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Simultaneous parameter estimation and semi-blind deconvolution in infinite-dimensional linear systems with unbounded input and output

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Simultaneous parameter estimation and semi-blind deconvolution in infinite-dimensional linear systems with unbounded input and output

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Content Simultaneous Parameter Estimation and Semi-Blind Deconvolution in Innite-Dimensional Linear Systems with Unbounded Input and Output by Moses Alexander Rubin Wintner A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Mathematics) December 2018 Acknowledgements I would like to thank my advisor Gary Rosen for his unbounded patience, careful guidance, and helpful discussions over the course of my study and work for this dissertation, and for aiding me in my maturation from a green graduate student into a mathematician. I would also like to thank Aravind Asok, Susan Montgomery, and the Dornsife College of Letters, Arts, and Sciences for the Dornsife Doctoral Fellowship, which funded my studies and research. I would like to make a further dedication to all my friends and colleagues that I have had the fortune to think and talk about mathematics with over the years, without whom I would have had no fun in this line of work and surely would not have come to undertake research; to my wife Jillian, who endured my preoccupation during nights and weekends for the past several years with patience and kindness; and foremost to my parents Lisa and Chuck and my brother Miles, for their unconditional love and support, and who taught me to love challenging myself. ii Table of Contents Acknowledgements ii List of Tables v List of Figures vi Abstract vii Chapter 1: Introduction 1 Chapter 2: Mathematical model for transdermal alcohol transport and detection 7 2.1 Derivation of PDE model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Transdermal alcohol model as abstract parabolic system . . . . . . . . . . . . . . . . . . . . . 10 2.3 Approximate semi-blind deconvolution problem . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Chapter 3: Preliminary results and background 13 3.1 Functional analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Linear semigroup theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Chapter 4: The estimation problem for a class of abstract parabolic systems with input and output on the boundary 20 4.1 Gelfand triple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.2 Parabolic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.3 Boundary input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.4 Boundary input for parabolic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.5 Continuous dependence on system parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.6 Transdermal alcohol model as abstract parabolic system with boundary input and output . . 28 4.6.1 Sesquilinear form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.6.2 Input and output operators for transdermal alcohol model . . . . . . . . . . . . . . . . 31 Chapter 5: Approximate semi-blind deconvolution in discrete time parabolic systems with input and output on the boundary 33 5.1 Abstract discrete time parabolic boundary input system . . . . . . . . . . . . . . . . . . . . . 33 5.2 Well-posedness of discrete time parabolic boundary input model . . . . . . . . . . . . . . . . 35 5.3 Finite dimensional approximation of parabolic boundary input model . . . . . . . . . . . . . . 39 5.4 Finite dimensional approximation of discrete time parabolic boundary input model . . . . . . 41 5.5 Discrete time transdermal alcohol model approximation and convergence . . . . . . . . . . . . 45 5.5.1 Linear spline approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.5.2 Finite dimensional approximation of discrete time transdermal alcohol model . . . . . 45 5.6 Approximate solution of estimation problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 iii Chapter 6: Approximate semi-blind deconvolution in delay systems with unbounded input and output 48 6.1 Homogeneous delay evolution system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 6.2 Inhomogeneous delay evolution system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6.3 Finite dimensional approximation of inhomogeneous delay evolution system . . . . . . . . . . 52 6.4 Linear spline approximation of delay system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.5 Results of Banks-Burns-Cli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.6 Approximate semi-blind deconvolution in delay systems . . . . . . . . . . . . . . . . . . . . . 59 Chapter 7: Approximate semi-blind deconvolution in continuous time parabolic systems with input and output on the boundary 62 7.1 Well posedness of continuous time parabolic boundary input model . . . . . . . . . . . . . . . 62 7.2 Finite dimensional approximation of continuous time parabolic boundary input model . . . . 67 7.2.1 Stability of continuous time parabolic boundary input model approximation . . . . . . 68 7.2.2 Consistency of continuous time parabolic boundary input model approximation . . . . 70 7.3 Finite dimensional approximation of continuous time transdermal alcohol model . . . . . . . 73 Chapter 8: Numerical results 76 8.1 Adjoint method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 8.2 Computation of derivatives of matrix exponentials . . . . . . . . . . . . . . . . . . . . . . . . 81 8.3 Numerical results for semi-blind deconvolution in parabolic systems . . . . . . . . . . . . . . . 81 8.3.1 Varying state discretization N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 8.3.2 Varying estimated input signal resolution P . . . . . . . . . . . . . . . . . . . . . . . . 84 8.3.3 Varying estimated system parameter complexity M . . . . . . . . . . . . . . . . . . . 86 8.4 Numerical results for semi-blind deconvolution of blood/breath alcohol concentration from transdermal alcohol concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 8.5 Numerical results for semi-blind deconvolution in delay systems . . . . . . . . . . . . . . . . . 90 Chapter 9: Conclusions 96 Bibliography 98 iv List of Tables 8.1 Average BrAC deconvolution error results, patient 5122, test episodes 1-4, varying resolution of state discretization, 30-minute mesh, spatially constant diusivity q 1 . . . . . . . . . . . . 84 8.2 Trained q 1 N , patient 5122, drinking episodes 1-4, simulated TAC (actual = 1) . . . . . . . . . 85 8.3 Trained q 2 N , patient 5122, drinking episodes 1-4, simulated TAC (actual = 0:3) . . . . . . . . 85 8.4 Average BrAC deconvolution error results, patient 5122, test episodes 1-4, varying input signal mesh resolution, N = 8, spatially constant diusivity q 1 . . . . . . . . . . . . . . . . . . . . . 85 8.5 Trained q 1 N , patient 5122, drinking episodes 1-4, N = 8, varying estimated signal resolution, simulated TAC (actual = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 8.6 Trained q 2 N , patient 5122, drinking episodes 1-4, N = 8, varying estimated signal resolution, simulated TAC (actual = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 8.7 Average BrAC deconvolution error results, patient 5122, test episodes 1-4, 30-minute mesh, N = 8, quadratically varying diusivity q 1 (x) = 3(x:5) 2 +:1 . . . . . . . . . . . . . . . . . 87 8.8 Trained q 2 N , patient 5122, test episodes 1-4, varying parameter complexity M, 30-minute mesh, N = 8, actual q 2 = 0:3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.9 Average BrAC deconvolution error results, patient 5122, test episodes 1-9, varying numbers of training episodes on 20-minute mesh, N = 16, assumed spatially constant diusivity q 1 . . . . 88 8.10 Trained q 1 N , patient 5122, drinking episodes 1-9, estimated for 20-minute mesh, N = 16, measured TAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 8.11 Trained q 2 N , patient 5122, drinking episodes 1-9, estimated for 20-minute mesh, N = 16, measured TAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 8.12 Average delay system deconvolution error results, test episodes 1-4, varying levels of simulta- neous state and input signal discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 8.13 Trained a 0 N , varying levels of discretization, test episodes 1-4, delay model (actual =0:5). . 94 8.14 Trained ` N , varying levels of discretization, test episodes 1-4, delay model (actual = 5). . . . 94 v List of Figures 1.1 Giner WrisTAS TM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 8.1 Patient 5122 BrAC drinking episodes 1-4 estimated using linear splines on 30-minute mesh from simulated TAC (using q 1 (x) = 1, q 2 =:3) for varying resolution of state discretization, 1 =:0002, 2 =:00001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 8.2 Patient 5122 BrAC drinking episodes 1-4 estimated using linear splines on meshes of varying resolution from simulated TAC (using q 1 (x) = 1, q 2 = :3) for state discretization N = 8, 1 =:0002, 2 =:00001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 8.3 Patient 5122 BrAC drinking episodes 1-4 estimated using linear splines on 30-minute mesh from simulated TAC (using q 1 (x) = 3(x:5) 2 +:1, q 2 =:3) for state discretization N = 8, varying levels of assumed system parameter complexity, 1 =:0002, 2 =:00001; 3 =:003 . . 86 8.4 Trained [q M 1 ] N (x), test episodes 1-4, varying parameter complexityM, 30-minute mesh,N = 8, actual q 1 (x) = 3(x:5) 2 +:1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.5 Patient 5122 BrAC drinking episodes 1-9 estimated using linear splines on 20-minute mesh from measured TAC, constant q 1 (x), N = 16, 1 =:006, 2 =:0004. . . . . . . . . . . . . . . 89 8.6 Deconvolved input signals, test episodes 1-4, delay system, varying levels of resolution of state and input mesh, 1 =:003, 2 =:003. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 vi Abstract A theory for approximate deconvolution and parameter estimation in certain innite-dimensional linear dynamical systems with unbounded input and output is presented. The theory is exposed by frameworks for approximate parameter estimation and deconvolution in parabolic systems in discrete and continuous time with input and output on the boundary, and to delay systems with unbounded input and output, e.g. systems with delays in the input. Numerical results demonstrating the performance of the approximations are presented. Finally, the theory is applied to a data from transdermal alcohol model and sensor together with training data in the form of breath alcohol measurements to calibrate the sensor and approximately deconvolve blood alcohol content over time. vii Chapter 1 Introduction The idea of measuring blood alcohol concentration (BAC) by sampling transdermal alcohol concentration (TAC) dates back to the 1930s, when Nyman and Palmlov determined in [27] that after similar subjects ingested one identical drink each, the ethanol concentration in the subjects' perspiration was highly correlated with their ethanol concentration in the blood. After correcting for the water content of each sample, the sample correlation between BAC and TAC was found to be r = 0:97; see [10]. An accurate transdermal alcohol sensor would be preferable to a breath alcohol sensor for a few reasons, a chief reason being continuous blood alcohol monitoring, which could enable more nuanced and exible alcoholism recovery programs. A few companies have developed and continue to develop transdermal alcohol sensors. A few examples are the SCRAM CAM from Scram Systems, worn on the ankle; the Transdermal Alcohol Sensor/Recorder (WrisTAS) from Giner, Inc., worn on the wrist; and Proof TM , from Milo Sensors, Inc., who have recently developed an enzymatic electrochemical sensor which is reportedly cheaper, more accurate, and provides more granular data than previous TAC sensors. The sensor in Figure 1.1, the Giner WrisTAS TM 7, is made to measure transdermal alcohol for weeks at a time. It continuously measures transdermal alcohol by detecting local ethanol vapor concentration above the skin surface at 5-minute intervals. The threshold BAC detectable by the WrisTAS is estimated to be 10-20 mg/dL. Details on the development and performance of this device can be found in [39], [40], [37], and [38]. Transdermal alcohol sensors such as those listed above are less than perfect for inferring BAC for two chief reasons: rst, the complex, variable, subject-dependent paths that ethanol molecules take from the blood-rich dermis to the TAC sensor; and second, the extant technological diculties in designing and manufacturing a 1 Figure 1.1: Giner WrisTAS TM relatively low-cost, highly accurate TAC sensor. Due to the diculties of reliably inferring BAC from TAC via a transdermal alcohol sensor, most commercially available sensors, of which the aforementioned are but a few examples, have seen use mainly in abstinence programs{they can reliably produce results with low false positive rates. One goal of this thesis is to develop theoretical foundations and algorithms to provide accurate quantitative estimates of BAC from TAC. Of course, much work has been done in studying diusion properties of ethanol in breath, blood, and sweat. The chemistry of the relationship between the blood alcohol concentration and breath alcohol concentration (BrAC) is described by Henry's law, an algebraic relation which is well-understood and depends on one real-valued parameter known as a partition coecient. This number is known to vary slightly with many variables such as subject weight, age, sex, time elapsed since eating, etc. Governments decide on partition coecients with which to calibrate breathalyzers and prosecute based on these estimates. The evident sequence of boundary crossings from blood to skin surface is artery! dermis or sweat gland! epidermis or sweat duct! skin surface: The sweat duct penetrates through the epidermis to the skin's surface. Water is also lost through the skin via so-called insensible perspiration, that is, passive diusion through the skin. 2 The nal layer of skin, the stratum corneum, is at the wrist composed of approximately 13 layers of keratinous cells [24], where it is thinner than most other parts of the body. The stratum corneum is hygroscopic, meaning it retains water. Furthermore, its permeability properties are highly variable, for example, in thickness [11], [24]; with temperature [19], [42], [20], [1], [43]; and hygroscopicity increases exponentially with humidity, especially above 90%, holding temperature constant [34]. Ethanol has been experimentally shown to rapidly and completely equilibrate across the skin, likely due to its small size and miscibility in water [10]. However, it is possible that as the stratum corneum retains water, its swelling obstructs the sweat ducts and/or attenuates the transdermal alcohol signal coming from eccrine sweat. Needless to say, the diusive properties of the media in this problem are highly variable. The problem of determining blood alcohol concentration from transdermal alcohol concentration is known as a deconvolution problem. In particular, deconvolution problems arise upon detecting a signal that has passed through a lter, whence a reconstruction of the original signal is desired. Mathematically, a typical deconvolution problem is one of estimating u(s) in the equation y(t) = Z t 0 K(ts)u(s)ds; u(0) =u 0 ; given y and K. Deconvolution problems are common in the literature, being a subset of the important class of inverse problems, that is, determining the causal factors of a set of observations. Deconvolution problems are common most everywhere in the sciences, e.g. in seismiology, optics, radio astronomy, biology, etc. Perhaps the most elementary method for deconvolution in theory relates to Fourier series: the Fourier transform of a convolution is a product and vice versa. Thus a deconvolution in the time domain is a quotient in the frequency domain: Fu = Fy FK : Two-dimensional versions of this Fourier deconvolution are used, for example, in microscopy and astronomy to de-noise images. However, this method is not generally robust as-is. In absence of domain restrictions for FK, i.e. information about frequencies not aected by the lter, this simple division provides inaccurate measurements for those frequencies which are barely present in the kernel K, as can readiliy be seen from 3 the formula. Furthermore, if, as is often the case,FK is estimated from data and therefore noisy, the above deconvolution estimate will be inaccurate. There are workarounds for these problems, but they are often ad-hoc and depend on the system under investigation. In practice, the impulse response function K is often not known. Thus, in most deconvolution problems, there are two unknowns: K and u. Systems for which K is wholly unknown, that is, for which one only knows the observed output data y, are known in the literature as blind deconvolution problems. However, often one has some knowledge of the causal process transforming u to y and/or paired input and output samples u j and y j , j = 1;:::;n (i.e. training data) with which to infer K. Such problems are semi-blind deconvolution problems. For example, if we possess some such knowledge of the lter K represents, we may be able to model it up to a small number of parameters. Then K is a member of a parametric family with, say, parameter q2Q, written K(;q). A classical deconvolution problem for linear time-invariant systems is as follows: let (Q;d) be some metric space of admissible parameters, and consider a nite-dimensional vector space H,m;p2N, and suppose that for each q2Q, A(q)2L(H), B(q)2L(R m ;H), C(q)2L(H;R p ). Then consider the linear time-invariant dynamical system for t 0, q2Q dened by _ '(t) =A(q)'(t) +B(q)u(t); y(t) =C(q)'(t); '(0) =' 0 2H: (1.0.1) The solution to this system exists for t 0 and is given by the variation of constants formula y(t) =C(q)e A(q)t ' 0 + Z t 0 C(q)e A(q)(ts) B(q)u(s)ds; where e A(q)t denotes the matrix exponential of A(q). Thus the convolution kernel K takes the form K(t;q) =C(q)e A(q)t B(q). Our goal is to infer an unknown input signal u(t) from sampled output signal y(t), given some paired input/output training data. 4 In this thesis we will consider this problem in much greater generality than above. In particular, we will consider H to be a Hilbert space (i.e. innite-dimensional), we will allow A(q) to be an unbounded operator on H (i.e. only densely dened), we will allow B(q) to map into a space properly containing the Hilbert space H, and we will allow C(q) to be dened only on a distinguished proper subset of H. In other words, the input operator B(q) and the output operator C(q) will be unbounded operators on H. In order to solve our problem in this generality, we will make regularity assumptions. Even once we have exposed the theory allowing for solutions to this innite dimensional problem, there still remains the task of computing it. Because computers are generally unable to work with innite-dimensional systems, we will solve our deconvolution problems approximately. In particular, we will take a sequence of nite-dimensional subspaces H N H, consider a sequence of dynamical systems on H N related to (1.0.1), and prove that solutions to these nite-dimensional problems will converge, in some sense, to a solution of the innite-dimensional problem. We will also apply similar methods to obtain approximate semi-blind deconvolution and parameter estimation results for delay dynamical systems. Previous work on the theory of innite-dimensional linear systems with boundary input can be found in [12], [17], and [18]. Work on extending domains of unbounded output operators was conducted in [44] and [45]. Work on well-posed linear systems with unbounded input and unbounded output can be found in [9], [14], [13], and [16], and in the particular case of delay systems in [30] and [14]. Work on approximation of parabolic systems was conducted in [7] and [6], and on delay systems in [4], [5], [8], [21], [22], and [25]. Deconvolution of input signals in parabolic systems with application to transdermal alcohol deconvolution was conducted, for example, in [32]. The present thesis extends this research group's works, other examples of which include [15], [33], and [36]. In [32], the authors presented a discrete time transdermal alcohol transport model and provided a method for its approximate deconvolution. They performed their deconvolution in two separate stages. In the rst, the system parameter q was t to the parametric model using a single time series of known input/output data using a traditional t-to-data least-squares method. This \optimal" training parameter was then used to deconvolve future system output signals for which the input was unknown. The authors discovered that the \optimal" system parameters found in the rst stage were highly variable, even for a single subject-device 5 pair. Consequently, their method's estimates for desirable data such as peak BAC and time elapsed from rst alcohol detection to peak BAC were reported to have high variance. In this thesis we make some progress in mitigating these issues. We now outline the present work. In Chapter 2, we discuss the mathematical model for transport of ethanol through the epidermis and formulate blood alcohol signal deconvolution as an optimization problem. In Chapter 3, we present results from the mathematics literature that we will use in our solution. In Chapter 4, boundary control methods from [12] are adapted to parabolic systems to obtain satisfactory regularity results for deconvolution in innite-dimensional parabolic systems with unbounded input and output. In Chapter 5, we discretize the parabolic transdermal alcohol model in time and formulate approximation and convergence results for simultaneous deconvolution and parameter estimation, essentially proving consistency of a simultaneous parameter estimation-deconvolution analog of the methods in [32]. In Chapter 6, we turn our attention to delay systems with unbounded input and output, and adapt the usual delay system state-space method to include the input signal u as a component of the state. We then use this formulation to prove approximation and convergence results for simultaneous deconvolution and parameter estimation in delay systems with unbounded input and output subject to some regularity assumptions. In Chapter 7, we port the enlarged state space idea from the approximation methods of the delay systems in Chapter 6 to formulate and prove convergence results for approximate parameter estimation-deconvolution in continuous time parabolic systems with input and output on the boundary. In Chapter 8 we compile results of numerical experiments demonstrating our theorems in both the parabolic and delay cases and nally catalog the performance of our methods in deconvolving BrAC from actual eld TAC readings obtained from the Giner WrisTAS. 6 Chapter 2 Mathematical model for transdermal alcohol transport and detection 2.1 Derivation of PDE model We now introduce the model for transport of alcohol through the skin. Let L denote the thickness in cm of the skin between the dermal layer and skin surface, i.e. epidermis. Let '(t;x) denote the concentration in moles/cm 2 in the epidermal layer of the skin at depth x2 [0;L] cm at time t 0 seconds, and let u(t) denote the concentration of alcohol in the blood and blood-rich dermal layer of the skin (BAC) at time t 0. We model transport of ethanol through the skin as a deterministic diusion process @' @t (t;x) = @ @x D(x) @' @x ; where D(x), x2 [0;L], denotes the diusivity at depth x in units of cm 2 /s, and we assume that for some > 0, D(x) for all x2 [0;L]. The boundary conditions at the epidermis/transdermal alcohol content (TAC) sensor and dermis/epidermis interfaces are modeled by the Neumann boundary conditions D(0) @' @x (t; 0) ='(t; 0); t 0; D(L) @' @x (t;L) =u(t); t 0; 7 respectively, where the proportionality constants ; > 0 are measured in cm/s. We assume there is no alcohol in the epidermis at time t = 0: '(0;x) = 0; 0xL: The TAC sensor detects alcohol evaporated from the skin, which we model with a linear relationship y(t) = '(t; 0); where denotes the constant of proportionality in TAC units divided by moles/cm 2 . We therefore have the model @' @t (t;x) = @ @x D(x) @' @x (t;x) ; t 0; 0xL; D(0) @' @x (t; 0) ='(t; 0); t 0; D(L) @' @x (t;L) =u(t); t 0; '(0;x) = 0; 0xL; y(t) = '(t; 0); t 0: The model outlined above is determined by the parameters D(x);L;;; and . However, these parameters are not independent, nor uniquely identiable from input and output data. If y, u, and ' satisfy the above 8 outlined system, then the following transformed system with mostly dimensionless parameters and variables y, u, and ^ ' satisfy @ ^ ' @ ^ t ( ^ t; ^ x) = @ @^ x ^ D(^ x) @ ^ ' @^ x ( ^ t; ^ x) ; 0 ^ x 1; ^ t 0; ^ D(0) @ ^ ' @^ x ( ^ t; 0) = ^ '( ^ t; 0); ^ t 0; ^ D(1) @ ^ ' @^ x ( ^ t; 1) = ^ u( ^ t) = ^ u(t); ^ t 0; ^ '(0; ^ x) = 0; 0 ^ x 1; y(t) =y( ^ t) = ^ '( ^ t; 0); ^ t 0; where ^ x = x=L, ^ t = t=, ^ D(^ x) = D(L^ x)=L 2 , ^ = = , = L=, and ^ '( ^ t; ^ x) = '( ^ t;L^ x) for ^ x2 [0; 1], ^ t 0. Thus it is evidently sucient to t only the three parameters ^ D(x), ^ , and . However, for a given inputu(t), any change in the dilation factor for the diusion clock can be compensated for with appropriate changes in the other parameters so that the system produces the same output y(t). Thus, since we are not particularly interested in the values of the parameters themselves (we simply want a model that produces the observed output y from a given input u), we may without loss of generality set = 1 and ^ t =t. The model to be identied then takes the following form with unknown (dimensionless) parameters q 1 (x) and q 2 : @' @t (t;x) = @ @x q 1 (x) @' @x ; 0x 1; t 0; q 1 (0) @' @x (t; 0) ='(t; 0); t 0; q 1 (1) @' @x (t; 1) =q 2 u(t); t 0; y(t) ='(t; 0); t 0; '(0;x) = 0; 0x 1: (2.1.1) There are many variations on the above proposed system. For example, we could impose Dirichlet or Robin boundary conditions at either end of the domain. In previous work on calibration of transdermal alcohol sensors and blind deconvolution of TAC data, the skin was typically assumed to be homogeneous, in particular, that q 1 (x) =q 1 2R, e.g. in [32]. In generalizing previous work in [32] to a functional skin depth-dependent 9 system parameter q(x), we introduce more degrees of freedom to the model, allowing for the skin-alcohol diusivity to vary over the depth of the subject's epidermis. This also eectively allows for the system to model active transport with a certain restriction on the relationship between the advection and diusivity coecients. 2.2 Transdermal alcohol model as abstract parabolic system The transdermal alcohol system given above is an example of a distributed parameter system in the form of a PDE together with input and output on the boundary. We will now translate it into an abstract parabolic framework (see Chapter 4). First, let Q =f(q 1 (x);q 2 )gC 2 [0; 1;R + ]R + (with metric inherited from the norm) be a compact metric space of admissible parameters such that for all (q 1 (x);q 2 )2Q, 0<"q 1 (x); q 2 "; for some "> 0 and for all x2 [0; 1]. Because any compact metric space must be bounded, we may assume that if (q 1 (x);q 2 )2Q, q 1 2C 1 [0; 1; [";]] and q 2 2 [";] for some "<. Let V =H 1 [0; 1] with its ordinary norm, and let H =L 2 [0; 1], being the natural state space in which to formulate (2.1.1) above. Then because H is a Hilbert space, we may identify it with its dual space via the Riesz representation theorem to obtain a sequence of dense and continuous embeddings V ,!H ,!V ; where V denotes the dual space of V (see section 4.1). For each q 2 Q, dene a sesquilinear form a(q;;) :VV !R by a(q;'; ) :='(0) (0) + Z 1 0 q 1 (x)' 0 (x) 0 (x)dx: (2.2.1) 10 Now for each q2Q, dene a linear form b(q; ) :V !R by b(q; ) =q 2 (1) and a linear form c(q;') :V !R by c(q;') =c(') ='(0): Then we can write (2.1.1) in so-called weak form as h _ '; i +a(q;'; ) =b(q; )u; 2V; y =c('); 'j t=0 =' 0 : (2.2.2) The weak forms above can be used to dene linear operators which give rise to an alternative strong form of (2.1.1). In particular, a sesquilinear form on a(q;;) on VV denes a linear operator A(q) :V !V via a(q;'; ) =hA(q)'; i V ;V ; whereh;i V ;V denotes the duality pairing. Similarly, b(q;) andc(q;) dene linear operatorsB(q) :R!V and C(q)2V via b(q;')v =hB(q)v;'i V ;V ; c(q;') =C(q)'; 11 for v2R. Then the so-called strong form of (2.1.1) is then given by the inhomogeneous unbounded Cauchy problem _ '(t) =A(q)'(t) +B(q)u(t); t 0; y(t) =C(q)'(t); t 0; '(0) =' 0 : (2.2.3) 2.3 Approximate semi-blind deconvolution problem Our goal is, given TAC data, to infer BAC, so that continuous blood alcohol monitoring is possible with the TAC sensor. We will have access to some data with which to calibrate the TAC sensor/subject pair. We will be interested in simultaneously tting the model parameters q and inferring the input u(t) of (2.1.1) which produced the samples ~ y j , j = 1;:::;n. To do this, we formulate an optimization problem: minimize a cost functional J(q;u) over a compact set QU, where Q is a compact metric space of admissible parameters and U is a compact function space of admissible input signals. Our given data will be f(~ u (i) j ; ~ y (i) j ) n j=1 = (~ u (i) ; ~ y (i) ) : i = 1;:::g; where each tuple comprises a paired time series of in- put/output samples of u(t) and y(t), respectively; a time series of sampled outputs ~ y = (~ y 1 ;:::; ~ y n ) to be deconvolved; a choice of continuous regularization functionR :QU!R. Suppose that for each q2Q, :QU!R n is a continuous function simulating the transdermal alcohol model, i.e. the causal relationship between (q;u) and y in (2.1.1). Then, lettingkk n denote the Euclidean norm onR n , our cost functional J(q;u) has the t-to-data form J(q;u) = 1 X i=1 (q; ~ u (i) ) ~ y (i) 2 n +k(q;u) ~ yk 2 n +R(q;u): (2.3.1) 12 Chapter 3 Preliminary results and background In this section, we will present without proof some theorems from the literature necessary for our results. Throughout this thesis, when X and Y are vector spaces,L(X;Y ) will denote the set of bounded linear operators from X to Y . L p [a;b;S] denotes the usual L p spaces of equivalence classes of functions [a;b]!S; W k;p [a;b;S] denote equivalence classes of absolutely continuous functions whose derivatives exist as elements of W k1;p [a;b;S], with W 0;p =L p . 3.1 Functional analysis Theorem 3.1. (Banach-Steinhaus). Let X be a Banach space and let Y be a normed vector space. Suppose F is a set of bounded linear transformations from X to Y , and that for all x2X, sup A2F kAxk Y <1: Then sup A2F kAk L(X;Y ) <1: Proof. [31], Theorem III.9. Theorem 3.2. (Bounded linear extension). Suppose SX is a dense embedding of normed vector spaces, that Y is a complete, normed vector space, and that T2L(S;Y ). Then T can be uniquely extended to a bounded linear transformation T2L(X;Y ). 13 Proof. [31], Theorem I.7. Theorem 3.3. (Sobolev embedding). Suppose 2m>n, j 0, and that R n has nite volume. Then W j+m;2 ( )!C j b ( ) W j+m;2 ( )!W j;q ( )!L q ( ) are sequences of dense and continuous embeddings for all q2 [1;1], where C j b ( ) denotes the space of functions having bounded, continuous derivatives up to order j on . Proof. [2], Theorem 4.12. We will typically make use of the above for n = 1, = [0;T ]. Theorem 3.4. (Young's inequality). Let 1p;q;r1 satisfy 1 p + 1 q = 1 r : If k2L p (R) and f2L q (R), then the convolution kf of k and f is in L r (R) and kkfk r kkk p kfk q : Proof. [3], Prop. 1.3.2(a). Denition 3.5. C2L(X;Y ) is a compact operator if it takes each bounded set (equivalently, the unit ball) in X to a precompact set in Y , i.e. a set with compact closure. Example 3.6. Suppose C2L(X;R p ) for some p2N. Then C is a compact operator, because the image of the unit ball in X is bounded in R p . Lemma 3.7. Let X and Y be Banach spaces and let C2L(X;Y ) be a compact operator. Then if x N !x in X , Cx N !Cx in Y . Proof. [31], Theorem VI.11. 14 3.2 Linear semigroup theory Throughout this subsection, let X denote a Banach space unless stated otherwise. Denition 3.8. A collection T (t), t 0, of bounded linear operators on X is a C 0 semigroup if it satises 1. T (0) =I, 2. T (t +s) =T (t)T (s), and 3. lim t!0 +T (t)x =x for each x2X. Denition 3.9. Each C 0 semigroup T(t) : X! X, t 0 is uniquely determined by its densely dened innitesimal generator (A;D(A)), i.e. D(A) := x2X : lim t!0 + T (t)xx t exists. Ax := lim t!0 + T (t)xx t ; x2D(A): Throughout this paper, the notation (A;D(A)) refers to a (possibly) unbounded linear operator on some Banach space X which is dened on the dense subset D(A)X. Lemma 3.10. If (A;D(A)) is the innitesimal generator of aC 0 semigroup andx2D(A), thenT (t)x2D(A) and for all t 0, d dt T (t)x =T (t)Ax =AT (t)x: Proof. [16], Lemma II.1.3(ii). Example 3.11. SupposeX is a nite dimensional vector space andA :X!X is linear. ThenA is bounded, and it is the innitesimal generator of a C 0 semigroup T (t) on X given for t 0 by the matrix exponential T (t)x =e tA x := 1 X k=0 (tA) k k! x: 15 Example 3.12. For any function f dened on a subinterval [a,b] of the real line, with ;t2R, dene the history function f t () :=f(t +) when t +2 [a;b] and f t () = 0 otherwise. Consider the semigroup S(t) on L p [0;1) dened for t 0 by S(t)f =f t : It is easy to see that S(0) =I and S(t +s) =S(t)S(s). Moreover, for right continuous f, lim t!0 + S(t)f = lim t!0 + f t =f: Therefore S(t) is a C 0 semigroup on L p [0;1), called the left shift semigroup. Its innitesimal generator D is given by Df = lim t!0 + S(t)ff t = lim t!0 + f t f t = d + f dt ; i.e. it is the right derivative operator dened on the dense subset of right dierentiable functions in L p [0;1). Denition 3.13. Let (A;D(A)) be a (possibly unbounded) linear operator on X. The resolvent set of (A;D(A)) is the set (A)C comprising all 2C such that the resolvent operator R (A) := (IA) 1 exists as a bounded linear operator on X. Theorem 3.14. Suppose (A;D(A)) is the innitesimal generator of a C 0 semigroup T (t) on a Hilbert space X, and that 2(A). Then there exist M 1, !2R such that 1. kT (t)kMe !t . 2. (A)f2C : Re>!g andkR (A)k M Re! . 16 Proof. 1. [16], Theorem I.5.5 2. [16], Theorem II.1.10. Denition 3.15. When the hypotheses of Theorem 3.14 hold for (A;D(A)) on X, we write A2G(M;!) on X. Theorem 3.16. (Feller-Miyadera-Phillips). A2G(M;!) on a Banach space X if and only if for each Re >!, we have 2(A) and kR (A) n k X M (Re!) n for all n2N. Proof. [16], Theorem II.3.8. Corollary 3.17. A2G(1;!) on a Banach space X if and only if for each Re>!, we have 2(A) and kR (A)k X 1 Re! : Denition 3.18. A densely dened operator (A;D(A)) on a Hilbert space is said to be dissipative if RehAx;xi 0 for all v2D(A). Theorem 3.19. (Lumer-Phillips). Let (A;D(A)) be a densely dened and dissipative linear operator on a Hilbert space. The following are equivalent. 1. The closure (A;D(A)) of (A;D(A)) in X is in G(1; 0), i.e. it generates a contraction semigroup. 2. Range(IA) is dense in X for some (equiv. all) 2C with Re > 0. 3. (A ;D(A )) is dissipative. Proof. [16], Theorem II.3.15 and Corollary II.3.17. Note that if X is a Hilbert space, then A2 G(1;!) on X if for all x2 X,hAx;xi < !kxk 2 X , since this is equivalent to 0 <hx; (!IA)xi =h(!IA) x;xi, so that A!I satises the hypotheses of the Lumer-Phillips theorem. 17 Denition 3.20. A subspaceD of the domain D(A) of a linear operator A :D(A)X!X is called a core for A ifD is dense in D(A) under the graph norm of A. Proposition 3.21. If (A;D(A)) is the innitesimal generator of aC 0 semigroupT (t) onX, then a sucient condition for a subspaceD of D(A) to be a core for A is thatD is dense in X and invariant under T (t). Proof. [16], Proposition I.1.7. Theorem 3.22. (Trotter-Kato). Suppose A N 2 G(M;!) generate the C 0 semigroups T N (t) on X for N = 1; 2;:::, and that A is the innitesimal generator of the C 0 semigroup T (t) on X. Assume that either 1. there exists a coreD for A such that for each x2D there exists x N 2D(A N ) such that x N !x and A N x N !Ax in X; or 2. R (A N )x!R (A)x for all x2X and some 2C with Re >!. Then for each x2X, T N (t)x!T (t)x uniformly for t in compact intervals. Proof. [16], Theorem III.4.8. Following the terminology of the well-known Lax equivalence theorem, the hypothesis A N 2 G(M;!) is often referred to as the stability hypothesis of the Trotter-Kato theorem; either latter hypothesis as consistency. We can extend the argument of a semigroup from the positive real axis to the complex plane in order to obtain various regularity properties. The most important such example of this is the analytic semigroup. Denition 3.23. Let 2 (0;=2] and dene :=f2C :j argj< 2 +gnf0g: 18 A family of bounded operators T (z)2L(X), z2 [f0g is called an analytic semigroup (of angle ) if 1. T (0) =I 2. T (z 1 +z 2 ) =T (z 1 )T (z 2 ) for all z 1 ;z 2 2 . 3. The map z7!T (z) is analytic in . 4. lim 03z!0 T (z)x =x for all x2X, 0 2 (0;). Theorem 3.24. Let (A;D(A)) generate an analytic semigroup T (t), t 0, on X. Then T (t) :X!D(A ) for every t> 0 and 0. Proof. [28], Theorem 6.13(a). 19 Chapter 4 The estimation problem for a class of abstract parabolic systems with input and output on the boundary 4.1 Gelfand triple Let V ,! H be a dense and continuous linear embedding of Hilbert spaces. By the Riesz representation theorem, the Riesz mapH!H which sends an element to its dual is an isometric isomorphism. The adjoint of the dense and continuous linear inclusion V ,!H is a linear map H ,!V which embeds H densely and continuously in V . Identifying H = H , we obtain a sequence of dense and continuous embeddings V ,!H ,!V called a Gelfand triple. Throughout this thesis, when referring to a Gelfand triple, letjj denote norm in H, letkk denote norm in V , and letkk denote norm in V . Because the inclusion maps above are linear and bounded, there exist constants c;c 0 > 0 such that k'k c 0 j'j; j jck k; for each '2H, 2V . 20 4.2 Parabolic systems Suppose that V ,!H ,!V is a Gelfand triple of Hilbert spaces and let Q be a metric space with metric d. Suppose for each q2Q we have a sesquilinear form a(q;;) :VV !R (orC) and that there exist ;;> 0, 0 2R independent of q2Q such that for each q2Q, a(q;'; )k'kk k '; 2V (4.2.1) Rea(q;';') + 0 j'j 2 k'k 2 '2V; (4.2.2) ja(p;'; )a(q;'; )jd(p;q)k'kk k '; 2V; p;q2Q: (4.2.3) We will refer to sesquilinear forms satisfying the above conditions as parabolic forms. Proposition 4.1. Suppose for each q2Q, a(q;;) is a parabolic form. Then there exists a bounded linear operator A(q)2L(V;V ) uniquely determined by a(q;'; ) =hA(q)'; i V ;V which is the innitesimal generator of an analytic semigroup T (t;q) on V . Furthermore, if we let D(A(q)) H :=f 2V : A(q) 2Hg; then (A(q);D(A(q)) H ) is the innitesimal generator of an analytic semigroup H, also denoted T (t;q). Proof. [41], Lemma 3.6.1. Proposition 4.2. If we let D(A(q)) V =f 2V : A(q) 2Vg, then (A(q);D(A(q)) V ) is the innitesimal generator of an analytic semigroup on V . Proof. Shown in the proof of Theorem 2.3 of [7]. 21 Proposition 4.3. The operator A(q) arising from a parabolic form is dissipative. Proof. We may assume without loss of generality that 0 c , wherec> 0 is a constant such thatjjckk. By (4.2.2), RehA(q)';'i =a(q;';')k'k 2 0 j'j 2 c 0 j'j 2 0: Theorem 4.4. Suppose A is the linear operator arising from a parabolic form. Then there exists a constant > 0 such that for every t> 0, f2H or V , kT (t)fkt 1 kfk kAT (t)fkt 3=2 jfj jAT (t)fjt 3=2 kfk : Proof. [41], Lemma 3.6.2. 4.3 Boundary input The framework presented here is inspired by [12]. LetQ be a compact metric space and assumeV ,!H ,!V is a Gelfand triple. Now assume that W ,!V is a sequence of dense and continuous injections of Hilbert spaces so that W is dense in V and in H. Let m;p2N, and let Y =R p . Assume that for each q2Q, we have bounded operators (q)2L(W;H), (q)2L(W;R m ) which is assumed surjective, and C(q)2L(V;Y ), all of which we assume to be continuous in q2Q in their respective operator norms. Further assume that ker (q)W is dense in V for each q2Q. Consider the boundary input system for t2 [0;T ] given by _ '(t) = (q)'(t); '(0) =' 0 ; (q)'(t) =u(t); (4.3.1) 22 with output equation y(t) =C(q)'(t): (4.3.2) We will use the following classical result. Proposition 4.5. For each q 2 Q, the operator (q)j ker (q) : ker (q) H ! H is the innitesimal generator of a C 0 semigroup on H which corresponds to the solution to (4.3.1) with u(t) 0. Proof. [29]. Denition 4.6. Let q2 Q, u2 C[0;T;R m ], ' 0 2 W , (q)' 0 = u(0). A function ' = '(;q;u;' 0 )2 C[0;T;W]\C 1 [0;T;H] satisfying (4.3.1) for each t2 [0;T] is said to be a strong (or classical) solution of (4.3.1). 4.4 Boundary input for parabolic systems Suppose we have a parabolic form a(q;;) on VV , with the inclusion map V ,!H dense and continuous so V ,!H ,!V is a Gelfand triple, so we obtain a linear operator A(q)2L(V;V ). Now assume for each q2Q, ((q); ker (q)) = (A(q);D(A(q)) H ); (4.4.1) where two linear operators are dened to be equal if their domains are equal and the operators agree there. For each q2 Q, let + (q)2L(R m ;W) be any right inverse of the surjection (q) and dene B(q)2 L(R m ;V ) by B(q) = ((q)A(q)) + (q); (4.4.2) where we regard A(q)2L(V;V ). Note that B(q) is linear and well-dened regardless of the choice of + (q). Indeed, for two choices + 1 (q) and + 2 (q), and any q2Q, Range( + 1 (q) + 2 (q)) ker((q)). Since A(q) = (q) on ker((q)), B 1 (q) =B 2 (q). 23 Furthermore, for eachq2Q,v2R m ,kB(q)vk is nite. To see this, note that because (q) was assumed to be surjective, there exists a constant K 0 > 0 such that for every v2R m , there exists x2W such that (q)x =v; kxk W K 0 jvj m : Because A(q)2L(V;V ), we also have A(q)2L(W;V ) by continuity of the embedding W ,!V , since then for some a;a 0 > 0, kA(q)xk akxka 0 kxk W : Note that a and a 0 can be made independent of q in the case that Q is compact, since then a(q) and a 0 (q) attain their suprema at some q2Q. Therefore if v2R m , then there exists x2W such that kB(q)vk k(q)A(q)k L(W;V ) kxk W K 0 k(q)A(q)k L(W;V ) jvj m ; so B(q)2L(R m ;V ). Assume that '(t) is a (strong) solution of (4.3.1). Then because B(q)(q) = (q)A(q), we have that _ '(t) = (q)'(t) = (B(q)(q) +A(q))'(t) =A(q)'(t) +B(q)u(t): Because the input operator B(q) is unbounded with respect to the norm on H, and because the output operatorC(q) is as well, we say that'(t) is a solution to the following inhomogeneous Cauchy problem in V with unbounded output dened for t 0 by _ '(t) =A(q)'(t) +B(q)u(t); y(t) =C(q)'(t); '(0) =' 0 : (4.4.3) 24 Denition 4.7. Let A(q)2L(V;V ) be dened by a parabolic form and B(q)2L(R m ;V ) dened as in (4.4.2). Let ' 0 2V and u2L 2 [0;T ;R m ]. Then '2C[0;T ;V ]\H 1 [0;T ;V ] is a weak solution of (4.3.1) if it satises (4.4.3) for almost all t2 [0;T ]. Denition 4.8. Suppose we are given an inhomogeneous Cauchy problem on a Gelfand triple, and assume that A(q) is the innitesimal generator of a C 0 semigroup T (t;q) on V , H, and V . B(q) will be said to be psuedo p-admissible for A(q) if for some t 1 > 0, p2 [1;1], q2Q, the range of the linear map B(t;q) :W 1;p [0;t 1 ;R m ]!V u7! Z t1 0 T (t 1 s;q)B(q)u(s)ds is contained in V . Lemma 4.9. Assume that in (4.3.1), for each q2Q, ((q); ker (q)) = (A(q);D(A(q) H ). Also assume that '2H and v2R m satisfy A(q)' +B(q)v2H for B(q) dened by (4.4.2). Then '2W , (q)' =v, and (q)' =A(q)' +B(q)v. Furthermore, there exists K(q)> 0 such that k'k W K(q) (j'j +jvj m +jA(q)' +B(q)vj): Proof. [12], Lemma 4.3. Proposition 4.10. Let q 2 Q. Suppose B(q) is pseudo 2-admissible for A(q). Then for all ' 0 2 W , u2H 2 [0;T;R m ] with (q)' 0 =u(0), there exists a unique solution '2C[0;T;W]\C 1 [0;T;H] of (4.3.1) depending continuously on ' 0 and u. 25 Proof. Suppose ' 0 2W and u2C 2 [0;T;R m ] satisfying (q)' 0 =u(0). Then '2C[0;T;V ]\C 1 [0;T;V ] and for t2 [0;T ], _ '(t) =A(q)'(t) +B(q)u(t) =T (t;q)[A(q)' 0 +B(q)u(0)] + Z t 0 T (ts;q)B(q) _ u(s)ds =T (t;q)(q)' 0 + Z t 0 T (ts;q)B(q) _ u(s)ds: The rst term is in H. Using pseudo 2-admissibility and the fact that u2C 2 [0;T;R m ], we see the second term is in V . Therefore we have _ '2C[0;T ;H]. Furthermore sup t2[0;T] j _ '(t)j sup t2[0;T] jT (t;q)jk(q)k L(W;H) k' 0 k W +c Z t 0 T (ts;q)B(q) _ u(s)ds sup t2[0;T] M(q)e !(q)t k(q)k L(W;H) k' 0 k W +cM B (q)k _ uk H 1 [0;T;R m ] ; where we have usedjjckk. Applying Lemma 4.9 to the term A(q)'(t) +B(q)u(t) = _ '(t)2H, we see that '2C[0;T ;W ]\C 1 [0;T ;H] satises (4.3.1). Because '(t) =T (t;q)' 0 + Z t 0 T (ts;q)B(q)u(s)ds; it is the unique solution of (4.3.1). Note sup t2[0;T] j'(t)j sup t2[0;T] jT (t;q)jkIdk L(W;H) k' 0 k W +cM B (q)kuk H 1 [0;T;R m ] and, using (q)' 0 =u(0), sup t2[0;T] ju(t)j m k(q)k L(W;R m ) k' 0 k W + p Tk _ uk L2[0;T;R m ] : Usingkk L2 kk H 1 , we obtain continuous dependence on ' 0 and u. 26 The previous propositions allow us to replace the boundary input model (4.3.1)-(4.3.2) with the inhomo- geneous Cauchy problem (4.4.3); '(t) is a solution of (4.3.1) if and only if it is also a solution of (4.4.3). If '(t)2V for t2 [0;T ] as in the previous proposition, then we may apply C(q) to '(t) to obtain our model y(t;q;u;' 0 ) =C(q)'(t;q;u;' 0 ) =C(q)T (t;q)' 0 +C(q) Z t 0 T (ts;q)B(q)u(s)ds: Later we will examine continuity properties of (q;u;' 0 )7!y(t;q;u;' 0 ). If 0<t 1 <t 2 <<t n =T are given sampling times, Y =R p , dene :QUW!Y n (q;u;' 0 ) := (y(t j ;q;u;' 0 )) n j=1 : We are interested in tting this abstract model to data by minimizing the cost functional J(q;u) = 1 X i=1 (q; ~ u (i) ) ~ y (i) 2 n +k(q;u) ~ yk 2 n +R(q;u) on QU, whereR is a continuous regularization term. Later we will see that (and J) are continuous and therefore amenable to optimization. 4.5 Continuous dependence on system parameter Because we are interested in simultaneous parameter estimation and deconvolution, we'll need our model to be continuous in its arguments, in particular, with respect to the parameter q. This will require the following assumptions. (A1) For each v2R m , q7! + (q)v is a continuous function Q!W . (A2) For each '2V , q7!C(q)' is a continuous function Q!R p . 27 Assumption (A1) (resp. (A2)) and compactness of Q imply that for all v2R m , sup q2Q k + (q)vk W (resp. for all '2V , sup q2Q jC(q)'j<1). Two applications of the Banach-Steinhaus theorem 3.1 then provide that sup q2Q + (q) L(R m ;W) =G<1; sup q2Q kC(q)k L(V;Y ) = <1: 4.6 Transdermal alcohol model as abstract parabolic system with boundary input and output We now apply the boundary input framework we have developed to the transdermal alcohol model. Recall from that Chapter 2 that our metric space of admissible parameters Q for the transdermal alcohol model was assumed to be a compact subset of C 1 [0; 1; [";]] [";] for some "; > 0. We have m = p = 1. Let H =L 2 [0; 1] and let V =H 1 [0; 1]H with its usual inner product h 1 ; 2 i V = Z 1 0 1 (x) 2 (x)dx + Z 1 0 0 1 (x) 0 2 (x)dx: Then V ,!H is dense and continuous by the Sobolev embedding theorem 3.3. 4.6.1 Sesquilinear form We will now show that the sesquilinear forma(q;'; ) given in (2.2.1) associated with the transdermal alcohol model is bounded, coercive, and depends continuously on the parameter q, i.e. it satises (4.2.1)-(4.2.3). Let '; 2V . We have a(q;'; ) ='(0) (0) + Z 1 0 q 1 (x)' 0 (x) 0 (x)dx: (4.6.1) Using the triangle and Cauchy-Schwarz inequalities, we obtain the following bound for the latter inner product term: jhq 1 ' 0 ; 0 i 2 j = Z 1 0 q 1 (x)' 0 (x) 0 (x)dx kq 1 k 1 Z 1 0 j' 0 (x)jj 0 (x)jdxkq 1 k 1 j' 0 jj 0 jkq 1 k 1 k'kk k: 28 Consider then '(0) (0). By the Sobolev embedding theorem 3.3, the identity map H 1 [0; 1]!L 1 [0; 1] is bounded. Thus, again using Cauchy-Schwarz, there exists a constant k such that j'(0) (0)j sup x2[0;1] j'(x) (x)j sup x2[0;1] j'(x)j sup x2[0;1] j (x)jk 2 k'kk k: Combining these estimates, we obtain that ja(q;'; )j (kq 1 k 1 +k 2 )k'kk k ( +k 2 )k'kk k: Therefore (4.2.1) holds. Now, to show coercivity, (4.2.2). Let 2V ,q = (q 1 (x);q 2 )2Q. Then becauseq 1 (x)2 [";] forx2 [0; 1], we havekq 1 k 1 ". Therefore for any 0 > 0, a(q; ; ) + 0 j j 2 = (0) 2 + Z 1 0 q 1 (x) 0 (x) 2 dx + 0 j j 2 "j 0 j 2 + 0 j j 2 min("; 0 )k k 2 : Finally, to show (4.2.3), let p;q2Q. ja(p;'; )a(q;'; )j = Z 1 0 [p 1 (x)q 1 (x)]' 0 (x) 0 (x)dx kp 1 q 1 k 1 Z 1 0 ' 0 (x) 0 (x)dx d(p;q)k'kk k: Therefore by Propositions 4.1 and 4.2, the linear operator (A(q);D(A(q)) H ) corresponding to the form a(q;;) is the innitesimal generator of an analytic semigroup T (t;q) onH, for anyH2fV;H;V g. Proposition 4.11. Consider the sets ~ D =f 2H 2 [0; 1] : q 1 (0) 0 (0) (0) = 0; q 1 (1) 0 (1) = 0gV D =D(A(q)) H =f 2V : A(q) 2Hg 29 and let D denote the derivative operator in this setting. Let ~ A(q) =Dq 1 D on ~ D. Then for each q2Q, ( ~ A(q); ~ D) = (A(q);D). Proof. Let '2 ~ D, 2V . Then, using integration by parts and the boundary conditions on ~ D, hDq 1 D'; i =(q 1 D') j 1 0 +hq 1 D';D i ='(0) (0) + Z 1 0 q 1 (x)' 0 (x) 0 (x)dx =hA(q)'; i: Since 2V was arbitrary, the two operators agree on ~ D, and because q 1 (x)2C 1 [0; 1], '2H 2 [0; 1] ~ D, we have A(q)'2L 2 [0; 1] =H, so ~ DD. Now suppose '2D, so that A(q)'2H. Then letting 2V , =A(q)', we have that h; i = Z 1 0 (x) (x)dx ='(0) (0) + Z 1 0 q 1 (x)' 0 (x) 0 (x)dx: Using integration by parts on the left hand side, we obtain (x) Z 1 x (t)dt 1 0 + Z 1 0 0 (x) Z 1 x (t)dtdx ='(0) (0) + Z 1 0 q 1 (x)' 0 (x) 0 (x)dx: Since the rst term ishA(q)'; 1i (0) ='(0) (0), we have Z 1 0 0 (x) Z 1 x (t)dtdx = Z 1 0 q 1 (x)' 0 (x) 0 (x)dx: Because 2H 1 [0; 1] and H 1 [0; 1]L 2 [0; 1] is dense by the Sobolev embedding theorem is dense in L 2 [0; 1], we have the equality Z 1 x (t)dt =q 1 (x)' 0 (x) 30 in L 2 [0; 1]. q 1 (1)' 0 (1) = 0 is then evident. Furthermore, this shows q 1 (x)' 0 (x)2 H 1 [0; 1], and because q 1 2C 1 [0; 1], we actually have '2H 2 [0; 1] and =Dq 1 D': Then, for any 2V , Z 1 0 Dq 1 (x)D'(x) (x)dx = Z 1 0 (x) (x)dx ='(0) (0) + Z 1 0 q 1 (x)' 0 (x) 0 (x)dx q 1 (x)D'(x) (x)j 1 0 + Z 1 0 q 1 (x)D'(x)D (x)dx ='(0) (0) + Z 1 0 q 1 (x)D'(x)D (x)dx q 1 (0)(D')(0) (0) ='(0) (0) (q 1 (0)' 0 (0)'(0)) (0) = 0; which implies '2 ~ D. ThereforeD = ~ D and ( ~ A(q); ~ D) = (A(q);D). 4.6.2 Input and output operators for transdermal alcohol model We will now investigate the operators B(q) and C(q) induced by the linear forms in the weak form of (2.1.1). First, let W =f 2H 2 (0; 1) : q 1 (0) 0 (0) (0) = 0g: W ,!V is dense and continuous by the Sobolev embedding theorem 3.3 with n = 1. Dene (q)2L(W;H) by (q) = d dx q 1 (x) d dx : For each q = (q 1 (x);q 2 )2Q, dene the boundary operator (q)2L(W;R) by (q)' = q 1 (1) q 2 0 (1): 31 Then (q) is clearly surjective for each q2Q and by Proposition 4.11 ker((q)) =f 2H 2 (0; 1) : q 1 (0) 0 (0) (0) = 0; 0 (1) = 0g =D =D(A(q)) H ; and is dense in V , by the Sobolev embedding theorem 3.3. For q2Q dene ( + (q)v)(x) =q 2 1 + Z x 0 dt q 1 (t) v for v2 R, x2 [0; 1]. Then + (q)2L(R;W), (q) + (q) = 1. Note also that the range of + (q) is in ker((q))V for some 2R and for each q2Q. We dene B(q) as in (4.4.2) and let C(q) =C = 0 2 L(V;R). The transdermal alcohol model's solution is given for t 0, q2Q, u2H 1 [0;T;R m ], ' 0 2W such that (q)' 0 =u(0), by y(t;q;u;' 0 ) =C(q)T (t;q)' 0 +C(q) Z t 0 T (ts;q)B(q)u(s)ds; with B(q) dened as in (4.4.2) and the semigroup T (t;q) generated by the linear operator A(q) associated to the parabolic form (2.1.1). 32 Chapter 5 Approximate semi-blind deconvolution in discrete time parabolic systems with input and output on the boundary Having formulated a class of abstract parabolic systems with unbounded input and output, we proceed with an approximation theory for such systems. However, we make the simplifying assumption that the input signal u is zero-order hold, i.e. piecewise constant, which serves to convert the system discussed in the previous chapter into a dynamical system with discrete timesteps. 5.1 Abstract discrete time parabolic boundary input system Take the abstract parabolic boundary input model (4.3.1) of the previous section. In particular, let Q be a compact metric space with metric d, and let V ,!H ,!V be a Gelfand triple withk'k c 0 j'j when '2H,j'jck'k when '2V . Also assume W ,!V is a dense embedding of Hilbert spaces. Further, we have for each q2Q, (q)2L(W;R m ), ker (q)V dense, + (q)2L(R m ;W ) an arbitrary but xed right inverse of (q), (q)2L(W;H), a(q;;) : VV ! R a parabolic form which denes a linear map A(q)2L(V;V ). We assume A(q) 2 H if and only if 2 ker (q), and (q) = A(q) for 2 ker (q). We also retain the continuity assumptions (A1) For each x2R m , q7! + (q)x is a continuous function Q!W . (A2) For each v2V , q7!C(q)v is a continuous function Q!R p . Throughout this section we regard A(q)2L(V;V ) and make the assumption that for each q2 Q, Range + (q) ker (q). 33 We formulate the discrete time analog of the inhomogeneous Cauchy problem with unbounded input and output given by (2.2.3). Fix > 0 and suppose each function u(t)2U is zero-order hold with timestep , so U is isomorphic to a subset of (R m ) n , i.e. u(t) =u i =u(i) for t2 [i; (i + 1)). Then let ' i ='(i), i = 0; 1; 2:::;n, and let i (t) ='(t) + (q)u i for t2 [i; (i + 1)). Then _ i (t) = _ '(t) =A(q)'(t) +B(q)u i =A(q) i (t) + + (q)u i +B(q)u i =A(q) i (t) + (A(q) +B(q)(q)) + (q)u i =A(q) i (t) + A(q) + ((q)A(q)) + (q)(q) + (q)u i =A(q) i (t) since we assumed (q) + (q) = 0 for each q2Q. Thus i satises the initial value problem _ i (t) =A(q) i (t); t2 [i; (i + 1)) i (i) =' i + (q)u i ; i = 0;:::;n 1: Therefore ' i+1 = b A(q)' i + b B(q)u i where b A(q) =T (;q)2L(V ), and using integration by parts and substituting for B(q) using (4.4.2), b B(q) = Z 0 T (s;q)B(q)ds = (I b A(q)) + (q)2L(R m ;V ): As in the continuous case, b B(q) is well-dened and does not depend on the choice of + (q). Because V is invariant under b A(q) by Lemma 3.10 and + (q) maps into WV , and since u is constant on each [i; (i + 1)), we actually have b B(q)2L(R m ;V ). If we make the assumption that ' 0 2 V (resp. ' 0 2H), it follows that we can regard b A(q)2L(V ) (resp. inL(H)) and therefore inductively that ' i 2V (resp. H) for all i = 0;:::;n. In fact, even if ' 0 2 V , then due to Theorem 3.24, we have b A(q)' 0 2 V . Therefore even in this case ' i 2V for i = 1;:::;n. 34 The discrete time model is therefore given for (q;u;')2QUV by (q;u;' 0 ) = C(q) b A(q) j ' 0 + j1 X k=0 C(q) b A(q) j1k b B(q)u k !n j=1 = C(q) b A(q) j ' 0 + j1 X k=0 C(q) b A(q) j1k (I b A(q)) + (q)u k !n j=1 2Y n : 5.2 Well-posedness of discrete time parabolic boundary input model Let Y =R p . We wish to show that the above discrete time model is a continuous map from QUV to Y n . We'll need the following lemmas. Lemma 5.1. Suppose for eachq2Q, Q compact, A(q) is dened by a parabolic form a(q;;), so it generates an analytic semigroup on eachH2fV;H;V g. Then the map q7!T(t;q)' is continuous for each '2H, uniformly in t for t in bounded intervals. Proof. This follows by taking H N =V , P N =I in [7], Theorem 2.3. Lemma 5.2. Under the hypotheses of Lemma 5.1, sup t2[0;T] q2Q kT (t;q)k H <1: Equivalently, theL(H)-valued function q7!T (t;q) is continuous. Proof. Because [0;T ]Q is compact and the map (t;q)7!T (t;q)' is continuous for all'2H,kT (t;q)'k L(H) attains its supremum on this set for each '2H. An application of the Banach-Steinhaus Theorem 3.1 proves the result. Corollary 5.3. IfH =V , H, or V , then for any > 0, sup q2Q b A(q) L(H) <1: 35 Lemma 5.4. LetH2fV;H;V g and assume (A1). Then for each u2R m , q7! b B(q)u is a continuous map from Q toH. Proof. Let q2Q, k = maxf1;c;c 0 ;cc 0 g. Then for any q2Q, 1 k [ b B(q) b B(q)]u H = 1 k [(I b A(q) + (q) (I b A(q)) + (q)]u H [(I b A(q) + (q) (I b A(q)) + (q)]u V [(I b A(q)) + (q) (I b A(q)) + (q)]u V + [(I b A(q) + (q) (I b A(q)) + (q)]u V = [ b A(q) b A(q)] + (q)u V + (I b A(q))[ + (q) + (q)]u V [ b A(q) b A(q)] + (q)u V + 1 + b A(q) L(V ) [ + (q) + (q)]u V By Lemma 5.1, the rst term can be made arbitrarily small by letting q!q. By Lemma 5.2 and assumption (A1), the second of these terms can be made arbitrarily small by taking q!q. Corollary 5.5. IfH denotes any of V , H, or V , b B(q) L(R m ;H) is uniformly bounded in q. Proof. Let u2R m ; the continuous map in the lemma attains its supremum at some q u 2Q by compactness of Q. Therefore sup q2Q b B(q)u H <1 for each u2R m , so by the Banach-Steinhaus theorem sup q2Q b B(q) L(R m ;H) <1: Now for each q2Q, let the nn matrix of operators b L(q) : (R m ) n !Y n be dened by ( b L(q)) ij =C(q) b A(q) j1i b B(q) 36 for j >i and b L(q) ij = 0 for ji. Thus b L(q) is a strictly upper triangular matrix of operators with (q;u;' 0 ) =C(q) b A(q) j ' 0 + b L(q)u: The following theorem holds true for in general, but because in our case ' 0 = 0, we present a simpler result. Theorem 5.6. (;; 0) is continuous. Proof. Let (q;u)2QU be given. Letkk n denote the norm on Y n . We bound b L(q)u b L(q)u n b L(q)u b L(q)u n + b L(q)u b L(q)u n = 1 + 2 : We have 2 1 = b L(q)u b L(q)u 2 n = n X j=1 j1 X i=0 C(q) b A(q) j1i b B(q)(u i u i ) !2 n X j=1 j1 X i=0 C(q) b A(q) j1i b B(q)(u i u i ) !2 We bound the absolute value terms for each j: j1 X i=0 C(q) b A(q) j1i b B(q)[u i u i ] j1 X i=0 C(q) b A(q) j1i b B(q)[u i u i ] Applying (A2),jC(q) j k k. Thus it suces to show the following can be made arbitrarily small: j1 X i=0 b A(q) j1i b B(q)[u i u i ] L(V ) j1 X i=0 b A(q) j1i L(V ) b B(q) L(U;V ) ku i u i k U : Taking suprema and passing to the corollaries above, b A(q) j1i L(V ) and b B(q) L(V ) are uniformly bounded in q and j. Taking u!u suces to then show 1 ! 0 as u!u. 37 Now we examine 2 = b L(q)u b L(q)u n . To make it arbitrarily small it suces to show that q7! b L(q)u is continuous for all u2 (R m ) n . To show this, it suces to show that for every x2 R m , q7! b L(q) ij x is continuous for all i;j = 1;:::;n. Letting x2R m , q!q in Q, jC(q) b A(q) j1i b B(q)xC(q) b A(q) j1i b B(q)xjjC(q) b A(q) j1i ( b B(q) b B(q))xj +jC(q)( b A(q) j1i b A(q) j1i b B(q)xj +j(C(q)C(q)) b A(q) j1i b B(q)xj b A(q) j1i ( b B(q) b B(q))x + [ b A(q) j1i b A(q) j1i ] b B(q)x +j(C(q)C(q)) b A(q) j1i b B(q)xj: Applying Lemma 5.4, Lemma 5.1, and assumption (A2), respectively, to these three terms proves the theorem, taking q!q. Theorem 5.7. Let J(q;u) = 1 X i=1 (q; ~ u (i) ) ~ y (i) 2 n +k (q;u) ~ yk 2 n +R(q;u); (5.2.1) whereR is a regularization term continuous in (q;u). Then J :QU!R is continuous. Proof. Suppose (q N ;u N )! (q;u). Then jJ(q N ;u N )J(q;u)j 1 X i=1 b L(q N )~ u (i) ~ y (i) 2 n b L(q)~ u (i) ~ y (i) 2 n + b L(q N )u N ~ y 2 n b L(q)u ~ y 2 n + R(q N ;u N )R(q;u) After factoring these dierences of squares, we see the rst and second terms go to 0 as N!1 by Theorem 5.6. The third vanishes by assumed continuity ofR. 38 Because Q and U were assumed to be compact, QU is compact. Thus J attains its global minimum (q;u) on QU and our global optimization problem has a solution. 5.3 Finite dimensional approximation of parabolic boundary input model We now consider a sequence of approximating semi-blind deconvolution problems, the solutions of which converge in some sense to a global minimizer of (5.2.1). Assume there exists a sequence of nite dimensional spaces H N V for N = 1; 2;::: such that For all '2V;9' N 2H N such that ' N ' ! 0 as N!1: (5.3.1) Now suppose that for any convergent sequence q M !q in Q, A N (q M ) is the sequence of linear operators on the spaces H N obtained from the restriction of our parabolic form a(q M ;;) to H N H N . Each A N (q M ) generates an analytic semigroup T N (t;q M ) given by the matrix exponential. Let P N : H! H N be the standard Hilbert space projections. We will use the following application of the Trotter-Kato theorem. Theorem 5.8. Suppose that the sesquilinear form a(q;;) :VV !C satises (4.2.1), (4.2.2), and (4.2.3), so that it generates a semigroup T (t;q) on each of V;H; V , for each q2Q. Suppose also that H N V are nite dimensional for N = 1; 2;:::, with Hilbert space projections P N :H!H N ; and for each 2V , there exists a sequence N 2H N such that N ! in V . Let T N (t;q) denote the analytic semigroup on H N with innitesimal generator A N (q) for each q2Q. Then for each convergent sequence q N !q in Q, '2H, T N (t;q N )P N '!T (t;q)' in V , uniformly for t in compact intervals. Proof. [7], Theorem 2.3. 39 Note that we can extend the domain of T N (t;q) to all of H via the denition T N (t;q)x =T N (t;q)P N x +xP N x: Proposition 5.9. jT N (t;q)j and T N (t;q) are uniformly bounded over N2N, t2 [0;T ], q2Q. Proof. Let'2H N . From coercivity (4.2.2), we have 0 2R,> 0 independent ofq andN such that for all 0 , q2Q, h(A N (q))';'ik'k 2 : Letting R (A N (q)) =', we haveh ;R (A N (q)) )i R (A N (q)) 2 , from which follows by Cauchy- Schwarz c 2 k k R (A N (q)) ; c 2 j jjR (A N (q)) j: By Theorem 3.16, A N (q)2G(c 2 =; 0 ) on H and on V ; therefore for t2 [0;T ] T N (t;q) c 2 e j0jT jT N (t;q)j c 2 e j0jT : Because for each q2Q, + (q)v maps into WV , there exists a sequence +N (q)2H N approximating + (q) in the V norm. We assume that the map q7! +N (q)v is a continuous map in the V norm for each v2R m and each N = 1; 2;:::. We are also interested, in some cases, in estimating an innite-dimensional parameter q2Q. To this end, suppose also that there exist metric spaces Q M Q for M = 1; 2;::: with metrics induced by the metric on Q. Assume we have p M :Q!Q M for M = 1; 2;:::, a continuous and surjective sequence of functions such 40 that8q2Q, p M q!q in Q. Note that since p M is continuous and surjective and Q is assumed compact, Q M is compact. 5.4 Finite dimensional approximation of discrete time parabolic boundary input model Recall that the input space U is composed of zero-order hold samples of functions with timestep . In other words, if T =n, we can identify U (R m ) n : To facilitate dimension reduction of our optimization, suppose that for integers P we have spacesU P U of dimensionk(P ) with ordered basesf P j;i : j = 1;:::;k(P ); i = 1;:::;ng. The components of the zero-order hold input signals u2U as approximated for increasing P by u P i = k(P) X j=1 a P j P j;i ; i = 0;:::;n for some coecients a P j in a xed compact subset ofR k(P) . Dene for each q2Q b A N (q) =T N (;q)2L(H N ;H N ) b B N (q) = (I b A N (q)) +N (q)2L(R m ;H N ): Now dene a vector of convolutions b L N (q) = ( b L N j (q)) n j=1 : (R m ) n !Y n for each q2Q by b L N j (q)u := j1 X i=0 C(q) b A N (q) j1i b B N (q)u i : Note that since H N V , C(q) b A N (q) is well-dened. Let N (q;u; 0) :Q N (R m ) n !Y n be dened by N (q;u; 0) = b L N (q)u: 41 We would like to show that N is continuous, and that for any convergent (q M ;u P ; 0)! (q;u; 0) in QU, N (q M ;u P ; 0)! (q;u; 0). Lemma 5.10. Let q M !q in Q, x2R m . 1. b B(q)x b B N (q M )x ! 0 as M;N!1. 2. For each N, sup q2Q b B N (q) L(R m ;V ) <1. Proof. 1. b B(q)x b B N (q M )x = (I b A(q)) + (q)x (I b A N (q M )) +N (q M )x + (q)x +N (q M )x + b A N (q M ) +N (q M )x b A(q) + (q)x = 1 + 2 : 1 is bounded by + (q)x +N (q)x + +N (q)x +N (q M )x : The quantity on the left goes to 0 as N!1 since we assumed +N (q) approximates + (q) in the V norm. The quantity on the right vanishes asN!1 because we assumedq7! +N (q)x was continuous in the V norm. Then, 2 b A(q) + (q)x b A(q) + (q M )x + b A(q) + (q M )x b A(q) +N (q M )x + b A(q) +N (q M )x b A N (q M ) +N (q M )x = 4 + 5 + 6 4 is bounded by b A(q) + (q)x + (q M )x : 42 which goes to 0 by Lemma 5.2 and (A1). 5 is bounded by b A(q) + (q M )x +N (q M )x ! 0; which goes to 0. Finally, 6 goes to 0 by Theorem 5.8. 2. In light of Proposition 5.9, it suces to show that +N (q) is uniformly bounded inQ, but this follows from the assumed continuity property together with the Banach-Steinhaus theorem 3.1. Lemma 5.11. For any q M !q in Q, u2 (R m ) n , b L N (q M )u b L(q)u n ! 0 as N!1. Proof. We examine the j th component. ( b L(q)u b L N (q M )u) j j1 X k=0 jC(q) b A(q) j1k b B(q)u k C(q M ) b A N (q M ) j1k b B N (q M )u k j We estimate the k th summand by j[C(q)C(q M )] b A(q) j1k b B(q)u k j +jC(q M )[ b A(q) j1k b A N (q M ) j1k ] b B(q)u k j +jC(q M ) b A N (q M ) j1k [ b B(q) b B N (q M )]u k j The rst term goes to 0 by (A2). The second term goes to 0 by (A2) and Theorem 5.8. The third term goes to 0 by (A2), Theorem 5.8, and Lemma 5.10. Proposition 5.12. For anyq M !q inQ,u P !u inU, N (q M ;u P ;' 0 ) (q;u;' 0 ) n ! 0 asN!1. Proof. b L N (q M )u P b L(q)u n b L N (q M )u P b L N (q M )u n + b L N (q M )u P b L(q)u n 43 The second term goes to 0 by Lemma 5.11. To show the rst goes to 0, it will suce to show that b L N (q M ) : (R m ) n !Y n is uniformly bounded in N. For each j, j b L N j (q M )j j1 X k=0 C(q M ) b A N (q M ) j1k b B N (q M ) ; which is uniformly bounded in N by (A2), Proposition 5.9, and Lemma 5.10. For each multi-index [M;N;P ], dene a sequence of cost functions on Q M U P , J N (q M ;u P ) = 1 X i=1 N (q M ; ~ u (i) ; 0) ~ y (i) 2 n + N (q M ;u P ; 0) ~ y 2 n +R(q M ;u P ) (5.4.1) There exist q N 2Q and u N 2U minimizing J N (q;u) by exactly the same arguments used to show J(q;u) has a minimizer in QU in the proof of (5.7); in particular, J N is continuous. Because QU is a compact metric space, the sequence of solutions (q N ;u N )2QU is bounded, so it has a convergent subsequence. Proposition 5.13. For any sequence (q M ;u P )! (q;u) in QU, lim [M;N;P]!1 J N (q M ;u P ) =J(q;u). Proof. jJ N (q M ;u P )J(q;u)j 1 X i=1 b L N (q M )~ u (i) ~ y (i) 2 n b L N (q)~ u (i) ~ y (i) 2 n + b L N (q M )u P ~ y 2 n b L N (q)u ~ y 2 n + R(q M ;u P )R(q;u) = 7 + 8 + 9 7 goes to 0 as N !1 if b L N (q M )~ u (i) b L N (q)~ u (i) n ! 0 for each i, which follows from the proof of Proposition 5.12. 8 goes to 0 i b L N (q M )u P b L N (q)u ! 0, which is estimated by b L N (q M )u P b L N (q)u b L N (q M )u P b L N (q M )u + b L N (q M )u b L(q)u + b L(q)u b L N (q)u which goes to 0 by Lemma 5.11 and Proposition 5.12. 9 vanishes by continuity ofR. 44 5.5 Discrete time transdermal alcohol model approximation and convergence 5.5.1 Linear spline approximation Denition 5.14. Let f2L p [0; 1] for some p 1. Let N f denote the linear interpolant of f on the uniform mesh of size 1=N; in other words, it is the unique function which is a linear polynomial on each interval [(i 1)=N;i=N), i = 1;:::;N and agrees with f at each i=N. Another characterization of the space of linear splines is as the span of the \pup tent" functionsf N j g N j=0 which are supported on [max(0; j1 N ); min(1; j+1 N )], are linear on either side of j N , and N j (j=N) = 1. Lemma 5.15. Suppose p = 2 and f2H 1 [0; 1], and let D denote the derivative operator. Then f N f 2 1 N kDfk 2 D(f N f) 2 kDfk 2 : If f2H 2 [0; 1], then D(f N f) 2 2 hf N f;D 2 fi 2 f N f 2 D 2 f 2 : Proof. [35], Theorems 2.3 and 2.4. 5.5.2 Finite dimensional approximation of discrete time transdermal alcohol model LetH N = N H, and letP N :H!H N denote the orthogonal projections onto H N in the linear spline basis f N j g N j=0 , with respect to the H inner product. Then (A N (q)) ij =[h N i ; N j i] 1 [a(q; N i ; N j )] 45 for a(q;;) dened as in (2.2.1). Let +N (q) = N + (q) for + (q) as dened in the previous section, which converges in V to + (q) by Lemma 5.15. As a convolution, for (q;u)2QU, the approximating outputs take the form y N (q;u;' 0 ) =C(q) b A N (q) j ' 0 +C(q) j1 X k=0 C(q) b A N (q) j1k I b A N (q) +N (q)u k : (5.5.1) 5.6 Approximate solution of estimation problem In this chapter, and indeed models throughout this thesis, involve operators on a Hilbert space; in particular, the state of our system is innite dimensional. In order to employ a computer to t our model to data, we will seek to approximate a minimizer of J by a sequence of minimizers of related cost functionals J N (q;u) :Q M U P !R for N = 1; 2;::: derived from a nite-dimensional approximating model. Lemma 5.16. Suppose that for M;N;P = 1; 2;:::, Q M U P QU, that J :QU!R J N :Q M U P !R are continuous, and for each (q;u), J N (q;u)!J(q;u) as N!1. Let (q M ;u P )! (q;u) be any convergent sequence in QU. Then J N (q M ;u P )!J(q;u) in R as [M;N;P ]!1. Proof. jJ N (q M ;u P )J N (q;u)jjJ N (q M ;u P )J N (q;u)j +jJ N (q;u)J(q;u)j; which goes to 0 by hypothesis. Proposition 5.17. Assume thatQU is compact and that the hypotheses of Lemma 5.16 hold. Let (q M ;u P ) be a global minimizer ofJ N inQ M U P . Then a subsequence of the (q M ;u P ) converges to a global minimizer of J in QU. 46 Proof. Because QU is compact,f(q M ;u P )gQU has a subsequential limit (q;u)2QU. Then for all (q;u)2QU, J(q;u) = lim k!1 J N k (q N k ;u N k ) lim k!1 J N k (q;u) =J(q;u): 47 Chapter 6 Approximate semi-blind deconvolution in delay systems with unbounded input and output We now turn our focus to a dierent class of dynamical systems, namely delay or delay dynamical systems. Such dynamical systems deal with a state which changes based on its past. In this example, we build upon the work of [25] and [5] to solve the parameter estimation and deconvolution problem in delay systems with unbounded input and output where the delay times may be unknown. 6.1 Homogeneous delay evolution system We assume that there exists a xed, given h> 0 and compact convex set R , and dene the compact convex set of admissible parameters QR + by Q = R, where R :=fr = (h 1 ;h 2 ;:::;h )2R : 0h i h i+1 h; i = 1;:::; 1g: Suppose v2L 2 [h; 0;R m ] and consider the linear retarded functional dierential equation of the form _ x(t) =L(q)x t + ^ B(q)v t y(t) = ^ C(q)x t (6.1.1) for t 0, where x(t)2 R n , v(t)2 R m , y(t)2 R p , and x t and u t are dened by the history function x t (s) =x(t +s) and v t (s) =u(t +s) for s2 [h; 0], and x t (s) =u t (s) = 0 if s +t = 2 [h; 0]. The bounded 48 linear operators L(q), ^ B(q), and ^ C(q) are maps from spaces of real-valued continuous functions to real nite-dimensional vector spaces, dened for q = (;r)2Q by L(q)' = X i=0 A i ()'(h i ) + Z 0 h A 01 (s;)'(s)ds; ^ C(q)' = X i=0 C i ()'(h i ) + Z 0 h C 01 (s;)'(s)ds; '2C[h; 0;R n ]; ^ B(q) = X i=0 B i ()(h i ) + Z 0 h B 01 (s;)(s)ds; 2C[h; 0;R m ] whereA i ()2R nn ,B i ()2R nm ,C i ()2R pn ,i = 0;:::;, andA 01 (;)2L 2 [h; 0;R nn ],B 01 (;)2 L 2 [h; 0;R nm ],C 01 (;)2L 2 [h; 0;R pn ]. We assume that the above matrices and functions are continuous in q2Q. Let Z = R n L 2 [h; 0;R n ] , V = L 2 [h; 0;R m ], and assume that we are given an initial data set SZV which is closed, convex, and bounded, and dene =SQ =S R: Also assume that is compact, so that the above matrices and functions are in fact uniformly continuous and thus uniformly bounded over q2Q. Elements 2 will be denoted in one of several ways: = (' 0 0 ;' 1 0 ;' 2 0 ;q) = (' 0 0 ;' 1 0 ;' 2 0 ;;r) = (' 0 0 ;' 1 0 ;' 2 0 ;;h 1 ;:::;h ); where q = (;r) = (;h 1 ;:::;h ). LetX =ZV . Consider the solution of (6.1.1) inX with initial data x(0) =' 0 ; x(s) =' 1 (s); v(s) =' 2 (s); s2 [h; 0); where ' = (' 0 ;' 1 ;' 2 )2X . Standard results guarantee existence, uniqueness, and continuous dependence on initial data (see e.g. [30]); the state of the system inX is then given by (x(t);x t ;v t )2X for t 0. 49 The solution semigroup T(t;q) corresponding to the free motion of (6.1.1), i.e. v(t) = 0 for t 0, onX providing the state has innitesimal generator D(A(q)) =f'2X : ' 1 2H 1 [h; 0;R n ]; ' 2 2H 1 [h; 0;R m ]; ' 1 (0) =' 0 ; ' 2 (0) = 0g; A(q)' = (L(q)' 1 + ^ B(q)' 2 B 0 (q)' 2 (0); _ ' 1 ; _ ' 2 ); '2D(A(q)): We are interested in simultaneously estimating and an unknown input u from sampled input-output training data to the system (6.1.1), i.e. for known training episodes ~ u (i) (t), i = 1;:::;, and sampling times t j , j = 1;:::;n, we assume that we sample our system's output ~ y (i) j =y(t j ;), and that we are given another sampled system output y j , j = 1;:::;n for which the input u is unknown. We are interested primarily in minimizing the cost functional J(;u) = 1 X i=1 n X j=1 j~ y (i) j y(t j ;; ~ u (i) )j 2 + n X j=1 jy j y(t j ;;u)j 2 +R(;u) over a xed compact subset ~ ~ U of L 2 [0;T ;R m ]. 6.2 Inhomogeneous delay evolution system Now assume that WXW are dense and continuous inclusions of Hilbert spaces, and assume that T(t;q) restricts/extends to a C 0 semigroup onW andW for each q2Q. For each q2Q dene the input operator B(q) :R m !W by B(q)v = (B 0 (q)v; 0; 0 0 v); 50 where 0 0 denotes the Dirac distribution centered at 0; implicitly this means elements of this form 0 0 must be contained inW. Also dene the output operator C(q)2L(W;R p ) by C(q)' = ^ C(q)' 1 ; where here it is implied that the second components of elements ofW must be continuous functions. To reconcile the unbounded input and output, we will make the following regularity assumptions. (H1) For some u2L 2 [0;T ;R m ], t2 (0;T ] Z t 0 T (ts;q)B(q)u(s)ds2W; and there exists a positive constant M B independent of t and q such that Z t 0 T (ts;q)B(q)u W M B kuk 2 : (H2) There exists a positive constant c such that for each '2W, kC(q)T (;q)'k 2 ck'k W : Now suppose u2L 2 [0;T ;R m ]. Then our object of study is the abstract inhomogeneous Cauchy problem _ '(t) =A(q)' t +B(q)u(t); t 0; (6.2.1) '(0) = (' 0 0 ;' 1 0 ;' 2 0 ); (6.2.2) y(t) =C(q)' t ; t 0: (6.2.3) 51 6.3 Finite dimensional approximation of inhomogeneous delay evolution system We will take the approach of the previous chapter in attempting to approximate a minimizer of J(;u) by a sequence of nite-dimensional approximating spaces X N to Z and related approximating problems posed on those spaces. However, the natural state space for the solution with q N = ( N ;h N 1 ;:::;h N ) is Z N =Z N (q N ) =R n L 2 [h N ; 0;R n ]: This space varies with N and is only considered a subspace of Z =R n L 2 [h; 0;R n ] via the Moore-Penrose pseudo-inverse, which in this case realizes a map Z N !Z that extends by zero the function in the second argument to the entire interval [h; 0]. LetX N be the closed subspace of linear splines onZ N and let N :Z N !X N be the canonical projection of Z N onto X N along (X N ) ? . Let 0 denote projection onto the rst coordinate, and letI N :Z!Z N be the surjective mapping withjI N 'j Z Nj'j 0 W for '2 Z that takes ' = (' 0 ;' 1 )2 Z into ~ ' = (' 0 ; ~ ' 1 ), where ~ ' 1 is the restriction of ' 1 to [h N ; 0]. Finally, let P N :Z!X N denote the composition of these two mappings, i.e. P N = N I N . Evidently the approximation spaces X N = X N (q) and Z N = Z N (q) depend on the parameter q; in particular, the spaces depend on the greatest delay time h N . Let P N (q) :Z!X N (q) be as in the previous paragraph and dene N = [ q2Q P N (q)Sfqg : We will also assume that we have projections p N (q N ) : V ! V N (q N ) for some Hilbert spaces V N L 2 [h N ; 0;R m ] that satisfy (p N u)(0) = 0. As in previous work on delay systems, we dene a weighted norm on the spaces Z N and X N . To this end, dene the weighting function g N (;q N ) on [h N ; 0) for given q N 2 Q by g N (s;q N ) := k + 1 52 when s2 [h N k ;h N k1 ), k = 0;:::; 1, where we take h N 0 = 0. Then we let the inner product on Z N =Z N (q N ) =Z N (g N ) be dened by h( 1 ; 1 ); ( 2 ; 2 )i g N :=h 1 ; 2 i R n + Z 0 h N 1 2 g N : Then we let N :Z N (g N )!X N (g N ) be the orthogonal Hilbert space projections, and let P N = N I N . We similarly dene a weighted norm on V N (q N ). DeneX N (q N ) =X N (q N )V N (q N ) with the weighted inner product in both components; denote its norm byjj N . Let P N (q N ) := (P N (q N );p N (q N )) :X!X N (q N ): ThenP N (q N ) is uniformly bounded in q N in both components, and in N. We can therefore extend the domain ofP N (q N ) for each N and q to a map P N (q N ) :W!X N (q N ) becauseX ,!W is dense and continuous, allowing application of the bounded extension theorem, Theorem 3.2. Note that the image ofP N (q N ) is contained in D(A(q)), for any q N ;q2Q, after possibly extending functions by zero to [h; 0] using the Moore-Penrose pseudoinverse. For N = (' N 0 ;q N )2 N , letA N (q N ) = P N (q N )A(q N )P N (q N ) :X N (q N )!X N (q N ), and dene the approximating input operator B N (q N ) = B 0 (q N );B 0 (q N ) N 0 ; 2N h N N 0 : (6.3.1) 53 Then for any q N !q in Q, B N (q N )!B(q) inL(R m ;W). Then for u(t)2L 2 [0;T;R m ], ' N (t)2X N (q N ) we can consider the inhomogeneous Cauchy problem _ ' N (t) =A N (q N )' N t +B N (q N )u(t); t 0; ' N (0) =P N (q N )(' 0 0 ;' 1 0 ;' 2 0 ) y N (t) =C(q N )' N t ; t 0; with solution ' N (t; N ;u) =T N (t;q N )' N 0 + Z t 0 T N (ts;q N )B N (q N )u(s)ds; where T N (t;q N ) is the semigroup onX N (q N ) generated by A N (q N ). The corresponding output is given by y N (t; N ;u) =C(q N )' N (t; N ). 6.4 Linear spline approximation of delay system We now examine our particular choice ofX N as the spaces of linear splines. However, here we will not have the luxury of a uniform mesh because the delay times are unknown. In particular, for givenq N = ( N ;h N 1 ;:::;h N ), let t N j = (j (k 1)N)(h N k h N k1 ) N +h N k1 ; j = (k 1)N;:::;kN, k = 1;:::;. Then we take X N (q N ) :=f(x(0);x) : x is a linear spline with knots at t N j g: For any q N 2Q, dene A(q N ) :D(A(q N ))Z N !Z N , by D(A(q N )) =f(x(0);x;v)2Z N : x2H 1 [h N ; 0;R n ]; v2H 1 [h N ; 0;R m ]g A(q N )(x(0);x;v) = (L(q N )x + ^ B(q N )vB 0 (q N )v(0);Dx;Dv): 54 More generally, we can dene A(q) :DZ!Z by A(q)(x(0);x;v) = (L(q)x + ^ B(q)vB 0 (q)v(0);Dx;Dv); where in the latter equation the derivative operator D is dened for functions on [h; 0] and in the former, for functions on [h N ; 0]. Note that X N (q N )D(A(q N )) for all q N 2Q. Proposition 6.1. Assume Q is compact. Then for each q N 2 Q and '2 D(A(q N )), there exists !2R independent of q N 2Q such that hA(q N )';'i g N!j'j 2 g N: Proof. Let q N 2Q, ' = (' 0 ;' 1 ;' 2 )2D(A(q N )). Then if D denotes the derivative operator, hA(q N )';'i g N =h(L(q N )' 1 ;D' 1 ;D' 2 );'i g N +h ^ B(q N )' 2 B 0 (q N )' 2 (0); 0; 0);'i g N: A direct application of [8], Lemma 2.3 shows that the rst of these terms is bounded by ! 0 (q N )j'j 2 g N , where ! 0 (q N ) = + 1 2 +jA 0 ( N )j + 1 2 X i=1 jA i ( N )j 2 + 1 2 Z 0 h N jA 01 ( N ;s)j 2 ds: Due to the continuity assumptions on the operators and compactness of Q, there exists q 0 2Q such that ! 0 =! 0 (q 0 )! 0 (q N ) for all q N 2Q. The second term is * X i=1 B i ( N )' 1 (h N i ) + Z 0 h N B 01 (s; N )' 1 (s)ds;' 1 (0) + R n ; which due to the continuity assumptions on the operators, compactness ofQ, and continuity of' 1 , is bounded by ! 1 j'j 2 g N for some ! 1 2R. Proposition 6.2. Assume the semigroups T N (t;q N ) are given onX N (q N ), with innitesimal generators A N (q N ) :=P N (q N )A(q N )P N (q N ) for q N 2Q. Then A N (q N )2G(M;!) onX N (q N ). 55 Proof. hP N (q N )A(q N )P N (q N )';'i g N =hA(q N )P N (q N )';P N (q N )'i g N!jP N (q N )'j 2 g N!j'j 2 g N: Therefore A(q N )!I satises the hypotheses of the Lumer-Phillips theorem and A N (q N )2 G(M;!) on X N (q N ) for some M independent of N, ! as in the previous proposition. Proposition 6.3. Let q N !q be any convergent sequence in Q and letD =D(A(q) 3 ), so thatD is dense inX and R (A(q))DD. ThenjA N (q N )P N (q N )'P N (q N )A(q)'j N ! 0. Proof. Write A N =A N (q N ), A =A(q), and let z = (x(0);x;v)2D. Then jA N P N (q N )zP N (q N )Azj N j N (L(q N )xL(q)x;D(x N x);D(v N v))j N +j N ([ ^ B(q N ) ^ B(q)]v + [B 0 (q)B 0 (q N )]v(0); 0; 0)j N ; where we understand D(x N x) and D(v N v) to be functions on [h N ; 0]. That the rst term vanishes as N!1 is the content of the argument given on p.813-814 of [5]. The second vanishes beause we assumed continuity in q of the operators ^ B and B 0 . Now let U N be a sequence of nite-dimensional subspaces of a compact set U L 2 [0;T;R m ]. The approximating cost functionals are dened on N L 2 [0;T ;R m ] by J N ( N ;u N ) = 1 X i=1 n X j=1 jy N (t j ; N ; ~ u (i) ) ~ y (i) j j 2 + n X j=1 j ~ y j y N (t j ; N ;u N )j 2 +R( N ;u N ): 6.5 Results of Banks-Burns-Cli Theorem 6.4. LetX ,X N , andP N be given as above, and suppose that for some M, !, we have A N 2 G(M;!) onX N and A2G(M;!) onX . Further suppose there existsDD(A),D dense inX such that R (A)DD for Re > ! and for each '2D,jA N P N 'P N A'j N ! 0 as N!1. Then for every '2X , jT N (t)P N 'P N T (t)'j N ! 0 56 as N!1. Proof. This is the content of [5], Theorem 3.1, using the extended operatorP N :W!X N in place of the projectionP N :X!X N . Theorem 6.5. Suppose u2 U is given, and that N = (' N 0 ;q N ) is a sequence of minimizers to J N (;u). Assume further that there exists2 such that N ! in the sense thatq N !q inR + and (I N ) y ' N 0 !' 0 in Z, where they denotes the Moore-Penrose pseudo-inverse toI N , which maps Z N !Z. Also suppose that A N (q N ) = A N , A(q) = A, and that the hypotheses of Theorem 6.4 hold. Then jP N '(t;;u)' N (t; N ;u)j N ! 0 as N!1 uniformly for t on compact intervals, where '(t;;u) =T (t;q)' 0 + Z t 0 T (ts;q)B(q)u(s)ds: If in addition, P N :Z!X N satises 0 (P N ')! 0 ' in R n for each '2Z, where 0 denotes projection onto the rst coordinate, then for each u2U, y N (t; N ;u)!y(t;;u) for each t. Proof. This follows from easy modications of [5], Theorem 4.1 and Corollary 4.1, replacingP N B(q) with B N (q N ), using (H1), and the fact that B N (q N )!B(q) inW. We now show a stronger result which we require to estimate u by way of minimizers to J N over all of N U N . Corollary 6.6. Assume that the hypotheses of the previous theorem hold, and that ( N ;u N ) is a sequence of minimizers to J N in N U N , and that there exists (;u)2 U such that ( N ;u N )! (;u) in the sense that q N !q in R + , (I N ) y ' 0 N !' 0 , andju N p N uj 2 ! 0. ThenjP N '(t;;u)' N (t; N ;u N )j N ! 0 as N!1 uniformly for t on compact intervals; ifP N :W!X N satises 0 (P N ')! 0 ' in R n for each '2W, then y N (t; N ;u N )!y(t;;u) 57 for each t. Proof. Write B N (q N ) =B N , B(q) =B. By the theorem, to showjP N '(t;;u)' N (t; N ;u)j N ! 0 it will suce to show that Z t 0 T N (ts)B N u N (s)dsP N Z t 0 T (ts)Bu(s)ds N ! 0: This quantity is estimated by Z t 0 [T N (ts)B N u N (s)P N T (ts)Bu(s)]ds N Z t 0 jT N (ts)B N [u N (s)u(s)]j N ds + Z t 0 jT N (ts)[B N B]u(s)j N ds + Z t 0 j[T N (ts)P N P N T (ts)]Bu(s)j N ds; where we have implicitly invoked the extended denition of T N (t) to all ofX via the formula T N (t)' =T N (t)P N '' +P N ': BecauseX N is nite-dimensional, a variation of constants formula holds for the system _ ' N (t) =A N ' N (t) +B N u(t) inX N ; it follows that u7! R t 0 T N (ts)B N u(s)ds is a bounded map fromL 2 [0;T ;R m ]!X N . In particular, due to the previous theorem, it is uniformly bounded in N. Therefore the rst term vanishes as N!1. 58 Continuity of q7! B(q) and the stability assumption A N 2 G(M;!) show the second term vanishes as N!1. Theorem 6.4 provides convergence for the third and nal term. Then y(t;;u) =C()'(t;;u) y N (t; N ;u N ) =C( N )'(t; N ;u N ): Due to (H1) and (H2), we may write the output of our system jy N (t; N ;u N )y(t;;u)jjy N (t; N ;u N )y(t;;u N )j +jy(t;;u N )y(t;;u)j: The rst term vanishes by Theorem 6.5. The second vanishes due to (H1) and continuity of ^ B(q) and ^ C(q). We therefore may apply the results of Section 5.6 to approximate a minimizer of J(;u) by a subsequence of minimizers of J N (;u) over N U N . 6.6 Approximate semi-blind deconvolution in delay systems For the calculation of our approximating systems, we use methods similar to those in [8], p.507-509. Let q N = ( N ;h N 1 ;:::;h N ) and let N = ( N 0 ; N 1 ;:::; N N ) denote the n (N + 1) matrix function of linear splines on the uniform mesh of size h N =N on [h N ; 0], and let N = ( N 1 ;:::; N N ) be the pN matrix function of linear splines on same mesh, so that each (x N (0);x N ;v N )2X N can be written (x N (0);x N ;v N ) = b N w N := ( N w N 1 ; N w N 2 ) for some coecient 2N + 1-vector w N = (w N 1 ;w N 2 ). 59 We computeP N (q N )(' 0 ;' 1 ;' 2 ) =e N in this basis for given (' 0 ;' 1 ;' 2 )2X ,q N 2Q. BecauseP N (q N ) is uniquely determined by the orthogonality relationship inX N (q N ), we have P N (' 0 ;' 1 ;' 2 ) (' 0 ;' 1 ;' 2 ) ? X N (q N ), which implies 0 =h b N e N (' 0 ;' 1 ;' 2 ); b N i X N (q N ) ; which in turn implies M N (q N )e N =f N (' 0 ;' 1 ;' 2 ), where M N (q N ) =h b N ; b N i X N (q N ) = N (0) T N (0) + Z 0 h N N (s) T N (s)g N (s)ds + Z 0 h N N (s) T N (s)g N (s)ds f N (' 0 ;' 1 ;' 2 ) =h b N ; (' 0 ;' 1 ;' 2 )i X N (q N ) = N (0) T ' 0 + Z 0 h N N (s) T ' 1 (s)g N (s)ds + Z 0 h N N (s) T ' 2 (s)g N (s)ds: Note that M N (q N ) 1 exists becauseP N (q N )(' 0 ;' 1 ;u) is uniquely determined by (' 0 ;' 1 ;' 2 ). Now, write A N (q N ) =P N (q N )A N (q N )P N (q N ) =A N , and suppose ' N = (x N (0);x N ;v N ) = b N w N 2 X N . Recalling thatX N (q N )D(A(q N )), we have v N (0) = 0. We write ~ A N 2L(R 2N+1 ) for the matrix representation of A N in the basis f(( N j (0); N j ); 0); ((0; 0); N k ) : j = 0;:::;N; k = 1;:::;Ng forX N (q N ). Thus for some coecient vector a N 2R 2N+1 , we have M N (q N )a N =A N ' N =P N A' N =f N (A' N ) =h b N ;A b N iw N =:K N (q N )w N : We have shown that ~ A N = (M N (q N )) 1 K N (q N ). One solves M N (q N )a N = K N (q N )w N directly to obtain a N = ~ A N w N . 60 In this basis, the approximate input operator of (6.3.1) has the form ~ B N (q N ) :=M N (q N ) 1 B N (q N ) = (B 0 (q N ); [B 0 (q N ); 0;:::; 0]; [2N=h N ; 0;:::; 0]) and output ~ C(q N ) =C(q N )M N (q N ). Therefore our approximating system then takes the form of a dynamical system on R 2N+1 prescribed for t 0 and given q N 2Q, u(t)2L 2 [0;T ] by _ w N (t) = ~ A N (q N )w N (t) + ~ B N (q N )u(t); w N (0) =P N (w 0 0 ;w 1 0 ;w 2 0 ); y N (t) =M N (q N )w N (t): 61 Chapter 7 Approximate semi-blind deconvolution in continuous time parabolic systems with input and output on the boundary We draw inspiration from the product space formulation of the previous section which includes the input as a component of the state to formulate and solve the parameter estimation and deconvolution problem of Chapter 4 in continuous time. We can leverage the regularity aorded by analyticity of the evolution semigroup in the parabolic case somewhat. However, we will make a time regularity assumption for our system; in particular, we formulate our system as a homogeneous one on V H 1 [0;T ;R m ], where the input signal is assumed in H 1 [0;T ;R m ] rather than in L 2 [0;T ;R m ], as it often is in the literature. 7.1 Well posedness of continuous time parabolic boundary input model We recall the setup of Chapter 4. LetQ be a compact metric space, letV ,!H ,!V be a Gelfand triple, and suppose for each q2Q, a(q;;) is a parabolic form on VV , so it denes a linear operator A(q)2L(V;V ) which is the innitesimal generator of an analytic semigroup on V , H, and V . Again dene Y =R p for some p2N. Suppose also that W ,! V is continuous, and that W is dense in V and in H. Further, for each q2Q, (q)2L(W;H), (q)2L(W;R m ) is surjective with right inverse + (q)2L(R m ;W ), ker (q) is dense in V , Range + (q) ker (q), B(q) = ((q)A(q)) + (q)2L(R m ;V ); 62 and C(q)2L(V;Y ). Also assume that UH 2 [0;T ;R m ] is a compact set of admissible input signals. Then consider again the nohomogeneous Cauchy problem with unbounded output on V (or, replacing A(q) with domain V by A(q) with domain D(A(q)) H = ker (q), the nonhomogeneous Cauchy problem on H with unbounded input and unbounded output) for given q2 Q, u2 H 1 [0;T;R m ], ' 0 2 W with (q)' 0 =u(0), _ '(t) =A(q)'(t) +B(q)u(t); t 0; (7.1.1) '(0) =' 0 ; (7.1.2) y(t) =C(q)'(t); t 0: (7.1.3) Dene X =V H 1 [0;T ;R m ] E =VH 2 [0;T ;R m ]: For each q2Q, consider the closed and densely dened linear operator D(A(q)) =E A(q) = 0 B B @ A(q) B(q) 0 0 D 1 C C A : WhenA(q) is the innitesimal generator of a C 0 semigroupT (t;q) onX , we have T (t;q) = 0 B B @ T (t;q) B(t;q) 0 S(t) 1 C C A (7.1.4) where T (t;q) is the semigroup generated by A(q), S(t) denotes the left shift semigroup on H 1 [0;T ;R m ], and for u2H 1 [0;T ;R m ], B(t;q)u = Z t 0 T (ts;q)B(q)u(s)ds2V : 63 Conversely, it is easy to see that ifT (t;q) dened by (7.1.4) is aC 0 semigroup onX , then its innitesimal generator must be given byA(q). Substituting our expression for B(q) and integrating by parts, we obtain for t 0, u2H 1 [0;T ;R m ], Z t 0 T (ts;q)B(q)u(s)ds = Z t 0 T (ts;q)((q)A(q)) + (q)u(s)ds = Z t 0 T (ts;q)A(q) + (q)u(s)ds =T (t;q) + (q)u(0) + (q)u(t) + Z t 0 T (ts;q) + (q) _ u(s)ds; by integrating by parts, which is in V by Lemma 3.10. We have shown the following. Proposition 7.1. LetH2fV;H;V g. For each q2Q,B(t;q)2L(H 1 ;V )L(H 1 ;H), andA(q) is the innitesimal generator of a C 0 semigroup onHH 1 [0;T ;R m ]. Proof. The rst claim follows from the above calculation; the second follows because for each q2Q,T (t;q), t 0 is clearly a semigroup of bounded operators onHH 1 [0;T ;R m ]. The only nontrivial conditions in the denition of a semigroup to check are that as t! 0 + , 1. S(t) converges strongly to the identity on H 1 [0;T ;R m ], which we know to be the case; 2. for each q2Q, T(t;q) converges strongly to the identity onH, which is true by Propositions 4.2 and 4.1; and 3. B(t;q) converges strongly to the zero operator onL(H 1 ;H), which holds by the calculation preceding the proposition statement. Proposition 7.2. For each u2H 1 [0;T ;R m ], the functionB(t;)u :Q!V is continuous. Proof. We have B(t;q)u =T (t;q) + (q)u(0) + (q)u(t) + Z t 0 T (ts;q) + (q) _ u(s)ds: 64 Then let q N !q in Q, u2H 1 [0;T ;R m ]. B(t;q)uB(t;q N )u = [T (t;q) + (q)T (t;q N ) + (q N )]u(0) + [ + (q N ) + (q)]u(t) + Z t 0 T (ts;q) + (q) _ u(s)ds Z t 0 T (ts;q N ) + (q N ) _ u(s)ds = 1 + 2 + 3 : We have k 1 k [T (t;q)T (t;q N )] + (q)u(0) + T (t;q N )[ + (q) + (q N )]u(0) ; which goes to 0 uniformly in t2 [0;T ] by Lemma 5.1, Lemma 5.2, and assumption (A1).k 2 k! 0 by (A1). Then, since + (q)x2V for all q2Q, x2R m , k 3 k Z t 0 [T (ts;q)T (ts;q N )] + (q) _ u(s) + T (ts;q N )[ + (q) + (q N )] _ u(s) ds: By Lemma 5.1, the rst term approaches 0, and the second is bounded by sup [0;T]Q kT (t;q)k ! Z t 0 [ + (q) + (q N )] _ u(s) ds<1; which approaches 0 by (A1) and Lemma 5.2. Corollary 7.3. There exists M B 0 independent of q2Q such that for each u2H 1 [0;T ;R m ], kB(t;q)ukM B kuk H 1 uniformly for t2 [0;T ]. 65 Proof. By Proposition 7.2, for each u2 H 1 [0;T;R m ], the map q7!kB(t;q)uk V attains its supremum at some q u 2Q, and an application of the Banach-Steinhaus theorem gives us that sup q2Q kB(t;q)k L(H 1 [0;T;R m ];V ) =:M B <1: Proposition 7.4. (q;u)7!B(t;q)u2V is continuous, uniformly for t2 [0;T ]. Proof. Let (q N ;u N )! (q;u) be convergent in QH 1 . B(t;q N )u N B(t;q)u [B(t;q)B(t;q N )]u + B(t;q N )[uu N ] The rst term goes to 0 by the previous proposition; the second goes to 0 by Proposition 7.3. LettingC(q) = (C(q); 0), ' 0 = 0, so that (q)' 0 =u(0) = 0, the output is given by y(t;q;u) =C(q)T (t;q)(0;u) =C(q)B(t;q)u: Given n sampling times 0<t 1 <<t n =T , dene the vector of observations (q;u) = (y(t j ;q;u)) n j=1 : As before, we want to minimize the cost functional J(q;u) = 1 X i=1 (q; ~ u (i) ) ~ y (i) 2 n +k(q;u) ~ yk 2 n +R(q;u) whereR is a continuous regularization term. 66 7.2 Finite dimensional approximation of continuous time parabolic boundary input model For each N = 1; 2;:::; assume H N V is a sequence of nite dimensional subspaces of H with norm jj N =jj and corresponding Hilbert space projections P N :H!H N satisfyingjP N xxj! 0 as N!1 for each x2H. Assume for each q2Q, the linear operator A N (q) is obtained by restricting the parabolic form a(q;;) to H N H N . Note that for all N, P N : H V ! H N is bounded (in the V norm), that is, because we assumed kk c 0 jj, that if x2H, jjP N xjj =c 0 jP N xjc 0 jxj: Therefore by Theorem 3.2 we may continuously extend each P N to a bounded map P N : V ! H N . In particular, P N A(q) :V !H N is a bounded map for each q2Q. Now dene for each q2Q B N (q) : =P N B(q) C N (q) : =C(q)j H N : Consider the approximating system on X N :=H N H 1 [0;T ;R m ] given by _ ' N (t) =A N (q)' N (t) +B N (q)u(t) y N (t) =C N (q)' N (t) ' N (0) =' N 0 2H N : (7.2.1) 67 This system has solution ' N (t)2H N if and only if the closed, densely dened linear operator D(A N (q)) :=H N H 2 [0;T ;R m ] A N (q) : = 0 B B @ A N (q) B N (q) 0 0 D 1 C C A is the innitesimal generator of the solution semigroupT N (t;q) to (7.2.1). This is easily seen to be true, because for each q2Q, (' N 0 ;u)2H N H 1 [0;T ;R m ] and t 0, a solution to (7.2.1) is given by (' N (t);u t ) =T N (t;q)(' N 0 ;u) = 0 B B @ T N (t;q) B N (t;q) 0 S(t) 1 C C A 0 B B @ ' N 0 u 1 C C A ; where for u2H 1 [0;T ;R m ], B N (t;q)u := Z t 0 T N (ts;q)B N (q)u(s)ds2H N : LetP N = diag(P N ; N ). We can again extend the domain of denition ofT N (t;q) fromH N H 1 [0;T ;R m ] to all (';u)2X =V H 1 [0;T ;R m ] via the formula T N (t;q)(';u) =T N (t;q)P N (';u) + (';u)P N (';u): 7.2.1 Stability of continuous time parabolic boundary input model approximation Now assume there exist nite-dimensional vector spaces U N H 2 [0;T;R m ] and maps N :U!U N such that N f!f in H 1 [0;T ;R m ] when f2H 2 [0;T ;R m ]. For example, we may take N to be the linear spline approximations of Lemma 5.15. 68 Now, by restricting the domain ofT N (t;q), we obtain a linear operator d T N (t;q) :H N U N !H N U N given for each t 0, N = 1; 2;:::, by d T N (t;q) := 0 B B @ T N (t;q) B N (t;q) 0 N S(t) 1 C C A = 0 B B @ I 0 0 N 1 C C A T N (t;q): One may think of d T N (t;q) as a modication of the semigroupT N (t;q) which discretizes the space of input signals U. Our goal will be to prove, for each convergent sequence q N !q in Q, strong convergence of the maps d T N (t;q N ) toT (t;q), uniformly in t for t2 [0;T ]. To do this, we will use the Trotter-Kato theorem 3.22. We proceed to show that its hypotheses hold for our system, beginning with the following stability lemma. Lemma 7.5. There exists M B 0 such that for all q2Q, N 1, u N 2L 2 [0;T ;R m ]H 1 [0;T ;R m ], jB N (t;q)u N j N M B u N 2 Proof. Let q2Q be given. jB N (t;q)u N j N = Z t 0 T N (ts;q)B N (q)u N (s)ds N Z T 0 Me !(Ts) jP N B(q)j N ju N (s)jds MkB(q)k Z T 0 e !(Ts) ju N (s)jds MkB(q)k c T u N 2 where in the last line we have used Cauchy-Schwarz; c T is a constant depending only on T . Thus it remains only to show thatkB(q)k is indeed nite and can be made independent of q. We have kB(q)k k(q)A(q)k L(W;V ) + (q) W : 69 The latter term is uniformly bounded in q by (A1). By the continuity property (4.2.3) of the parabolic form and compactness of Q, the hypotheses of the Banach-Steinhaus theorem apply to the collection of operators (q)A(q), q2Q; we see that it is uniformly bounded in q2Q in the operator topology. Proposition 7.6. There exists an absolute constantM 1 such that for each t 0, N 1, and for all convergent sequences q N !q2Q, T N (t;q N ) X N Me max(!;0)t : Proof. Let (';u)2H N H 1 [0;T ;R m ]. Then there exist M and ! such that T N (t;q N )(';u) X N jT N (t;q N )' +B N (t;q N )uj N +kS(t)uk H 1 Me !t j'j N +cM B kuk H 1 +kuk H 1 (M +cM B + 1)e max(!;0)t k(';u)k H N H 1: Corollary 7.7. There exists b !2R such that d T N (t;q) Me b !t : 7.2.2 Consistency of continuous time parabolic boundary input model approximation We now aim to show the consistency hypothesis of the Trotter-Kato theorem holds under our assumptions. First note thatD :=E =VH 2 [0;T ;R m ] =D(A(q)) is a core forA(q). Proposition 7.8. Assume that q N !q is a convergent sequence in Q, and that (';u)2D. Let ' N 2H N be a sequence converging to ' in V . Then as N!1, A N (q N )(' N ; N u)A(q)(';u) X ! 0: 70 Proof. A N (q N )(' N ; N u)A(q)(';u) = 0 B B @ P N A(q N )' N A(q)' +P N B(q N )( N u)(0)B(q)u(0) D[ N I]u 1 C C A We then estimate the rst entry by P N A(q N )' N A(q)' = P N A(q N )' N P N A(q N )' +P N A(q N )'P N A(q)' +P N A(q)'A(q)' A(q N )' N A(q N )' + A(q N )'A(q)' + P N A(q)'A(q)' : Because A(q) arises from a parabolic form, the continuity assumption (4.2.3) ensures that q7! A(q) is a continuousL(V;V )-valued map. BecauseQ is assumed compact, the quantity sup q2Q kA(q)k L(V;V ) is nite. Therefore the rst term vanishes as N!1. The continuity property (4.2.3) yields vanishing of the second term. The third vanishes because P N !I strongly in H, therefore in V . Then, P N B(q N )( N u)(0)B(q)u(0) P N B(q N )[ N I]u(0) + P N [B(q N )B(q)]u(0) + [P N I]B(q)u(0) : We have that q7!B(q) is continuous in the operator norm by assumption (A1) and the continuity property of parabolic forms (4.2.3); compactness of Q yields uniform boundedness of sup q2Q kB(q)k . Therefore the rst term goes to 0 since we assumed N u!u in H 1 [0;T;R m ] for u2H 2 [0;T;R m ]. The second vanishes by continuity of q7!B(q). Finally, the third vanishes because P N !I strongly in H, hence in V . Finally, D[ N I]u! 0 by Lemma 5.15. Proposition 7.9. Suppose q N ! q is a convergent sequence in Q. Then for each (';u)2X = V H 1 [0;T ;R m ], T N (t;q N )(';u)!T (t;q)(';u) inX , uniformly in t. 71 Proof. This follows from directly applying the Trotter-Kato theorem 3.22 under the hypotheses veried by Proposition 7.6 and Proposition 7.8. Corollary 7.10. Under the hypotheses of Proposition 7.9, if q N !q in Q, [ d T N (t;q N )T (t;q)](';u) X ! 0 as N!1, uniformly for t in bounded intervals. Proof. [ d T N (t;q N )T (t;q)](';u) X [ d T N (t;q N )T N (t;q N )](';u) X + [T N (t;q)T (t;q)](';u) X : The second term approaches 0 by the proposition. We use the extended denition ofT N (t;q), the estimate kS(t)k H 1 1, and uniform boundedness in N for N L(H 1 ) ensured by Lemma 5.15 to estimate [ N S(t) N S(t)]u H 1 N H 1 [ N I]u H 1 + [ N I]S(t)u H 1 ; which approaches 0 as N!1 by Lemma 5.15. Corollary 7.11. Under the hypotheses of Proposition 7.9, if q N ! q in Q and u N ! u in H 1 [0;T;R m ], then b T N (t;q N )(';u N )!T (t;q)(';u) inX =V H 1 [0;T ;R m ]. Proof. d T N (t;q N )(';u N )T (t;q)(';u) X d T N (t;q N )(';u N u) + d T N (t;q N )T (t;q)(';u) M (';u N u) X + [ d T N (t;q N )T (t;q)](';u) : where we have used Proposition 7.6 on the rst term. u N !u and the previous corollary, respectively, suce to prove vanishing of the two terms. 72 Because C(q N )x!C(q)x for each x2V by (A2), we nally obtain our model approximation for xed sampling times t j , j = 1;:::;n as N (q;u) = (y N j ) n j=1 =C(q N ) d T N (t j ;q N )(0;u): Proposition 7.12. Suppose q N !q in Q and u N !u in U. Then N (q N ;u N )! (q;u) as N!1. Proof. First, for each t 0, jy N (t;q N ;u N )y(t;q;u)j = [C(q)C(q N )]T (t;q)(0;u) +C(q N )[T (t;q) d T N (t;q N )](0;u) +C(q N ) d T N (t;q N )(0;uu N ): The rst term converges to 0 by (A2), and the third by (A2) and Proposition 7.6. For the second, note that the ranges ofT (t;q) andT N (t;q N ) are contained in VH 1 [0;T ;R m ] (so that C(q N )T (t;q)(0;u) is dened), butT (t;q) is only a semigroup on V H 1 [0;T ;R m ]. Because Y is nite-dimensional, C(q N )2L(V;Y ) has nite rank and is therefore a compact operator for each q N 2Q. Therefore in light of Lemma 3.7, it will suce to prove that d T N (t;q N )(0;u)!T (t;q)(0;u) in X =V H 1 [0;T ;R m ], but this was shown in Proposition 7.9. 7.3 Finite dimensional approximation of continuous time transdermal alcohol model We recall the operators from section 4.6. Let the space of admissible parameters Q =fq 1 (x);q 2 gC 1 [0; 1; [";]] [";] be compact, and let H =L 2 [0; 1], V =H 1 [0; 1], m = 1, Y =R. 73 For each q2Q, our spaces and operators will be of the form (q) := d dx q 1 (x) d dx W :=f 2H 2 (0; 1) : q 1 (0) 0 (0) (0) = 0g (q) : W!R (q) :=q 1 2 q 1 (1) 0 (1) ker((q)) =f 2H 2 (0; 1) : q 1 (0) 0 (0) (0) = 0; 0 (1) = 0gW C(q) :=C = 0 2L(V;Y ) ' 0 := 0: Then (q) is clearly surjective for each q2Q and ker((q)) is dense in V . Thus (q) is the innitesimal generator of a C 0 semigroup on H by the results of section 4.3. We again take ( + (q)v)(x) =q 2 1 + Z x 0 dt q 1 (t) v Then for each v2R, (q) + (q)v = v and q7! + (q)v is a continuous function Q! W. The additional condition Range( + (q)) ker (q) is also satised in our case. We saw previously that the sesquilinear form on V dened by a(q;'; ) ='(0) (0) + Z 1 0 q 1 (x)' 0 (x) 0 (x)dx: is bounded, coercive, and depends continuously on q. Now, for B(q) = ((q)A(q)) + (q); (A1) follows from strong continuity in q of + (q), which is clear, and strong continuity in q of A(q) by that property of the sesquilinear form a(q;;). (A2) is trivial because the output operator C does not depend on q. 74 We now proceed to dene our approximation setting. We again take our approximating subspaces H N to be the spaces of linear splines on [0; 1], namely H N = spanf N k g N k=0 . For the assumptions on the input signal and its approximants, we'll takeU to be a compact (i.e. uniformly bounded and equicontinuous, by the Arzela-Ascoli theorem) subset of H 2 [0;T ], andU N to be spaces of linear splines on [0;T ], with N the map taking a functionu2U to its linear spline interpolant on the uniform mesh of size =T=N; in other words, the piecewise linear function satisfying (j) =u(j) for j = 0;:::;N. U N and U share the H 2 [0;T;R m ] norm. Furthermore we constrain our optimization so that in our space U we x u(0) = (q)' 0 = 0. If Q is a function space, we again approximate it as well by Q N , the space of linear interpolating splines on [0; 1] of mesh size 1=N. Therefore, as before, the hypotheses of Lemma 5.16 are satised, so we may approximate a global minimizer of J(q;u) = 1 X i=1 (q; ~ u (i) ) ~ y (i) 2 n +k(q;u) ~ yk 2 n +R(q;u) in QU by a subsequence of global minimizers to J N (q N ;u N ) = 1 X i=1 N (q N ; ~ u (i) ) ~ y (i) 2 n + N (q N ;u N ) ~ y 2 n +R(q N ;u N ) in Q N U N . 75 Chapter 8 Numerical results We now present results of the implementation of our approximate deconvolution in the cases of Chapters 5 and 6. 8.1 Adjoint method We will solve the nite-dimensional optimization problems via gradient descent. This will require the computation of the gradient vectorrJ N (q M ;u P ) of the approximate cost J N with respect to the variables (q M ;u P )2Q M U P . An ecient way to computerJ N (q M ;u P ), since it depends on a dynamical system, is the adjoint method. Here we follow the exposition in [23]. Suppose we would like to compute the gradient of a function J with respect to a parameter p (in our case p = (q;u)), and suppose that the function J depeneds on a \state" (p), so that J(p) =F ((p);p). Further suppose that the state satises an algebraic constraint written G((p);p) = 0. Then rJ(p) =D p F (;p) = @F @ @ @p + @F @p and @G @ @ @p = @G @p : (8.1.1) From the equation on the right, @ @p = @G @ 1 @G @p : Directly calculating @ @p from this equation would require solving a large matrix system for each component of p. Such a task could easily be intractable, for example, in the case where represents the state of a nite 76 dimensional dynamical system andG encodes the dierence equations. Reducing the number of computations required for this is where the adjoint method is useful. Substituting into the left-hand equation of (8.1.1) and rearranging parentheses, rJ(p) = @F @ @G @ 1 ! @G @p + @F @p : If we denote the parenthesized term by , it satises @G @ T = @F @ T : In terms of , reintroducing the regularization term, we have @J N @p = T @G @p + @F @p : Thus to compute the gradient, we need only compute and the derivatives of F and G with respect to and p. We will demonstrate the use of the adjoint method for the discrete time transdermal alcohol model; it is much the same for the delay model. Suppose N is xed. Recalling from (5.4.1), J N (q M ;u P ) = 1 X i=1 L N (q M )~ u (i) ~ y (i) 2 n + L N (q M )u P ~ y 2 n +R(q M ;u P ) = X i=1 n X j=1 jC N (q M )' N j (q M ; ~ u (i) ) ~ y (i) j j 2 + n X j=1 jC N (q M )' N j (q M ;u P ) ~ y j j 2 +R(q M ;u P ): Let b A N (q M ) =T N (;q M )2R (N+1)(N+1) b B N (q M ) = (I N+1 b A N (q M )) +N (q M )2R (N+1)1 b C N (q M ) = [1; 0;:::; 0]2R N+1 77 For any single training episode, our dynamical system is described for all j = 1;:::n by ' j = b A N (q M )' j1 + b B N (q M )u P j1 ' 0 = 0 y j = b C N (q M )' j where ' j 2 R N+1 is a column vector which denotes the state of the system and u j = P P k=0 P k (j)c k , j = 1;:::;n. Let ~ ' = (' 1 ;' 2 ;:::;' n ) T , u = (u j ) j 2R n . For a xed drinking episode, our system can be described simultaneously for all timesteps j = 1;:::n by 0 B B B B B B B B B B B B B B @ I N+1 0 0 0 0 b A N (q M ) I N+1 0 0 0 0 b A N (q M ) I N+1 0 0 . . . . . . . . . . . . . . . 0 b A N (q M ) I N 1 C C C C C C C C C C C C C C A 0 B B B B B B B B B B B B B B @ ' 1 ' 2 . . . . . . ' n 1 C C C C C C C C C C C C C C A (I n b B N (q M ))u P = 0 n(N+1) : Alternatively, Tridiag( b A N (q M );I N+1 ; 0)~ ' + (I n b B N (q M ))u P = 0; where Tridiag denotes a block tridiagonal matrix and denotes the Kronecker product of matrices. Put more succinctly, ~ A N (q M )~ ' + ~ B N (q M )u P = 0 with ~ A N (q M )2R n(N+1)n(N+1) and ~ B N (q M )2R n(N+1)n . Now consider the individual dynamical systems corresponding to the training episodes. ' (i) j = b A N (q M )' (i) j1 + b B N (q M )~ u (i) j1 ; y (i) j = b C N (q M )' (i) j ; ' (i) 0 = 0; 78 for episodes i = 1; 2;::: and timesteps j = 1;:::n. Write ~ ' = ( ~ ' j ) n j=1 for the state corresponding to the test episode, with ~ ' j 2R n the state at time j. ' j = b A N (q M )' j1 + b B N (q M )u P j1 y j = b C N (q M )' j ' 0 = 0 and let = [' (1) ;:::;' () ; ~ '] T 2R (+1)n(N+1) ~ U = [~ u (1) ;::: ~ u () ] T 2R n A N (q M ) =I +1 ~ A N (q M )2R (+1)n(N+1)(+1)n(N+1) B N 1 (q M ) = 0 B B @ I ~ B N (q M ) 0 (N+1)nn 1 C C A 2R (+1)n(N+1)n B N 2 (q M ) = 0 B B @ 0 (N+1)nn ~ B N (q M ) 1 C C A 2R (+1)n(N+1)n ; Then an equation describing our total system is G(;q M ;u P ) =A N (q M ) +B N 1 (q M ) ~ U +B N 2 (q M )u P = 0: Now let ~ C N (q M ) =I n b C N (q M )2R nn(N+1) ~ ~ C N (q M ) =I +1 ~ C N (q M )2R (+1)n(+1)n(N+1) Y = [~ y (1) ;:::; ~ y () ; ~ y] T 2R (+1)n 79 where ~ y denotes output data for the episode to be deconvolved. Then, writing p = (q M ;u P ), and using the fact that in our case b C N (q M ) = b C N , F ((p);p) :=J N (q M ;u P ) = ~ ~ C N Y 2 R (+1)n +R(p): We seek argmin p F ((p);p) subject to G(;p) = 0. This is a problem appropriate for the adjoint method. The equation for in this case is easily seen to be [A N (q M )] T = ( ~ ~ C N Y )( ~ ~ C N ) T : Recall that our system described byG((p);p) \contains" all training episodesi = 1;:::, the test episode i = + 1, and all timesteps j = 1;:::n. For a given episode i, the equation for above takes the form [ ~ A N (q M )] T (i) = ( ~ C N ' (i) ~ y (i) )( ~ C N ) T = ((I n b C N )' (i) ~ y (i) )(I n b C N ) T : Since ~ A N (q M ) = Tridiag[ b A N (q M );I N+1 ; 0], its transpose is Tridiag[0;I N+1 ; b A N (q M ) T ]. Substituting this and multiplying out the equation for (i) , we obtain an equation for each timestep j: (i) j1 = b A N (q M ) T (i) j + ( b C N ' (i) j1 ~ y (i) j1 )( b C N ) T for each j =n;:::; 1, and (i) n = ( b C N ' (i) n ~ y (i) n )( b C N ) T : These equations are also valid for the test episode, that is, if we remove the superscripts (i). Now write u P = P c, where P ij =h P i ; P j i L2[0;T] is the linear spline basis matrix. Returning to the gradient computation, we haverJ N (p) = h @J N @q M ; @J N @c i . @J N @q M k = n X j=1 " 1 X i=1 [ (i) j ] T @ b A N (q M ) @q M k ' (i) j1 + @ b B N (q M ) @q M k ~ u (i) j1 ! + [ j ] T @ b A N (q M ) @q M k ~ ' j1 + @ b B N (q M ) @q M k u j1 !# @J N @c j = T j b B N (q M ) P : 80 8.2 Computation of derivatives of matrix exponentials We can compute the derivatives @ @q M i b A N (q M ) at the same time as b A N (q M ) = T N (q M ;) = e A N (q M ) by making use of a technique drawn from [26]. For t 0 and q M 2 Q M , set T N (t;q M ) = e tA N (q M ) . Then T N (t;q M ), t 0, is the unique principal fundamental matrix solution to the initial value problem for t 0 given by _ z(t) =A N (q M )z(t); z(0) =I: (8.2.1) Then setting N k (q M ;t) = @T N (t;q M )=@q M k , dierentiating (8.2.1) with respect to q M k , interchanging the order of dierentiation, and using the product rule, we obtain for each k _ N k (t;q M ) =A N (q M ) N k (t;q M ) + (@A N (q M )=@q M k )T N (t;q M ); N k (0;q M ) = 0: Combining the two initial value problems, we obtain for each k 2 6 6 4 @ @q M e tA N (q M ) e tA N (q M ) 3 7 7 5 = 2 6 6 4 N k (t;q M ) T N (t;q M ) 3 7 7 5 = exp 0 B B @ 2 6 6 4 A N (q M ) @A N (q M )=@q N k ) 0 A N (q M ) 3 7 7 5 t 1 C C A 2 6 6 4 0 I 3 7 7 5 8.3 Numerical results for semi-blind deconvolution in parabolic systems Before testing our deconvolution scheme on actual TAC data, we x articial system parameters and use the discrete time model y j (q M ;u P ) = j1 X k=0 C N (q M ) b A N (q M ) jk1 b B N (q M )u P k : An alcohol researcher, referred to from here on as patient 5122, wore the Giner WrisTAS TM 7 sensor for 18 days, during which she collected breathalyzer measurements at 30-minute intervals and maintained a drinking diary for all drinking episodes. The WrisTAS TM took measurements every 5 minutes, so we set = 1=12 hr. We use the discrete time model and xed parameters to generate articial TAC data from her 81 measured breath alcohol concentration (BrAC) data in order to test our deconvolution scheme's performance and demonstrate our convergence theorems. All operators in this section will be as in Chapter 4. In testing our deconvolution method on the transdermal alcohol model, we xed the measured BrAC and articially generated TAC from drinking episode 1 as the sole training episode for all tests on articially generated TAC. We have three approximation indices: N, which denotes discretization of the state; P, which denotes discretization of the input signal; and M, which denotes discretization of the system parameter q, if it is functional and hence merits approximation. Othewrise Q M = Q. Linear splines were used for all approximations. In particular, for each j = 1;:::;n, the state ' j in X N measured at time j is represented by anN + 1-vector of spline coecients for linear splines on the uniform mesh of size 1=N on [0; 1]. The entire input signal u2U P is represented by a P + 1-vector c of linear spline coecients, so that u(t) = P (t)c, where P (t) = ( P i (t)) i is a row vector of linear spline basis functions on a uniform mesh of size T=P on [0;T], where T is a xed stopping time. Similarly, when M > 0, q 1 (x) will be represented in Q M by an M + 1-vector of linear spline coecients on a uniform mesh of size 1=M on [0; 1]. We examine several testing scenarios. In each, we minimize for some xed number of training episodes, J N (q M ;u P ) = 1 X i=1 N (q M ; ~ u (i) ) ~ y (i) 2 n + N (q M ;u P ) ~ y 2 n +R(q M ;u P ) with regularization R(q M ;u P ) = 1 kuk 2 2 + 2 k _ uk 2 2 + 3 jq M j over Q M U P , where U P = ( P X i=0 c P i P i : c P i 2 [0; 100] ) ; where the P i , i = 0;:::;P are linear splines on [0;T] on a uniform mesh of size T=P. We perform our minimizations using MATLAB's fmincon routine. In previous work on this approximate deconvolution problem (see e.g. [32]), a two-stage training approach was used: rst, the optimal model parameter q was t to known input/output data using the approximate model N ; then, the input signal u producing the sampled output from the system with parameter q was 82 determined via optimization. That approach, though not technically optimal since it solves two optimization problems in succession, allows one to t optimal regularization parameters 1 , 2 , 3 , during the rst phase. Our solution does not allow for this systematic inference, because the system parameter and input signal are t simultaneously. Put heuristically, the derivatives of J N with respect to the regularization parameters are always positive, so any gradient-based scheme will drive them to 0. Our regularization parameters are therefore determined through pilot tests conducted on a portion of generated data not used in testing. Furthermore, because typical values of BAC and TAC, measured in %, are on the order of hundredths, we multiply them by 10 to bring them to the same scale as the typical system parameter values to improve the optimization routine. The optimization routines were found to be highly sensitive to the initial search values of q and u. 8.3.1 Varying state discretization N For the rst few tests, we let q = (q 1 ;q 2 ) = (1;:3), so that Q =Q M R 2 is nite-dimensional. We then let N = 128 and generate observations y (i) = N (q; ~ u (i) ) = (y j (q; ~ u (i) )) n j=1 as dened in Chapter 5, where ~ u (i) denotes the actual sampled BrAC signal in the rst drinking episode logged by the subject wearing the WrisTAS bracelet. In this testing paradigm, we use only the input/output data (~ u (1) ;y (1) ) as our training data. We choose P so that U P is the space of linear splines on a uniform 30-minute mesh of a 12-hour interval. We then x 1 =:0002, 2 =:00001, and optimize J N (q;u) over QU P with initial search values (q 0 ;u 0 ) = (0:85; 0:5; 0;:::; 0), where the test episode ~ y is taken to be the simulated TAC data from actual BrAC episodes i = 1; 2; 3; 4. We perform our approximate optimization for increasing values of N, namely N = 4; 8; 16; 32; 64. The deconvolved signals for these increasing values of N, i.e. for increasing discretization of the (simulated) epidermis, are pictured in Figure 8.1. Even the coarsest level of discretization appears sucient to capture the system's dynamics. 83 Figure 8.1: Patient 5122 BrAC drinking episodes 1-4 estimated using linear splines on 30-minute mesh from simulated TAC (using q 1 (x) = 1, q 2 = :3) for varying resolution of state discretization, 1 = :0002, 2 =:00001 In Table 8.1 are tabulated mean values of various error measures comparing the actual BrAC to the deconvolved BrAC, computed across test episodes 1 through 4. The trained parameter values, tabulated below in 8.2 and 8.2, also appear to have converged at the coarsest level of discretization. N L 2 error L 1 error jAUC errorj jpeak time errorj jpeak height errorj 4 0.0185 0.0850 0.0270 0.3125 0.0334 8 0.0194 0.0894 0.0160 0.3125 0.0271 16 0.0192 0.0894 0.0141 0.3125 0.0273 32 0.0188 0.0872 0.0203 0.3125 0.0302 64 0.0183 0.0863 0.0157 0.3125 0.0250 Table 8.1: Average BrAC deconvolution error results, patient 5122, test episodes 1-4, varying resolution of state discretization, 30-minute mesh, spatially constant diusivity q 1 8.3.2 Varying estimated input signal resolution P Now we x N = 8 and vary the mesh size, so that U P are spaces of linear splines on uniform meshes of size 10, 15, 20, and 30 minutes, respectively. Depicted results of the deconvolution and tabulated error measures are in Figure 8.2 and Table 8.4; trained parameter values are given in Tables 8.5 and 8.6. 84 N 1 2 3 4 4 1.110 1.037 1.128 1.056 8 1.051 1.040 1.129 1.097 16 1.067 1.030 1.140 1.101 32 1.067 1.039 1.074 1.111 64 1.055 1.032 1.131 1.049 Table 8.2: Trained q 1 N , patient 5122, drinking episodes 1-4, simulated TAC (actual = 1) N 1 2 3 4 4 0.302 0.302 0.304 0.306 8 0.304 0.302 0.304 0.305 16 0.304 0.302 0.304 0.305 32 0.304 0.302 0.306 0.304 64 0.304 0.302 0.304 0.307 Table 8.3: Trained q 2 N , patient 5122, drinking episodes 1-4, simulated TAC (actual = 0:3) Figure 8.2: Patient 5122 BrAC drinking episodes 1-4 estimated using linear splines on meshes of varying resolution from simulated TAC (using q 1 (x) = 1, q 2 = :3) for state discretization N = 8, 1 = :0002, 2 =:00001 Samples/hr L 2 error L 1 error jAUC errorj jpeak time errorj jpeak height errorj 2 0.0194 0.0894 0.0160 0.3125 0.0271 3 0.0199 0.0927 0.0288 0.1042 0.0240 4 0.0251 0.1077 0.0752 0.1458 0.0347 6 0.0197 0.0847 0.0205 0.1042 0.0270 Table 8.4: Average BrAC deconvolution error results, patient 5122, test episodes 1-4, varying input signal mesh resolution, N = 8, spatially constant diusivity q 1 Spl/hr 1 2 3 4 2 1.0505 1.0402 1.1292 1.0968 3 1.1242 1.0327 1.0898 1.0628 4 1.1364 1.0729 1.1773 1.1276 6 1.0896 1.0376 1.1176 1.0733 Table 8.5: Trained q 1 N , patient 5122, drinking episodes 1-4, N = 8, varying estimated signal resolution, simulated TAC (actual = 1) Spl/hr 1 2 3 4 2 0.3040 0.3019 0.3041 0.3046 3 0.3009 0.3022 0.3053 0.3060 4 0.3000 0.2993 0.3023 0.3028 6 0.3025 0.3020 0.3043 0.3056 Table 8.6: Trained q 2 N , patient 5122, drinking episodes 1-4, N = 8, varying estimated signal resolution, simulated TAC (actual = 1) 85 Figure 8.3: Patient 5122 BrAC drinking episodes 1-4 estimated using linear splines on 30-minute mesh from simulated TAC (using q 1 (x) = 3(x:5) 2 +:1, q 2 = :3) for state discretization N = 8, varying levels of assumed system parameter complexity, 1 =:0002, 2 =:00001; 3 =:003 8.3.3 Varying estimated system parameter complexity M Now suppose that our simulated TAC is no longer generated using a constant value of q 1 (x) =q 1 , and instead is dened by q 1 (x) = 3(x:5) 2 +:1 for x2 [0; 1]. We use this functional diusivity parameter to simulate TAC data in the same manner as before. We then x a level of system parameter complexity M with which to run our algorithm to deconvolve the signal u, allowing increasing numbers of linear splines to estimate the parameter q 1 (x). In other words, we take Q M = Q M 1 [";], where Q M 1 are spaces of linear splines on a uniform mesh of size 1=M on [0; 1] with each coecient in [";]. The algorithm does a poor job of determining the quadratic parameter, as evidenced in 8.4, indicating that it may not be identiable, but the introduction of these extra degrees of freedom into the model appears to improve the deconvolution algorithm's performance on deconvolving test output data. 86 M L 2 error L 1 error jAUC errorj jpeak time errorj jpeak height errorj 1 0.0427 0.1842 0.0668 2.9167 0.1059 2 0.0802 0.2047 0.4625 1.4792 0.1705 4 0.0257 0.1160 0.0642 2.6875 0.0755 8 0.0155 0.0676 0.0275 0.1667 0.0175 Table 8.7: Average BrAC deconvolution error results, patient 5122, test episodes 1-4, 30-minute mesh, N = 8, quadratically varying diusivity q 1 (x) = 3(x:5) 2 +:1 Figure 8.4: Trained [q M 1 ] N (x), test episodes 1- 4, varying parameter complexity M, 30-minute mesh, N = 8, actual q 1 (x) = 3(x:5) 2 +:1 M 1 2 3 4 1 0.3109 0.3110 0.3082 0.3139 2 0.3900 0.3187 0.3093 0.3907 4 0.3026 0.3011 0.3051 0.3060 8 0.3024 0.3009 0.3038 0.3049 Table 8.8: Trained q 2 N , patient 5122, test episodes 1-4, varying parameter complexity M, 30-minute mesh, N = 8, actual q 2 = 0:3 87 8.4 Numerical results for semi-blind deconvolution of blood/breath alcohol concentration from transdermal alcohol concentration As stated, one of the committee members wore a WrisTAS TM 7 bracelet for 18 days, during which she collected breathalyzer measurements sampled at 30-minute intervals during normal drinking episodes. After partitioning the resultant data, we were left with 11 drinking episodes. Two episodes were excluded: one because the measured TAC signal did not return to 0 after 12 hours, and the other because the subject brie y removed the TAC device during the drinking episode. Therefore, after excluding these data, we were left with 9 episodes of paired input/output (BrAC/TAC) data. The sampled breathalyzer data were t to cubic splines and used as a proxy for blood alcohol content (BAC) ~ u (i) (t),i = 1;:::; 9,t2 [0; 12], wheret was measured in hours. We then resampled the splines at 5 minute intervals to obtain zero-order hold approximations to the BrAC signals. We tested our algorithm on actual paired BrAC/TAC data sampled from the skin using the WrisTAS in three paradigms, xing N = 16, choosing the constant-diusivity model, and deconvolving the BrAC as a sum of linear splines on a uniform 20-minute mesh on the interval t = 0 to t = 12 hours. We chose regularization parameters 1 =:006, 2 =:0004 and took (0:2;:::; 0:2; 0;:::; 0) as our initial values for the linear spline coecients, which were 0:2 for the rst four hours and 0 for the last eight. We then ran our deconvolution algorithm in three scenarios, diering the numbers of training episodes given to our algorithm. When taking as sampled test output ~ y to be deconvolved the actual TAC readings for drink- ing episodei, we used as training data in the rst scenario episodei+1 only (modulo 9), in the second scenario the 3 training episodesi+1,i+2, andi+3 (modulo 9), and in the third scenario all 8 episodes besides episodei. L 2 error L 1 error jAUC errorj jpeak time errorj jpeak height errorj 1 0.2179 0.5463 1.3242 4.6759 0.4905 3 0.1230 0.3456 0.6729 3.1852 0.3032 8 0.1295 0.3683 0.6309 3.3889 0.2975 Table 8.9: Average BrAC deconvolution error results, patient 5122, test episodes 1-9, varying numbers of training episodes on 20-minute mesh, N = 16, assumed spatially constant diusivity q 1 88 Figure 8.5: Patient 5122 BrAC drinking episodes 1-9 estimated using linear splines on 20-minute mesh from measured TAC, constant q 1 (x), N = 16, 1 =:006, 2 =:0004. 1 2 3 4 5 6 7 8 9 1 4447.9203 0.2955 2.9491 16.9401 769.3717 0.6869 2.6579 0.6413 1.0924 3 0.5623 0.5461 2.0699 13.0445 0.7810 0.7841 0.7513 0.4396 0.6646 8 0.6016 0.5982 1.2590 0.4835 0.6215 0.6040 0.5955 0.6148 0.6977 Table 8.10: Trainedq 1 N , patient 5122, drinking episodes 1-9, estimated for 20-minute mesh,N = 16, measured TAC 1 2 3 4 5 6 7 8 9 1 0.98657 1.1879 1.4577 1.0386 0.71866 1.8344 1.6824 0.96312 1.2337 3 1.3158 1.3028 1.4419 1.3542 0.97915 1.0974 1.0931 1.1214 1.29 8 1.2545 1.2422 1.2996 1.2276 1.2407 1.2446 1.2402 1.2357 1.289 Table 8.11: Trainedq 2 N , patient 5122, drinking episodes 1-9, estimated for 20-minute mesh,N = 16, measured TAC 89 These data appear to conrm that our algorithm works as it should in mitigating wild variability in system parameters for isolated drinking episodes. When only one training episode was used as in [32], test episode i was trained on episode i + 1; a cursory look at the data suggests that episodes 2 and 6, used as the training episodes for deconvolving episodes 1 and 5, respectively, in the rst deconvolution paradigm, may have been anomalous in some non-obvious way. 8.5 Numerical results for semi-blind deconvolution in delay systems We now test our approximate deconvolution algorithm on the delay model from Chapter 6. Consider the delay system prescribed by _ x(t) =a 0 x(t) 0:1x(t`) + 0:1u(t`); x 0 = (0; 0); y(t) =x(t); for n = 1, a 0 ;`2R unknown parameters and u(t)2L 2 [0;T ;R m ]. Therefore our set of admissible initial data isS =f(0; 0; 0)g. So our compactness assumptions on the admissible parameter set are satised, we'll assume `2 [20; 0], a 0 2 [10; 0]. In the notation from Chapter 6, we have A 0 () =a 0 , A 1 () =0:1, B 1 () = 0:1, C 0 () = 1, and the rest of the constituent operators of L(q), ^ B(q), ^ C(q) zero. For ' = (' 0 ;' 1 ;' 2 )2D(A(q)), A(q)' = (a 0 ' 1 (0) 0:1' 1 (`) + 0:1' 2 (`); _ ' 1 ; _ ' 2 ) B(q)v = (0; 0; 0 0 v); 90 so that the dynamical system under study can be written as the inhomogenous Cauchy problem _ '(t) =A(q)' t +B(q)u(t); ' 0 = (0; 0; 0); y(t) =' 0 t : Now we examine the approximating systems to the above, letting q N = ( N ;` N ), N = 1; 2:::;. We have m =n =p = 1. Further, because = 1, we have g N (s) = 1 for all s. Let M N = ` N 6N Tridiag([1;:::; 1]; [2; 4;:::; 4; 2]; [1;:::; 1])2R (N+1)(N+1) ; and letM N 2 R NN be the lower right submatrix ofM N . Then in the notation of the nal section of Chapter 6, M N (q N ) = diag(M N 11 (q N );M N 22 (q N )), where (M N 11 (q N )) ij = i1 j1 + (M N ) ij (M N 22 (q N )) ij = (M N ) ij : 91 We proceed to compute K N (q N ) = diag(K N 11 (q N );K N 22 (q N )). Using g N (s) = 1, we have K N (q N )(w N 1 ;w N 2 ) =f N (L(q N ) N + ^ B(q N )u N B 0 (q N )u N (0); _ N ; _ u N ) =f N (L(q N ) N w N 1 + ^ B(q N ) N w N 2 B 0 (q N ) N (0)w N 2 ; _ N w N 1 ; _ N w N 2 ) =f N (L(q N ) N ; _ N ; 0)w N 1 +f N ( ^ B(q N ) N B 0 (q N ) N (0); 0; _ N )w N 2 = N (0) T L(q N ) N + Z 0 ` N N (s) T _ N (s)ds; N (0) T ^ B(q N ) N + Z 0 ` N N (s) T _ N (s)ds 0 B B @ w N 1 w N 2 1 C C A = N (0) T (a 0 N (0) 0:1 N (` N )) + Z 0 ` N N (s) T _ N (s)ds + N (0) T (0:1 N (` N )) + Z 0 ` N N (s) T _ N (s)ds 0 B B @ w N 1 w N 2 1 C C A : We have N (0) T = (1; 0;:::; 0), N (0) = (0;:::; 0). Dene K N := Z 0 ` N N i (s) _ N j (s)ds ij = 1 2 Tridiag([1;:::; 1]; [1; 0;:::; 0;1]; [1;:::;1]): We letK N denote the lower right submatrix ofK N . Then (K N 11 (q N )) ij = i;1 j;1 a 0 0:1 i;1 j;N+1 + (K N ) ij (K N 22 (q N )) ij = (K N ) ij : Then, as before, for given q N , we can write the above inhomogeneous Cauchy problem in the basis f(( N j (0); N j ); 0); ((0; 0); N k ) : j = 0;:::;N; k = 1;:::;Ng 92 forX N (q N ) as _ w N (t) = ~ A N (q N )w N (t) + ~ B N (q N )u(t); t 0; w N (t) = (0; 0; 0); y N (t) = ~ C N (q N )w N (t); t 0; where ~ A N (q N ) =M N (q N ) 1 K N (q N ), ~ B N (q N ) =M N (q N ) 1 B N (q N ), ~ C N (q N ) =C N (q N )M N (q N ),w N (t)2 R 2N+1 for each N. We will takeUL 2 [0;T ] to be compact and we takeU N U to be spaces of linear spline approximations to functions inU of on a uniform mesh of sizeT=N. We can then considerU N to be linear spline approximations to functions in U. We then generate articial data ~ y (i) from our approximate model withN = 64, for some articially chosen input signals ~ u (i) ,i = 1;:::; 4 and the parameter valuesa 0 =0:5,` = 5. We choose sampling timest j = jT 4N , j = 0;:::; 4N (so we capture the dynamics of the linear splines) and sample y (i) j at those times. Fixing m = 3, i.e. using all non-test episodes for training, we use MATLAB's constrained minimization routine to minimize the cost functionals J N ( N ;u N ) = 1 X i=1 n X j=1 jy N (t j ;q N ; ~ u (i) ) ~ y (i) j j 2 + n X j=1 jy N (t j ;q N ;u N ) ~ y j j 2 +R(q N ;u N ) over N U N for N = 2; 4; 8; 16; 32, where R(q N ;u N ) = 1 u N 2 2 + 2 _ u N 2 2 for 1 = 0:003, 2 = 0:003, and we use initial optimization search values a 0 =0:3,` = 3. The deconvolution results are depicted in Figure 8.6 and average error measures computed across episodes 1 through 4 are tabulated in Table 8.12. The mean absolute peak time error, measured in the parabolic case, is omitted here because it was skewed by the estimates of the step function input signal, test episode 4. The trained parameter values are tabulated in Tables 8.13 and 8.14. 93 Figure 8.6: Deconvolved input signals, test episodes 1-4, delay system, varying levels of resolution of state and input mesh, 1 =:003, 2 =:003. N L 2 MSE L 1 error jAUC errorj jpeak height errorj 2 1.0011 0.1252 1.3854 0.1203 4 0.5260 0.1165 1.0776 0.1017 8 0.2352 0.0917 0.6744 0.0664 16 0.1312 0.0916 0.5136 0.0399 32 0.1118 0.1122 0.6125 0.0278 Table 8.12: Average delay system deconvolution error results, test episodes 1-4, varying levels of simultaneous state and input signal discretization N 1 2 3 4 2 -1.0327 -1.0195 -0.9952 -0.9957 4 -0.7759 -0.7686 -0.3636 -0.7542 8 -0.6171 -0.6176 -0.4239 -0.6057 16 -0.5349 -0.5360 -0.4336 -0.5314 32 -0.4950 -0.4951 -0.4142 -0.4922 Table 8.13: Trained a 0 N , varying levels of dis- cretization, test episodes 1-4, delay model (actual =0:5). N 1 2 3 4 2 4.2442 4.1085 10.9996 4.5438 4 4.7958 4.6368 3.9693 5.1420 8 4.9729 4.7954 4.2894 4.9252 16 4.7072 4.7601 4.4606 4.7535 32 4.6938 4.6950 4.5107 4.7030 Table 8.14: Trained ` N , varying levels of dis- cretization, test episodes 1-4, delay model (actual = 5). 94 The convergence of the approximate deconvolved signal to the true deconvolved signal as N!1 is evidently slower in N in the delay case than in the transdermal alcohol case. That is, we can observe the convergence in both parameters and signal as N increases. 95 Chapter 9 Conclusions We have shown above a method for approximate semi-blind deconvolution and parameter estimation in well posed innite-dimensional dynamical systems with unbounded input and output. We demonstrated the successes and limitations of our results in parabolic systems and hereditary systems. Compared to previous methods, our method is more optimal, but is generally more sensitive to diculties accompanying optimization such as choosing regularization coecients and initial search values. However, the exibility to choose an appropriate cost functional to optimize, especially one allowing incorporation of arbitrary amounts of population data, makes our algorithm a unique approach. We have provided evidence that if accurate continuous blood alcohol monitoring is desired using a transdermal alcohol sensor, then our optimization method is a strong candidate for use in its rmware. It may be preferable to the previously used two-stage optimization procedure of [32] which separately deduce system parameters and input signal. Not only is our model more accurate, but it provides evidence that one can avoid the cost and inconvenience of patient-device pair laboratory calibration by by using population data. It also circumvents previously reported issues of highly variable and outlying skin diusivity parameters due to humidity, skin variability, temperature, etc. by estimating system parameters simultaneously with the blood alcohol signal at runtime, as it were. However, what our methods gain in accuracy they appear to lose somewhat in robustness, in the sense that the deconvolution procedure is highly sensitive to choice of optimization parameters. The problem of unbounded input and output operators, mitigation of which usually requires a degree of regularity in the operators as in (H1) and (H2) for the delay case, was circumvented in the parabolic boundary 96 input case by instead requiring/leveraging smoothness in time of the input signal u(t). This approach to dynamical systems with unbounded input and output is to our knowledge somewhat novel; we are only aware of a version of such a result in the boundary control frameworks of [17] and [18]. 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Abstract (if available)

Abstract A theory for approximate deconvolution and parameter estimation in certain infinite-dimensional linear dynamical systems with unbounded input and output is presented. The theory is exposed by frameworks for approximate parameter estimation and deconvolution in parabolic systems in discrete and continuous time with input and output on the boundary, and to delay systems with unbounded input and output, e.g. systems with delays in the input. Numerical results demonstrating the performance of the approximations are presented. Finally, the theory is applied to a data from transdermal alcohol model and sensor together with training data in the form of breath alcohol measurements to calibrate the sensor and approximately deconvolve blood alcohol content over time.

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Creator Wintner, Moses Alexander Rubin (author)

Core Title Simultaneous parameter estimation and semi-blind deconvolution in infinite-dimensional linear systems with unbounded input and output

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Mathematics

Publication Date 10/19/2018

Defense Date 06/22/2018

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag approximation algorithms,boundary input,deconvolution,delay systems,dynamical systems,linear semigroups,mathematical modeling,OAI-PMH Harvest,optimization and control,parabolic systems

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Advisor Rosen, Gary (committee chair), Luczak, Susan (committee member), Wang, Chunming (committee member)

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