, on the subsets U m in (6.10), and u thus is a solution to (6.8). The deconvolution problem and its nite-dimensional approximations are stated in (6.8) and (6.11). The corresponding optimization problems can be seen as quadratic programming problems, which are controlled by (4.12),(4.13), and the op- timization problems' nite-dimensional approximation, (5.21), (5.22). Some ideas from [1] can consequently be adapted to obtain stronger convergence results than those in Theorem 3.1. In the remainder of this section, we present stronger convergence results from [1, 31, 30]. Assume that U is a closed and convex subset of H 1 (0;T ), and for each m, letU m be a closed and convex subset ofU. U is in a nite-dimensional subspace of H 1 (0;T ). The maps u7! J(u) and u7! J N (u) from U into R and from U m intoR, respectively, are strictly convex. Solutions u and u N to the optimization problems (6.11) and (6.11) hence exist and are unique. Furthermore, the sequence fjju N jj R g is bounded. J Nj (u Nj )J Nj (u mj )!J(u)<1; (6.15) whereN j = (n j ;m j ). The sequencefu mj gU withu mj 2U mj in (6.15) represents 55 the approximations to u that exists because of the assumption on the subsets U m given in Assumption 6.1. Then sinceU was assumed to be closed and convex, there exists a weakly convergent subsequence,fu Nj g offu N g with u Nj *u , for some u 2U. Also, J and J N are weakly and sequentially lower semi-continuous. Then, with Assumptions 3.1 and 3.2, we have for each u2U J(u ) =J(w lim j!1 u Nj ) lim inf j!1 J(u Nj ) = lim inf j!1 J Nj (u Nj ) lim sup j!1 J Nj (u Nj ) lim sup j!1 J Nj (u mj ) =J( lim j!1 u mj ) =J(u); (6.16) where N j = (n j ;m j ). Furthermore, the sequencefu mj g U with u mj 2 U mj in (6.16) represents the approximations to u that exists because of the assumption on the subsets U m given in Assumption 6.1. Therefore, u is the solution to the approximation problem introduced in (6.8). Last, by the strict convexity of J(u) and J N (u), the sequencefu N g must converge weakly to the unique solution, u . Since the weighted norm in H 1 (0;T ) is bounded by J, the sequencefu N g must converge either strongly or, in the norm in H 1 (0;T ), to the unique solution u . 56 7 Application to the Alcohol Biosensor Problem In this chapter, we demonstrate that the models (3.10)-(3.14) satisfy all the assump- tions that we made in Chapters 4 and 5. First, we make the following identications. Apply the same setting stated in [31, 33, 30]. Let H =L 2 (0; 1) , V =H 1 (0; 1) and V = H 1 (0; 1) with regular inner products and norms. We have consequently the continuous and dense embeddings, and hence the Gelfand triple H 1 (0; 1) ,! L 2 (0; 1) ,! H 1 (0; 1). We dene the sesquilinear forms a 1 (q;;) : V V ! R a 2 (q;;) :HH!R as a 1 (q; 1 ; 2 ) = 1 (0) 2 (0) +q 1 Z 1 0 0 1 () 0 2 ()dx; 1 ; 2 2V; (7.1) and a 2 (q; 1 ; 2 ) =q 2 Z 1 0 1 () 2 ()dx; 1 ; 2 2V; (7.2) respectively, and b(q), c 1 (q), and c 2 (q) as b(q) = q 3 0 ( 1) and c 1 (q) = c 2 (q) = 0 (), respectively, where 0 denotes the Dirac distribution at = 0 in H 1 (0; 1). Note that in this case, hB 1 (q)u; i V ;V =q 3 (1)u; C 1 (q) =(0); and C 2 (q) _ = _ (0); (7.3) for u2R, 2V , 2V and _ 2V . Now, we demonstrate that sesquilinear forma 1 satises conditions (i), (ii), and (iii), and a 2 satises conditions (i), (ii'), (iii) in Section 4.1. (i) Boundedness:ja(q; 1 ; 2 )j 0 jj 1 jjjj 2 jj, 1 ; 2 2V , q2Q. 57 ja 1 (q; 1 ; 2 )jj 1 (0) 2 (0)j +q 1 j Z 1 0 0 1 () 0 2 ()dj j 1 (0) 2 (0)j +q 1 f Z 1 0 j 0 1 ()j 2 dg 1=2 f Z 1 0 j 0 2 ()j 2 dg 1=2 j 1 (0) 2 (0)j +q 1 k 1 k V k 2 k V (7.4) By Morrey's inequality, 0 > 0 exists such thatj 1 (0) 2 (0)j 0 k 1 k V k 2 k V . Hence, 0 > 0 exists such that ja 1 (q; 1 ; 2 )j 0 k 1 k V k 2 k V (7.5) Similarly, we have ja 2 (q; 1 ; 2 )jq 2 j Z 1 0 1 () 2 ()dj q 2 f Z 1 0 j 1 ()j 2 dg 1=2 f Z 1 0 j 2 ()j 2 dg 1=2 q 2 k 1 k H k 2 k H Since V ,!H; thenjj H <Kkk V q 2 k 2 k 1 k V k 2 k V (7.6) Therefore, 0 > 0 exists such that ja 2 (q; 1 ; 2 )j 0 k 1 k V k 2 k V (7.7) (ii) V-coercivity: a(q; ; ) + 0 j j 2 0 jj jj 2 , 2V , q2Q, For 2V and 0 > 0, we have a 1 (q; ; ) + 0 j j 2 H = (0) 2 +q 1 Z 1 0 j 0 j 2 d + 0 Z 1 0 j j 2 d 0 f Z 1 0 j 0 j 2 d + Z 1 0 j j 2 dg = 0 k k 2 V : (7.8) 58 (ii') H-semicoercivity: a(q; ; ) 1 j j 2 , 2V , q2Q. a 2 (q; ; ) + 0 j j 2 H =q 2 Z 1 0 j j 2 d + 0 Z 1 0 j j 2 d 1 Z 1 0 j j 2 d = 1 j j 2 H : (7.9) (iii) Continuity: For 1 ; 2 2V , the map q7!a(q; 1 ; 2 ) is continuous in the sense thatja(q ; 1 ; 2 )a(q ; 1 ; 2 )jd Q (q ;q )jj 1 jjjj 2 jj, for all q ;q 2Q. Letd Q be the Euclidean distance onR. By Cauchy-Schwarz inequality, for 1 ; 2 2 V and q;q 0 2Q, we have ja 1 (q; 1 ; 2 )a 1 (q 0 ; 1 ; 2 )j =jq 1 Z 1 0 0 1 () 0 2 ()dq 0 1 Z 1 0 0 1 () 0 2 ()dj =j(q 1 q 0 1 ) Z 1 0 0 1 () 0 2 ()dj j(q 1 q 0 1 )jj Z 1 0 0 1 () 0 2 ()dj j(q 1 q 0 1 )jf Z 1 0 j 0 1 ()j 2 dg 1=2 f Z 1 0 j 0 2 ()j 2 dg 1=2 =j(q 1 q 0 1 )jk 1 k V k 2 k V =d Q (q 1 ;q 0 1 )jk 1 k V k 2 k V (7.10) 59 ja 2 (q; 1 ; 2 )a 2 (q 0 ; 1 ; 2 )j =jq 2 Z 1 0 1 () 2 ()dq 0 2 Z 1 0 1 () 2 ()dj =j(q 2 q 0 2 ) Z 1 0 1 () 2 ()dj j(q 2 q 0 2 )jj Z 1 0 1 () 2 ()dj j(q 2 q 0 2 )jf Z 1 0 j 1 ()j 2 dg 1=2 f Z 1 0 j 2 ()j 2 dg 1=2 d Q (q 2 ;q 0 2 )j 1 j H j 2 j H Since V ,!H; thenjj H <Kkk V d Q (q 2 ;q 0 2 )jk 1 k V k 2 k V (7.11) (iv) Measurability: For all 1 ; 2 2V , the map q!a(q; 1 ; 2 ) is measurable on Q with respect to all measures dened in terms of the densities in F (Q), where R r is the set of feasible parameters. Consider an arbitrary measure represents a probability density functionf(q;).We can demonstrate the sesquilinear form is measurable with respect to this measure by showing that a 1 (; 1 ; 2 )f(;) is Lebesque measurable as a function of q for xed 1 and 2 . Since the sesquilinear form is a continuous function and f(;) is a measurable function, the function a 1 (; 1 ; 2 )f(;)is integrable over Q. There- fore, a 1 (; 1 ; 2 ) is measurable with respect to the probability measure () for any 1 ; 2 2V . Similarly, a 2 (; 1 ; 2 ), maps q!<B(q)u; > V ;V , q!C 1 (q) and q!C 2 (q) _ are also measurable with respect to the probability measure (). Assumption 5.1 Positive real number and exists such that for any 2 , we have 0< f(q;)<1 for ()-a.e. q2 Q. Since we assume the random variable q has a truncated bivariate normal distri- bution for the alcohol biosensor calibration problem, the joint probability density 60 functionf(q;; ) becomes a continuous function of q. Furthermore, sincef(q;; ) is always positive for any given q, m;M2 R exists such that 0 < m < f(q;) < M <1 for every q2 Q. In this case, our choice of the joint probability density function satises Assumption 5.1. Assumption 5.2 For all w2 V, u N 2W N exists such that u N F N w V N ! 0 as N!1. In the light of the similar proof showed in [33], we can demonstrate that this assumption stands. However, in [33], the proof indicated that this assumption is satised when the model is parabolic with two parameters. Hence, the proof must still be slightly modied to t our case. This assumption is based on our choice of nite-dimensional approximation functions. In this thesis, we use linear B-spline polynomials as the nite-dimensional approximation functions. Moreover, we use linear B-spline polynomialsf' n J g n j=0 for-discretization corresponding to the uniform meshfj=ng n j=0 on the interval [0; 1]. Dene the operatorI n :V N !V n as I n v(q;) = n X i=0 v(q; i )' n i (): (7.12) We dene the standard zero-order B-splinesf m1 1;j g m1 j=1 ,f m2 2;j g m2 j=1 andf m3 3;j g m3 j=1 corresponding to the uniform meshfa N i + (b N i a N i )j=m i g mi j=0 on the intervals [a N i ;b N i ] (i = 1; 2; 3). Let N = (n;m 1 ;m 2 ;m 3 ),s = (j;j 1 ;j 2 ;j 3 ), and we then use tensor products to denef N s g N s (;q 1 ;q 2 ;q 3 ) =f' n j () m1 1;j1 (q 1 ) m2 2;j2 (q 2 ) m3 3;j3 (q 3 )g N s . Hence, we can dene space W N by W N = spanf N s g N s where dim(W N ) = 2(n + 1)m 1 m 2 m 3 . Let m j = m1 1;j1 (q 1 ) m2 2;j2 (q 2 ) m3 3;j3 (q 3 ) with j = (j 1 1)m 1 m 2 + (j 2 1)m 2 + j 3 and R j represents the jth cuboid on which m j (q) is 1 and 0 else- where. Moreover, let p j = RRR Rj f(q; N )dq. Then, we can dene the opera- tor P m1;m2;m3 : H N ! O N m1;m2;m3 with O N m1;m2;m3 = f' 2 H N : '(q;) = 61 P m1m2m3 j=1 ' j () m J (q)gH N as P m1m2m3 v(q;) = m1m2m3 X j=1 ' j () m J (q) (7.13) where v j () = 1 p j ZZ Rj v(^ q;)f(^ q; N )d^ q (7.14) With these denitions, we arrive at P m1;m2;m3 I n : V N ! W N dened in the following way, P m1m2m3 I n v(q;eta) = m1m2m3 X j+1 1 p j ZZZ Rj ( n X i=0 v(^ q; i )' n i ())f(^ q; N )d^ q m j (q): (7.15) Lemma 7.1. O N m1;m2;m3 is a closed subspace of H N . Proof. Letfv k g 1 k=1 O N m1;m2;m3 be a convergent sequence. Then,fv k g is a Cauchy sequence in H N . Let v k = P m1m2m3 j=1 ' n;k J () m j (q). Sincef' n;k j g 1 k=1 is a Cauchy sequence in Hilbert space H,' n j 2H exists such thatj' n;k j !' n j j H ! 0 ask!1. Let v(q;) = P m1m2m3 j=1 ' n j () m j (q)2O N m1;m2;m3 , and then jv k vj H N =j m1m2m3 X j=1 (' n;k j ' n j ) m j j H N m1m2m3 X j=1 ZZZ Rj j' n;k j ' n j j H f(q; N )dq ! 0 k!1: (7.16) Hence, lim k!1 fv k g =v2O N m1;m2;m3 and O N m1;m2;m3 H N is closed. Lemma 7.2. Projection P m1;m2;m3 dened in (7.13) is orthogonal from H N to O N m1;m2;m3 . 62 Proof. For any v2cH N and w2O N m1;m2;m3 , we have <P m1;m2;m3 vv;w> H N = ZZZ Q N Z 1 0 ( m1m2m3 X j=1 v j () m j (q))( m1m2m3 X k=1 w j () m k (q))df(q; N )dq ZZZ Q N Z 1 0 v(q;)( m1m2m3 X k=1 w j () m k (q))df(q; N )dq = Z 1 0 m1m2m3 X j=1 m1m2m3 X k=1 ZZZ Q N v j ()w j () m j (q) m k (q)f(q; N )dqd Z 1 0 m1m2m3 X j=1 m1m2m3 X k=1 ZZZ Q N v(q;)( m1m2m3 X k=1 w j () m k (q))f(q; N )dqd = Z 1 0 m1m2m3 X j=1 p j v j ()w j ()d Z 1 0 m1m2m3 X k=1 p k v k ()w k ()d = 0: (7.17) Therefore, ProjectionP m1;m2;m3 dened in (7.13) is orthogonal fromH N toO N m1;m2;m3 , andjP m1m2m3 j H N 1. Since V N ,! H N ,! (V N ) , we can assume thatjj (V N ) k 1 jj H N and jj H Nk 2 kk V N for k 1 ;k 2 > 0.Hence kP m1m2m3 vk 2 V N =<P m1m2m3 v;P m1m2m3 v> V N jP m1m2m3 vj (V N ) kP m1m2m3 vk V N k 1 jP m1m2m3 vj H NkP m1m2m3 vk V N k 1 jvj H NkP m1m2m3 vk V N k 1 k 2 kvk V NkP m1m2m3 vk V N : (7.18) Therefore,c> 0 exists such thatkP m1m2m3 k V Nc. For anyv2C( Q;H 2 (0; 1)), we have 63 F N vP m1m2m3 I n F N v V N F N vP m1m2m3 F N v V N (7.19) + P m1m2m3 F N vP m1m2m3 I n F N v V N : (7.20) For the rst term on the right-hand side, we have F N vP m1m2m3 F N v 2 V N = ZZZ Q N jF N vP m1m2m3 F N vj 2 V f(q; N )dq = ZZZ Q N jF N v(q;) m1m2m3 X j=1 1 p j ZZZ Rj F N v(^ q;)f(^ q; N )d^ q m j (q)j 2 V f(q; N )dq = ZZZ Q N j m1m2m3 X j=1 1 p j ZZZ Rj (F N v(q;)F N v(^ q;))f(^ q; N )d^ q m j (q)j 2 V f(q; N )dq ZZZ Q N ( m1m2m3 X j=1 1 p j ZZZ Rj jF N v(q;)F N v(^ q;)j V f(^ q; N )d^ q m j (q)) 2 f(q; N )dq (7.21) Since v(q;) is continuous on a compact set, for any > 0, m 1 ;m 2 ;m 3 2N exists such thatjF N v(q;)F N v(^ q;)j< 2 . Then, (7.21) becomes F N vP m1m2m3 F N v 2 V N ZZZ Q N ( 2 m1m2m3 X j=1 m j (q)) 2 f(q; N )dq< 2 4 : (7.22) SincekP m1m2m3 k V Nc where c> 0, we have P m1m2m3 F N vP m1m2m3 I n F N v V N c 2 F N vI n F N v 2 V N =c 2 ZZZ Q N F N vI n F N v 2 V f(q; N )dq: (7.23) As we know, for any > 0, N 0 2 N exists such that F N vI n F N v V < 2c for 64 all n>N 0 .Therefore, P m1m2m3 F N vP m1m2m3 I n F N v 2 V N c 2 ZZZ Q N 2 4c 2 f(q; N )dq 2 4 : (7.24) By (7.19),(7.22) and (7.24), we can conclude that for any> 0, there existsN 0 = (n 0 ;m 0 1 ;m 0 2 ;m 0 3 ) such thatkvP m1m2m3 I n vk V N < for all (n;m 1 ;m 2 ;m 3 )>N 0 . Also, Since C( Q;H 2 (0; 1)) is dense in V, then for any ^ v 2 V and > 0, there exists v2C( Q;H 2 (0; 1)) such that F N ^ vF N v V N 3 . Consider the following inequality; F N vP m1m2m3 I n F N ^ v V N F N ^ vP m1m2m3 F N v V N + P m1m2m3 F N vP m1m2m3 I n F N v V N + P m1m2m3 I n F N vP m1m2m3 I n F N ^ v V N : (7.25) For a suciently large N, we will have F N vP m1m2m3 F N v V N 3 . Since for some c> 0, we havekP m1m2m3 k V N <c, we achieve the following equation P m1m2m3 I n F N vP m1m2m3 I n F N ^ v V N c I n F N vI n F N ^ v V N : (7.26) Then, by choosing a large-enough N, we will have I n F N vI n F N ^ v V N 3c . Thus, for any ^ v2 V and> 0,N 0 exists such thatnorm^ vP m1m2m3 I n F N ^ w V N < . Hence the Assumption 5.2 is satised. By now, we have demonstrated that the telegraph equation for the alcohol biosensor problem satises all the assumptions for the parameter estimation prob- lem. We can use a similar argument above to show that this model also satises the assumptions for the deconvolution problem. 65 8 Numerical studies 8.1 Computational Considerations In this section, we present our model estimation results when using both simulated data and clinical data. We used the telegraph model and optimization problems described in previous sections. For the deconvolution problem, we consider that the estimated BrAC signal either a) depends not only on the time t but also on the three random parameters q 1 , q 2 , and q 3 , or b) depends only on the time t. On the on hand, when dening the estimated BrAC as a function of parameters and time, the credible bands can be computed by sampling the distribution of parameters and applying the samples to the formula obtained from the system. This setting signicantly increases the eciency of computing the credible bands. On the other hand, when the estimated BrAC is dened as a function of the time t only, the credible bands must be obtained by sampling and simulation. The training time takes much longer under this setting because it involves solving the optimization problem and computing the optimal regularization parameters. To solve the optimization problems numerically, we use an iterative gradient- based scheme. The matrix form of the operators ^ A, ^ B, and ^ C can be computed based on the chosen basis for the space W N . The gradient of J N () with respect to all parameters can be computed manually with the adjoint method (See Section 6.1) or the MATLAB built-in command. The adjoint method is more ecient when the approximating system's dimension and the number of parameters are high. In our case, these two methods do not lead to signicant dierences in the results as we keep the approximating system's dimension small. We consider a one-dimensional telegraph equation with random parameters in- 66 troduced in Section 3.3. The equation is as follows: ' tt (t;) +q 2 ' t (t;) =q 1 ' (t;); 0<< 1; t> 0; (8.1) q 1 ' (t; 0)'(t; 0) = 0; t> 0; (8.2) q 1 ' (t; 1) =q 3 u(t); t> 0; (8.3) '(0;) =' 0 (); ' t (0;) =' 1 (); 0<< 1; (8.4) y(t) = ['(t; 0);' t (t; 0)] T t> 0; (8.5) To deal with the unbounded inputs and outputs, we apply the parabolic regular- ization term with = 0:0000001 in this chapter (See details in Section 5.3). Next, we assume that the density function of our parameters belongs to the truncated exponential family dened in the following way: Denition 8.1. Let '(q;), q2R n be a member of an exponential family, where are the vectors of parameters, and denotes the corresponding cumulative dis- tribution function. Let D R n be a bounded region where ' is restricted. Then D () = R D '(q;)dq, and the family of pdfs, f(;) is given in the following equa- tion f(q;) = '(q;) D (q) D () = 1 D () h(q)c() exp k X i=1 w i ()t i (q) D (q) where the domainD is described by the parameters a and b. This family of pdfs is called a truncated exponential family. The statex j ,j = 1; 2;::: in the model described by (5.38) is a function of and parametersq = [q 1 ;q 2 ;q 3 ]. The spaceW N is built by using linear B-splinesf' n j g n j=0 for the uniform meshfj=ng n j=0 on the interval [0; 1]. In the meantime, we dene the standard zero-order B-splinesf m1 1;j g m1 j=1 ,f m2 2;j g m2 j=1 andf m3 3;j g m3 j=1 for the uniform meshfa N i + (b N i a N i )j=m i g mi j=0 on the intervals [a N i ;b N i ] (i = 1; 2; 3). Let N = (n;m 1 ;m 2 ;m 3 ),s = (j;j 1 ;j 2 ;j 3 ); we then use tensor products to denef N s g N s = f' n j m1 1;j1 m2 2;j2 m3 3;j3 g N s . Hence, we can dene space W N by W N = spanf N s g N s , and 67 the nite-dimensional approximation of the state x j becomes x j (;q) = n X i=0 m1 X j1=1 m2 X j2=1 m3 X j3=1 x N i;j1;j2;j3 ' m i () m1 1;j1 (q 1 ) m2 2;j2 (q 2 ) m3 3;j3 (q 3 ) (8.6) for j = 0; 1;:::;nm 1 m 2 m 3 . Similarly, the input u is a function of time t and random parameters q = [q 1 ;q 2 ;q 3 ]. To construct a nite-dimensional subspace S M , we use standard lin- ear B-spline polynomialsf m j g m j=0 for the uniform meshfjT=mg m j=0 on the interval [0;T ]. We apply the same zero-order B-splinesf m1 1;j g m1 j=1 ,f m2 2;j g m2 j=1 andf m3 3;j g m3 j=1 for the uniform mesh on the intervals [a i ;b i ] (i = 1; 2; 3), where [a i ;b i ] (i = 1; 2; 3) are the optimal support by solving the parameter estimation problem. By dening M = (m;m 1 ;m 2 ;m 3 ) and p = (i;j 1 ;j 2 ;j 3 ), the tensor products become the basis for the approximation subspaces. Letf M p g M p =f m i m1 1;j1 m2 2;j2 m3 3;j3 g M p . Then, u(t;q) = n X i=0 m1 X j1=1 m2 X j2=1 m3 X j3=1 u N i;j1;j2;j3 m i (t) m1 1;j1 (q 1 ) m2 2;j2 (q 2 ) m3 3;j3 (q 3 ): (8.7) After choosing the approximating spaces for both parameter estimation and deconvolution problems, we can derive the matrix representations for both problems. Let L = (N;m 1 ;m 2 ;m 3 ); we then have the matrix-vector version of our system as follows M L X L k+1 =K L X L k +B L U L k ; Y L k =C L X L k ; k = 0; 1; ;K (8.8) where the vectors X L k represent coecients of the basis, the vectors U L k are inputs ,andY L k are outputs. In the parameter estimation problem, U L k is a two-by-one vector for each k = 0; 1;:::;K; Y L k is a scalar; and B L is a (n + 1)m 1 m 2 m 3 2 matrix. Furthermore, M L ,K L are both 2(n + 1)m 1 m 2 m 3 2(n + 1)m 1 m 2 m 3 matrices, whileX L is a (n + 68 1)m 1 m 2 m 3 2 matrix; andC L is a 2(n+1)m 1 m 2 m 3 matrices. Moreover,M L ,K L , X L , and C L have the same representations and dimensions in the deconvolution problems. However, in the deconvolution problem,B L becomes a 2(n+1)m 1 m 2 m 3 mm 1 m 2 m 3 matrix. As we demonstrated in Chapter 3 and 7, a 1 (q; 1 ; 2 ) = 1 (0) 2 (0) +q 1 Z 1 0 0 1 () 0 2 ()dx; 1 ; 2 2V; (8.9) and a 2 (q; 1 ; 2 ) =q 2 Z 1 0 1 () 2 ()dx; 1 ; 2 2V; (8.10) b(q), c 1 (q), and c 2 (q) as b(q) =q 3 0 ( 1) and c 1 (q) =c 2 (q) = 0 (), respectively, where 0 denotes the Dirac distribution at = 0 in H 1 (0; 1). Note that in this case we have hB 1 (q)u; i V ;V =q 3 (1)u; C 1 (q) =(0); and C 2 (q) _ = _ (0); (8.11) for u2R, 2V , 2V and _ 2V . Based on the information above, we can derive the formulation for each ma- trix. First, we re-number N i;j;k;h (;q 1 ;q 2 ;q 3 ) so that i;j;k;h = N l where l = [(j 1)m 1 m 2 +(k1)m 2 +h1](n+1)+i withi = 1; ;n+1,j = 1; ;m 1 ,k = 1;;m 2 and h = 1;;m 3 . Then, we have 69 M N s;r =< N r ; N s > H = Z b1 a1 Z b2 a2 Z b3 a3 Z 1 0 N r N s f(q;)ddq 3 dq 2 dq 1 = Z b1 a1 Z b2 a2 Z b3 a3 m1 1;j1 m2 2;k1 m3 3;h1 m1 1;j2 m2 2;k2 m3 3;h2 f(q;)dq 3 dq 2 dq 1 Z 1 0 ' n i1 ' n i2 d (8.12) K N s;r =a 1 (q; N r ; N s ) + Z b1 a1 Z b2 a2 Z b3 a3 N r (0;q) N s (0;q)f(q;)dq 3 dq 2 dq 1 = Z b1 a1 Z b2 a2 Z b3 a3 Z 1 0 q 1 @ N r @ @ N s @ ddq 3 dq 2 dq 1 +' n i1 (0)' n i2 (0) Z b1 a1 Z b2 a2 Z b3 a3 m1 1;j1 m2 2;k1 m3 3;h1 m1 1;j2 m2 2;k2 m3 3;h2 f(q;)dq 3 dq 2 dq 1 = Z b1 a1 Z b2 a2 Z b3 a3 q 1 m1 1;j1 m2 2;k1 m3 3;h1 m1 1;j2 m2 2;k2 m3 3;h2 f(q;)dq 3 dq 2 dq 1 Z 1 0 (' n i1 ) 0 (' n i2 ) 0 d +' n i1 (0)' n i2 (0) Z b1 a1 Z b2 a2 Z b3 a3 m1 1;j1 m2 2;k1 m3 3;h1 m1 1;j2 m2 2;k2 m3 3;h2 f(q;)dq 3 dq 2 dq 1 (8.13) E N s;r =a 2 (q; N r ; N s ) = Z b1 a1 Z b2 a2 Z b3 a3 Z 1 0 q 2 N r N s ddq 3 dq 2 dq 1 = Z b1 a1 Z b2 a2 Z b3 a3 q 1 m1 1;j1 m2 2;k1 m3 3;h1 m1 1;j2 m2 2;k2 m3 3;h2 f(q;)dq 3 dq 2 dq 1 Z 1 0 ' n i1 ' n i2 d (8.14) B N s = Z b1 a1 Z b2 a2 Z b3 a3 q 3 N s (1;q)f(q;)dq 3 dq 2 dq 1 =' n i2 (1) Z b1 a1 Z b2 a2 Z b3 a3 q 3 m1 1;j2 m2 2;k2 m1 3;h2 (q;)dq 3 dq 2 dq 1 (8.15) 70 C N r = Z b1 a1 Z b2 a2 Z b3 a3 N r (0;q)f(q;)dq 3 dq 2 dq 1 =' n i1 (0) Z b1 a1 Z b2 a2 Z b3 a3 m1 1;j1 m2 2;k1 m1 3;h1 f(q;)dq 3 dq 2 dq 1 (8.16) where r;s = 1; ; (n + 1)m 1 m 2 m 3 with r = [(j 1 1)m 3 m 2 + (k 1 1)m 2 +h 1 1] +i 1 ,s = [(j 2 1)m 3 m 2 + (k 2 1)m 2 +h 2 1] +i 2 fori 1 ;i 2 = 1; ;n + 1,j 1 ;j 2 = 1; ;m 1 ,k 1 ;k 2 = 1; ;m 2 and ,h 1 ;h 2 = 1; ;m 3 . Hence, the matrices stated in (8.8) are as follows: M L = 2 6 4 M N s;r 0 0 M N s;r 3 7 5 (8.17) K L = 2 6 4 0 M N s;r K N s;r K N s;r E N s;r 3 7 5 (8.18) B L = 2 6 4 0 B N s 3 7 5 (8.19) C L = c N r 0 (8.20) After computing those matrices, we use the MATLAB routine FMINCON to solve a constrained optimization problem so that we can obtain the optimal param- eters. Before solving the deconvolutio problem, routine FMINSEARCH is used to nd the optimal regularization parameters. Thereafter, the estimated inputs are obtained by solving the optimization problems with the help of routine LSQNON- 71 NEG. 8.2 Numerical Results with Simulated Data In this section, we present the numerical results when using simulated data. The simulated data were generated in the following way. First, we set up a target distribution and sample this distribution to obtain 103 samples of parameters q. Second, we choose the input signal to be u(t) =jcos(t)j [0;2] (t),t2 [0; 20] with a sampling interval = 0:1. Next, a B-spline based approximation to the system (8.1) - (8.5) using 128 equally spaced point grids on [0,1] is then computed corresponding to each sample q. Then, the rst 100 output signals are averaged at each time point, and this averaged output signal and the input are used to train our telegraph model and obtain the optimal distribution, while the last three output signals are used as a test set when we deconvolve our system with the optimal distribution. The distribution of parameters is assumed to be a truncated normal distribution. Set the parabolic regularization coecient to be 0.0000001, n = m 1 = m 2 = m 3 = 4 and x the regularization parameter r = (r 1 ;r 2 ) = (0:1; 1). Let the target distribution be a normal distribution with mean = [12; 10; 8] and covariance matrix = 2 6 6 6 6 4 9 3 1 3 5 2 1 2 1 3 7 7 7 7 5 (8.21) In Figures 8.1-8.3, we illustrate BrAC and TAC from the simulated data used for training and testing, respectively: 72 Figure 8.1: Input function u(t) Figure 8.2: Simulated TAC corresponding to u(t) for training 73 Figure 8.3: Simulated TAC corresponding to u(t) for testing We then use the averaged TAC of the rst 100 simulated TAC data and input function u(t) to train our telegraph model and obtain the optimal distribution of the parameters. The optimal distribution is as follows: q 1 2 [6:4652; 18:3337], q 2 2 [5:6841; 15:4382], q 3 2 [5:9579; 10:1780], = [11:3405; 9:6020; 7:8823] and covariance matrix = 2 6 6 6 6 4 14:3285 2:4949 0:8370 2:4949 8:1738 2:2986 0:8370 2:2986 0:7445 3 7 7 7 7 5 (8.22) Figure 8.4 presents the tting results of our model on the 100 training episodes.In the graph, the red dots represent the TAC data from each training episode. The black dashed lines are the 90% credible bands for TAC, and the green line represents the input function. 74 Figure 8.4: Training on 100 simulated episodes Figures 8.5-8.7 dsiplay the performance of our model on the test episodes. Our model's estimation depends on the input function u(t) and the distribution of pa- rameters obtained by training the averaged episode, which is the average of 100 training episodes. In these gures, the red line represents the input function. More- over, the blue dotted line denotes the TAC from each episode, while the black line indicates the estimated BrAC, and the gray area covers the 90% credible band. 75 Figure 8.5: Simulated Test Episode 1: Estimating the input function u(t) with TAC from Test Episode 1 and the optimal distribution of the parameters from the averaged training episodes Figure 8.6: Simulated Test Episode 2: Estimating the input function u(t) with TAC from Test Episode 2 and the optimal distribution of the parameters from the averaged training episodes 76 Figure 8.7: Simulated Test Episode 3:Estimating the input function u(t) with TAC from Test Episode 3 and the optimal distribution of the parameters from the aver- aged training episodes 8.3 Numerical Results with Alcohol Biosensor Data 8.3.1 Dataset 1 - Single Subject with 11 Drinking Episodes A single individual wore a WrisTAS TM 7 alcohol biosensor for 18 days. During this time, the biosensor was set to measure the TAC data at 5-minute intervals. We marked 11 drinking episodes in this period. The rst drinking episode was in the laboratory; from the start,BrAC data were measured every 15 minutes and stopped when the BrAC measurements returned to 0.000. For the second drinking episode, BrAC data were also measured every 15 minutes. For the remaining 9 episodes, the subject took a BrAC measurement every 30 minutes until it returned to 0.000. The TAC measurements are in milligrams per deciliter (mg/dl), and the BrAC measurements are percentage of alcohol. Figure 8.8 portrays the BrAC and TAC of all 11 episodes. 77 Figure 8.8: BrAC and TAC measurements for Dataset 1 We then assume the three parameters (q 1 , q 2 , and q 3 ) in the telegraph model satisfy a truncated multivariate normal distribution and use each episode to the t the model and compute the optimal distribution and optimal regularization param- eters. Table 8.1 lists all results of the optimal distribution for each episode. (Note: q 1 , q 2 and q 3 satisfy a truncated multivariate normal distribution. (q 1 ;q 2 ;q 3 )2 [q 1 left;q 1 right] [q 2 left;q 2 right] [q 3 left;q 3 right], variance = [ 11 ; 21 ; 31 ; 21 ; 22 ; 32 ; 31 ; 32 ; 33 ], regularization parameter r = (r 1 ;r 2 ).) 78 Table 8.1: Optimal distribution for each episode Episode Number q 1 -left q 1 -right q 2 -left q 2 -right q 1 mean q 2 mean 11 21 Episode 1 0.0000 43.8248 0.0000 2.4102 35.3838 1.2148 4.8764 0.0000 Episode 2 0.0000 7.7645 0.0000 1.9899 6.2458 1.0254 1.1356 0.0000 Episode 3 0.0000 57.5021 0.0000 3.3747 46.4713 1.7345 6.3171 0.0000 Episode 4 0.0000 1.9008 0.0000 2.0643 1.0302 1.0441 0.3968 0.0000 Episode 5 0.0000 62.7186 0.0000 2.7645 56.4592 1.3894 7.6010 0.0000 Episode 6 0.0000 5.1519 0.0000 1.4673 4.1210 0.7475 0.7874 0.0000 Episode 7 0.0000 27.2914 0.0000 1.1558 22.4360 0.6165 3.3118 0.0000 Episode 8 0.0000 19.0759 0.0000 3.0563 18.1687 1.5668 2.6783 0.0000 Episode 9 0.0000 40.7706 0.0000 2.7009 33.0547 1.3928 4.6047 0.0000 Episode 10 0.0000 36.6625 0.0000 2.8345 29.8222 1.4539 4.2233 0.0000 Episode 11 0.0000 25.9376 0.0000 1.6913 21.2655 0.8408 3.1304 0.0000 median 0.0000 27.2914 0.0000 2.4102 22.4360 1.2148 3.3118 0.0000 Episode Number 22 q 3 mean q 3 -left q 3 -right 33 31 32 r1 r2 Episode 1 0.3987 2.0518 0.0000 4.6166 0.5266 0.0000 0.0000 0.2118 0.8654 Episode 2 0.3304 1.6074 0.0000 3.0107 0.4188 0.0000 0.0000 0.0000 0.0167 Episode 3 0.5623 2.0259 0.0000 3.8565 0.7161 0.0000 0.0000 N.A. N.A. Episode 4 0.3440 1.5108 0.0000 3.6845 0.3998 0.0000 0.0000 0.2175 0.0191 Episode 5 0.4557 1.6545 0.0000 3.0861 0.6201 0.0000 0.0000 N.A. N.A. Episode 6 0.2440 1.1940 0.0000 2.5602 0.2970 0.0000 0.0000 0.1951 0.2041 Episode 7 0.2021 1.0434 0.0000 2.3229 0.2678 0.0000 0.0000 0.0858 0.2949 Episode 8 0.5094 2.2602 0.0000 4.6450 0.5699 0.0000 0.0000 0.1046 0.0588 Episode 9 0.4489 1.6047 0.0000 3.1614 0.5590 0.0000 0.0000 0.0000 1.2614 Episode 10 0.4754 1.4616 0.0000 3.2337 0.3724 0.0000 0.0000 0.0000 2.9932 Episode 11 0.2725 1.4344 0.0000 2.9984 0.4879 0.0000 0.0000 0.3772 0.8601 median 0.3987 1.6047 0.0000 3.1614 0.4879 0.0000 0.0000 0.1046 0.2949 79 Figure 8.9: Numerical results for each episode computed by a truncated multivariate normal distribution with each their own optimal parameters Since our aim is to nd an optimal population distribution, we rst apply the median of the parameters for 11 episodes to the telegraph model and compute the estimated BrAC and credible bands for each episode. The results are displayed in Figure 8.10. 80 Figure 8.10: Numerical results for each episode computed by a truncated multivari- ate normal distribution with the median of 11 sets of optimal distributions To increase our approach's eciency, we consider picking some similar episodes based on their TAC. The training process may take much time when the number of 81 episodes used in training is large. To reduce the processing time of training and keep the same level of accuracy of our model, we consider training our model with some small groups of selected episodes. We then obtain the parameters' optimal distribu- tion by training the model with those selected episode instead of all episodes. After comparing the graphs of the TAC of each episode, we nd two groups of episodes with similar characteristics. We put the episodes with low peak TAC measurement in one group, and the episodes with high peak TAC measurement in another group. Group 1 contains Episodes 2,6,7,and 8, and Group 2 is composed of Episodes 1,4,5, and 11. Episodes 3,9,and 10 thus remained and the comprise Group 3. These three episodes will be group 3. We take the Group 1 and Group 2 as training group. They are then used to train our model separately. After training, we use the opti- mal distribution of the two groups' parameters to test with the episodes in Group 3. We also list the optimal distribution obtained by training the model with all 11 episodes. The optimal distributions of the parameters from each group are as follows: All 11 episodes: We have q 1 2 [4:0245; 26:1133] q 2 2 [0:9326; 2:2715], q 3 2 [1:0324; 1:7089] = [20:8699; 1:5298; 0:9039] and covariance matrix = 2 6 6 6 6 4 38:7411 0:4908 1:5304 0:4908 0:1488 0:0761 1:5304 0:0761 0:1007 3 7 7 7 7 5 (8.23) The graphs in Figure 8.11 illustrate the density function of the optimal distribution: 82 Figure 8.11: The density function of the optimal distribution from 11 episodes Group 1: We haveq 1 2 [3:1641; 11:1560]q 2 2 [0:8797; 1:7281],q 3 2 [0:6465; 1:2041] = [7:4026; 1:3367; 0:8404] and covariance matrix = 2 6 6 6 6 4 14:7756 0:2548 0:1904 0:2548 0:1348 0:1389 0:1904 0:1389 0:1439 3 7 7 7 7 5 (8.24) The graphs in Figure 8.12 illustrate the density function of the optimal distribution: 83 Figure 8.12: The density function of the optimal distribution from Group 1 Group 2: We haveq 1 2 [4:4667; 30:3350]q 2 2 [1:0939; 2:2416],q 3 2 [1:1207; 1:4525] = [17:5610; 1:5763; 0:9174] and covariance matrix = 2 6 6 6 6 4 84:0615 0:6291 1:6580 0:6291 0:0803 0:0290 1:6580 0:0290 0:0440 3 7 7 7 7 5 (8.25) The graphs in Figure 8.13 illustrate the density function of the optimal distribution: 84 Figure 8.13: The density function of the optimal distribution from Group 2 We then show the estimated results for each episode in Group 3 using these three sets of the optimal distribution of parameters. 85 Figure 8.14: Estimated BrAC using the TAC from Episode 3 and the optimal distribution of the parameters from a) all 11 episodes (top left), b) Group 1 (top right), c) Group 2 (bottom ) Figure 8.15: Estimated BrAC using the TAC from Episode 9 and the optimal distribution of the parameters from a) all 11 episodes (top left), b) Group 1 (top right), c) Group 2 (bottom ) 86 Figure 8.16: Estimated BrAC using the TAC from Episode 10 and the optimal distribution of the parameters from a) all 11 episodes (top left), b) Group 1 (top right), c) Group 2 (bottom ) Figure 8.14-8.16 clearly illustrate that the estimation for Group 3 using the optimal distribution obtained by Group 2 is much better than the distributions ob- tained by either Group 1 or all 11 episodes . The main reason is that the episodes in Group 3 share similar characteristics to the episodes from Group 2 than those from Group 1. Alcohol researchers also study several statistics associated with the BrAC. The peak BrAC (both the value and the time of the peak BrAC) is one of the most important statistics. The peak BrAC can be used to obtain the BrAC absorption rate and the BrAC elimination rate (see [31, 30, 33]).[31] introduced that "the BrAC absorption rate equals the peak BrAC value divided by the amount of time that passes from the last zero BrAC measurement until the peak BrAC shows. The BrAC elimination rate is calculated by the peak BrAC value divided by the amount of time that passes from the peak BrAC to the rst zero BrAC." Table 8.3 shows the estimated peak BrAC for each episode in Group 3 using the optimal distribution from Group 1 and Group 2. 87 Table 8.2: Dataset 1:Estimation of the peak BrAC value Value of the Peak BrAC Optimal Distribution Trained by All 11 Episodes Optimal Distribution Trained by Group 1 Optimal Distribution Trained by Group 2 Episode Number Actual Value Estimated Value Credible Bands 90% Actual Value Estimated Value Credible Bands 90% Actual Value Estimated Value Credible Bands 90% Episode 3 in Group 3 0.0360 0.0272 [0.0233,0.0311] 0.0360 0.0488 [0.0392,0.0576] 0.0360 0.0163 [0.0141,0.0186] Episode 9 in Group 3 0.0260 0.0184 [0.0156,0.0212] 0.0260 0.0300 [0.0254,0.0344] 0.0260 0.0116 [0.0100,0.0133] Episode 10 in Group 3 0.0130 0.0061 [0.0052,0.0070] 0.0130 0.0130 [0.0106,0.0153] 0.0130 0.0032 [0.0028,0.0036] Table 8.3: Dataset 1:Estimation of the peak BrAC time Time of the Peak BrAC Optimal Distribution Trained by All 11 Episodes Optimal Distribution Trained by Group 1 Optimal Distribution Trained by Group 2 Episode Number Actual Peak Time Estimated Peak Value Actual Peak Time Estimated Peak Value Actual Peak Time Estimated Peak Value Episode 3 in Group 3 1.5 1.4667 1.5 1.15 1.5 1.75 Episode 9 in Group 3 2.0000 1.8667 2.0000 1.700 2.0000 2.0667 Episode 10 in Group 3 1.0833 1.1667 1.0833 0.9167 1.0833 1.3667 Table 8.2 clearly illustrate that the estimation of the peak BrAC for Group 3 using the optimal distribution obtained by Group 2 is better than the distributions obtained by either Group 1 or all 11 episodes. 8.3.2 Dataset 2 - Multiple Subjects, each with a Single Drinking Episode Dataset 2 was collected from 17 subjects wearing a WrisTAS TM 7 alcohol biosensor under the same protocol as in the rst episode in Dataset 1. For each subject, there is only one drinking episode. The participants were administered a adjusted dose (by gender an weight) of alcohol designed to reach a peak BrAC around 0.05%, which is neither too high nor too low. All subjects were monitored in the laboratory with breathalyzer reading taken every 15 minutes until BrAC returned to 0.000. The biosensor was set to measure the TAC data at 5-minute intervals. For this dataset, we divide these 17 episodes into two groups: the training and test group. We randomly pick 14 episodes (named Episode 1-14) and use them to train our system and obtain the optimal parameters. Then we deconvolve our system using the optimal parameters to estimate the remaining three episodes (named Episodes 15- 17) in the test group. The optimal parameters are as follows: q 1 2 [6:0746; 56:3077] 88 q 2 2 [0:2947; 3:2000], q 3 2 [0:4041; 2:2591] = [45:0429; 0:9574; 0:9439] and covariance matrix = 2 6 6 6 6 4 223:6090 8:1723 3:9502 8:1723 1:8330 0:4401 3:9502 0:4401 0:3624 3 7 7 7 7 5 (8.26) The graphs in Figure 8.17 illustrate the density function of the optimal distribution: Figure 8.17: The density function of the optimal distribution from Dataset 2 We then show the estimated results for the last three episodes using the optimal 89 distribution of the parameters in Figure 8.18. Figure 8.18: Numerical results for the three test episode computed by a truncated multivariate normal distribution with optimal distribution Table 8.4: Dataset 2:Estimation of the peak BrAC value Value of the Peak BrAC Episode Number Actual Value Estimated Value Credible Bands Episode 15 0.0421 0.0361 [0.0188,0.0541] Episode 16 0.0475 0.0251 [0.0129,0.0375] Episode 17 0.0511 0.0429 [0.0216,0.0640] Table 8.5: Dataset 2:Estimation of the peak BrAC time Time of the Peak BrAC Episode Number Actual Peak Time Estimated Peak Time Episode 15 1.41 1.2333 Episode 16 1.2900 1.0500 Episode 17 1.5900 1.43333 90 Based on Figure 8.14 and Tables 8.4-8.5, our estimation oers a reliable estima- tion of the peak BrAC and associated credible bands. However, further studies are necessary worth doing to narrow down the credible bands. Note: In this section, we set m 1 = m 2 = m 3 = 4 and n = 4. As we know, the larger m 1 ;m 2 ;m 3 ;n are, the more accurate the nal results are. 8.4 Comparison between the results of the hyperbolic model and parabolic model In [33], Sirlanci t the parabolic model with the same Dataset 1 in Section 8.3.1. We compare the estimated results from the two models by the peak BrAC and the area under the BrAC curve. (Note: the regularization parameters used in [33] were obtained by solving an optimization problem, whereas we x the values of the reg- ularization parameters in our results.) For the parabolic model, we use the optimal distribution from the stratied population stated in [33]; for the hyperbolic model, we consider the optimal distribution trained by Group 1 with q 1 2 [3:1641; 11:1560] q 2 2 [0:8797; 1:7281], q 3 2 [0:6465; 1:2041], mean = [7:4026; 1:3367; 0:8404] and covariance matrix = 2 6 6 6 6 4 14:7756 0:2548 0:1904 0:2548 0:1348 0:1389 0:1904 0:1389 0:1439 3 7 7 7 7 5 (8.27) The graphs in Figure 8.19 illustrate the density function of the optimal distribution: 91 Figure 8.19: The density function of the optimal distribution from Group 1 in Dataset 1 In the rst graph of Figure 8.15, the density is depicted by color. In the remain- ing four graphs, we use four dierent values ofq 2 (q 2 = 1:0409; 1:2106; 1:3803; 1:5499), and we let q 1 , q 2 and the density be the x,y and z axis, respectively. Table 8.6: Comparison between the hyperbolic and parabolic models- Estimation of the peak BrAC value Value of the Peak BrAC Optimal Distribution Trained by the Parabolic Model in [33] Optimal Distribution Trained by the Hyperbolic Model Episode Number Actual Value Estimated Value Credible Bands 75% Actual Value Estimated Value Credible Bands 90% Episode 3 in Dataset 1 0.0360 0.0468 [0.0362,0.0650] 0.0360 0.0488 [0.0392,0.0576] Episode 9 in Dataset 1 0.0260 0.0302 [0.0187,0.0506] 0.0260 0.0300 [0.0254,0.0344] Episode 10 in Dataset 1 0.0130 0.0195 [0.0119,0.0333] 0.0130 0.0130 [0.0106,0.0153] 92 Table 8.7: Comparison between the hyperbolic and parabolic models-Estimation of the peak BrAC time Time of the Peak BrAC Optimal Distribution Trained by the Parabolic model in [33] Optimal Distribution Trained by the Hyperbolic Model Episode Number Actual Peak Time Estimated Peak Value Actual Peak Time Estimated Peak Value Episode 3 in Dataset 1 1.5 1.1000 1.5 1.15 Episode 9 in Dataset 1 2.0000 2.6333 2.0000 1.700 Episode 10 in Dataset 1 1.0833 0.9333 1.0833 0.9167 Table 8.8: Comparison between the hyperbolic and parabolic models-Estimation of the peak BrAC value Area under the BrAC curve Optimal Distribution Trained by the Parabolic Model in [33] Optimal Distribution Trained by the Hyperbolic Model Episode Number Actual Value Estimated Value Credible Bands 75% Actual Value Estimated Value Credible Bands 90% Episode 3 in Dataset 1 0.0750 0.1086 [0.0638,0.1540] 0.0750 0.1343 [0.0992,0.1621] Episode 9 in Dataset 1 0.0554 0.0518 [0.0168,0.0948] 0.0554 0.0837 [0.0610,0.0973] Episode 10 in Dataset 1 0.0175 0.0184 [0.0062,0.0335] 0.0175 0.0216 [0.0164,0.0257] Tables 8.6-8.8 clearly illustrate that the estimation for the episodes in Group 3 using the hyperbolic model's optimal distribution is either better or similar to the distributions obtained by the parabolic model. Meanwhile, the condence bands of the estimation obtained by the hyperbolic model are much better than those from the parabolic model, which means that our hyperbolic model provides more accurate condence bands. On the one hand, the better results from the hyperbolic system are as expected. On the other hand, because we used three parameters in the system, which added an extra degree of freedom compared to the two-parameter parabolic model, the hyperbolic system is not as ecient as the parabolic system. The state space of the hyperbolic system is twice as large as that of the parabolic system. Furthermore, the optimal distribution in the hyperbolic system is deter- mined by 15 parameters instead of seven in the parabolic system. 93 9 Discussion and concluding remarks We have obtained meaningful results in this thesis; however some mathematical questions remain open, thus warranting further research. One of them is that we dene the measures in terms of a density, instead of measures directly. We have to assume the type of density function of the parameters in advance. In other words, we could apply the convergence theory directly to the class of problems based on the metric on a space of measures. Another direction is to estimate the shape of the density instead of parameters. Furthermore, the performance of the population model could be improved with higher dimensional parameterization. In this thesis, we kept the dimension of all parameters as small as possible due to the PC's com- puting ability. It would be interesting to verify the convergence theory numerically via a high-performance computing cluster. In Section 8.3.1, we divided Dataset 1 into subgroups according to their common characteristics (peak TAC or the shape of TAC graph). Grouping the episodes by their TAC improves the performance and eciency of the estimation; however, this is unlikely to occur in practice because we will not be able to see the peak TAC or the full graph of TAC when estimating in real time. Moreover, the peak TAC is not the only common characteristic that impacts our estimation. Therefore, if we could identify other observable characteristics that would also aect the estimation, they could be used to improve the eciency and accuracy of the estimation, compared to the estimation using a single population model for all people. We could nd such common characteristics among observable personal covariates (e.g., height, weight, and gender) and environmental covariates (e.g., weather, location, skin hydration and ambient temperature). Finally, supposing that the alcohol biosensor can be incorporated into wearable devices such as an Apple Watch and a Fitbit, we would then be able to collect personal and environmental covariates while tracking TAC. In that case, our model could provide reliable estimated BrAC based on TAC simultaneously. 94 References [1] H. T. Banks and J. A. Burns. Hereditary control problems: Nu- merical methods based on averaging approximations, Mar 11 1976. URL: http://libproxy.usc.edu/login?url=https://www-proquest-com. libproxy2.usc.edu/docview/87604382?accountid=14749. [2] H. T. Banks and K. Ito. Unied framework for approximation in inverse problems for distributed parameter systems. Technical report, 01 1988. URL: http://libproxy.usc.edu/login?url=https://www-proquest-com. libproxy2.usc.edu/docview/86870969?accountid=14749. [3] H. T. Banks and K. Ito. A unied framework for approximation in inverse problems for distributed parameter systems. Control Theory Advanced Tech- nology, 4(1):73{90, 1988. URL:https://apps.dtic.mil/dtic/tr/fulltext/ u2/a193780.pdf. [4] H. T. Banks and Karl Kunisch. Estimation Techniques for Distributed Pa- rameter Systems. Springer Science & Business Media, 2012. URL: https: //www.springer.com/us/book/9780817634339. [5] R. F. Curtain and D. Salamon. Finite-dimensional compensators for innite-dimensional systems with unbounded input operators. SIAM Journal on Control and Optimization, 24(4):797{20, 07 1986. URL: http://libproxy.usc.edu/login?url=https://www-proquest-com. libproxy2.usc.edu/docview/926027128?accountid=14749. [6] Zheng Dai, I.G. Rosen, Chuming Wang, Nancy Barnett, and Susan Luczak. Us- ing drinking data and pharmacokinetic modeling to calibrate transport model and blind deconvolution based data analysis software for transdermal alcohol biosensors. Mathematical Biosciences and Engineering, 13:911{934, 07 2016. doi:10.3934/mbe.2016023. [7] Mehdi Dehghan and Ali Shokri. A numerical method for solving the hyper- bolic telegraph equation. Numerical Methods for Partial Dierential Equations, 24(4):1080{1093, 2008. URL: https://onlinelibrary.wiley.com/doi/abs/ 10.1002/num.20306, doi:10.1002/num.20306. [8] M. Dumett, Rosen I. G., Sabat J., Shaman A., Tempelman L. A., Wang C., and Swift R. M. Deconvolving an estimate of breath measured blood alco- hol concentration from biosensor collected transdermal ethanol data. Applied Mathematics and Computation, 196(1):724{743, 2008. [9] M.A. Dumett, I.G. Rosen, J. Sabat, A. Shaman, L. Tempelman, C. Wang, and R.M. Swift. Deconvolving an estimate of breath measured blood al- cohol concentration from biosensor collected transdermal ethanol data. Ap- plied Mathematics and Computation, 196(2):724 { 743, 2008. URL: http:// www.sciencedirect.com/science/article/pii/S0096300307007254, doi: https://doi.org/10.1016/j.amc.2007.07.026. [10] Claude Jeerey Gittelson, Roman Andreev, and Christoph Schwab. Optimality of adaptive galerkin methods for random parabolic partial dierential ewua- tions. Journal of Computational and Applied Mathematics, pages 189{201, 2014. 95 [11] E. Hille and R. S. Phillips. Functional Analysis and Semi-Groups. American Mathematical Society, 1957. URL: https://bookstore.ams.org/coll-31. [12] T. Hillen and K.P. Hadeler. Nonlinear hyperbolic systems and transport equa- tions in mathematical biology. Springer, Berlin, Heidelberg, 2005. URL: https://link.springer.com/book/10.1007/3-540-27907-5. [13] T. Kato. Perturbation Theory for Linear Operators. Springer, 1976. URL: https://www.springer.com/gp/book/9783540586616. [14] Tosio Kato. Perturbation Theory for Linear Operators, volume 132. Springer Science & Business Media, 2013. URL:https://www.springer.com/us/book/ 9783540586616. [15] A. F. J. Levi and I. G. Rosen. A novel formulation of the adjoint method in the optimal design of quantum electronic devices. SIAM Journal on Control and Optimization, 48(5):3191{3223, 2010. [16] J.L. Lions. Optimal Control of Systems Governed by Partial Dierential Equa- tions. Grundlehren der mathematischen Wissenschaften in Einzeldarstellun- gen mit besonderer Ber ucksichtigung der Anwendungsgebiete. Springer-Verlag, 1971. URL: https://books.google.com/books?id=aL9tlwEACAAJ. [17] S. E. Luczak, I. G. Rosen, and T.L. Wall. Development of a real-time repeated- measures assessment protocol to capture change over the course of a drinking episode. Alcohol and Alcoholism, 50(1):1{8, 2015. [18] Susan E. Luczak, I. Gary Rosen, and Jordan Weiss. Determining blood and/or breath alcohol concentration from transdermal alcohol data. 2013 American Control Conference, pages 473{478, 2013. [19] V. Mendez, S. Fedotov, and W. Horsthemke. Reaction-Transport Systems. Springer, 2019. URL: https://www.springer.com/gp/book/9783642114427. [20] Akira Okubo and Simon A. Levin. Diusion and Ecological Problems, Modern Perspectives, Second Edition. Springer, New York, 2001. [21] A. Pazy. Semigroups of Linear Operators and Applications to Partial Dier- ential Equations. Springer-Verlag, 1983. URL: https://link.springer.com/ book/10.1007/978-1-4612-5561-1. [22] A. Pazy. Semigroups of Linear Operators and Applications to Partial Dif- ferential Equations. Applied Mathematical Sciences. Springer, 1983. URL: https://books.google.com/books?id=80XYPwAACAAJ. [23] Anthony J Pritchard and Dietmar Salamon. The linear quadratic control problem for innite dimensional systems with unbounded input and output operators. SIAM Journal on Control and Optimization, 25(1):121{144, 1987. URL: https://epubs.siam.org/doi/abs/10.1137/0325009, doi:https:// doi.org/10.1137/0325009. [24] I. G. Rosen, S. E. Luczak, W. Hu, and M. Hankin. Discrete-time blind de- convolution for distributed parameter systems with dirichlet boundary input and unbounded output with application to a transdermal alcohol biosensor. roceedings of 2013 SIAM Conference on Control and its Applications, 2013. 96 [25] I. G. Rosen, S. E. Luczak, and J. Weiss. Blind deconvolution for distributed parameter systems with unbounded input and output and determining blood alcohol concentration from transdermal biosensor data. Applied Math and Computation, 231(1):357{376, 2014. [26] I Gary Rosen, Susan E Luczak, and Jordan Weiss. Blind deconvolution for dis- tributed parameter systems with unbounded input and output and determining blood alcohol concentration from transdermal biosensor data. Applied Mathe- matics and Computation, 231:357{376, 2014. URL: https://www.ncbi.nlm. nih.gov/pmc/articles/PMC3972634/, doi:10.1016/j.amc.2013.12.099. [27] M. H. Schultz. Spline Analysis. Prentice-Hall Series in Automatic Computa- tion. Pearson Education, Limited, 1972. URL: https://books.google.com/ books?id=AdRQAAAAMAAJ. [28] MH. Schultz. Spline Analysis. Prentice Hall, Englewood Clis, 1973. [29] Melike Sirlanci, Susan E. Luczak, Catharine E. Fairbairn, Dayheon Kang, Ruoxi Pan, Xin Yu, and I Gary Rosen. Estimating the distribution of ran- dom parameters in a diusion equation forward model for a transdermal alcohol biosensor. Automatica, 2018. to appear, arXiv:1808.04058. URL: https://arxiv.org/abs/1808.04058. [30] Melike Sirlanci, Susan E. Luczak, and I. Gary Rosen. Estimation of the dis- tribution of random parameters in discrete time abstract parabolic systems with unbounded input and output: Approximation and convergence, 2018. arXiv:1807.04904. [31] Melike Sirlanci, I Gary Rosen, Susan E. Luczak, Catharine E. Fairbairn, Konrad Bresin, and Dayheon Kang. Deconvolving the input to random abstract parabolic systems; a population model-based approach to estimat- ing blood/breath alcohol concentration from transdermal alcohol biosensor data. Inverse problems, 34(12), 2018. arXiv:1807.05088v1. URL: http: //iopscience.iop.org/article/10.1088/1361-6420/aae791/pdf. [32] H. Tanabe. Equations of Evolution. Monographs and Studies in Mathematics. Pitman, 1979. URL: https://books.google.com/books?id=Dn6zAAAAIAAJ. [33] Melike Sirlanci Tuysuzoglu. Finite dimensional approximation and convergence in the estimation of the distribution of, and input to, random abstract parabolic systems with application to the deconvolution of blood/breath alcohol con- centration from biosensor measured transdermal alcohol level. PhD Disserta- tion, 2018. URL: http://digitallibrary.usc.edu/cdm/ref/collection/ p15799coll89/id/21622. 97
Abstract (if available)
Abstract
This research is motivated by a blind deconvolution problem involving the data from a transdermal alcohol biosensor, tracking the number of ethanol molecules in perspiration. To make these devices practicable, we developed a hyperbolic population model to convert biosensor-measured transdermal alcohol concentration (TAC) into blood alcohol concentration (BAC) or breath alcohol concentration (BrAC) consistently and reliably. Two steps are required to solving this problem: one is a parameter estimation problem, and the other is a deconvolution problem. All un-modeled sources of uncertainty observed in individual data were attributed to random effects by defining the parameters in the system as random variables with joint density function. The distribution was then estimated by estimating the joint density function. An abstract approximation framework and convergence theory was developed to estimate the distribution of random parameters in an infinite-dimensional discrete-time system. By taking expectation on both sides of the hyperbolic PDEs, the randomness of the parameters of a hyperbolic system was embedded in sesquilinear form or equivalently in random regularly dissipative operators. Then, each random variable could be treated as another space-like independent variable. In this way, the parameter estimation problem became an optimization problem over the space of all feasible distributions for the random parameters. In this optimization problem, aggregate population data, instead of individual data, were used to fit the model describing the population’s mean behavior. This process allowed us to estimate the distribution of the parameters instead of individual parameters. After solving the optimization problem and obtaining the optimal distribution, the model was inverted to produce credible bands for BAC/BrAC corresponding to TAC by solving a deconvolution problem. We demonstrated that the alcohol biosensor model satisfies all the assumptions we made in developing the approximation framework and convergence theory. Last, numerical results were obtained and presented when using simulated data, multiple episodes from the same subject, and a single episode from multiple subjects.
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Dai, Zheng
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Core Title
An abstract hyperbolic population model for the transdermal transport of ethanol in humans: estimating the distribution of random parameters and the deconvolution of breath alcohol concentration
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College of Letters, Arts and Sciences
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Doctor of Philosophy
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Applied Mathematics
Publication Date
09/11/2020
Defense Date
08/27/2020
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alcohol biosensor,BrAC estimation,hyperbolic,OAI-PMH Harvest,random parameters,telegraph equation,transdermal alcohol concentration
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English
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Rosen, Gary (
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), Luczak, Susan (
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), Wang, Chunming (
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dz919abc@gmail.com,zhengdai@usc.edu
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Tags
alcohol biosensor
BrAC estimation
hyperbolic
random parameters
telegraph equation
transdermal alcohol concentration