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In sum, both the storage requirement and the cost of contractions between tensors
represented in the low-rank CP format is dramatically reduced compared to the original
tensors. However, the contractions (except when one of the tensors is a two-index tensor)
lead to a rank increase. Thus, to preserve the low-rank representation, a rank reduction
procedure should be performed after each contraction.
Rank reduction. The rank-reduction procedure amounts to the minimization problem
of the difference between an original tensor A and its lower-rank approximation, ˜A
:
jjA ˜A
jj e (6.4)
where e determines the accuracy. The accelerated gradient (AG) algorithm22 has been
successfully employed in Ref. 12. The most time-consuming step in the AG algorithm is
the calculation of the gradient. The complexity of this step is O(d ˜R
n (˜R
+R)) where
d is the dimension of the tensor, and R and ˜R
are the ranks of A and ˜A
, respectively.
Thus, the cost of this step depends critically on the initial rank. The authors of
Ref. 12, who employed the trivial decomposition as the initial step for rank reduction,
have suggested that a better starting point can be provided by using RI or Cholesky
decomposed integrals.
To further reduce the cost of the rank reduction algorithm, the sliced factorization
procedure has been developed12, in which the initial reduction is performed for smaller
blocks of data. This algorithm can be trivially parallelized.
We plan to begin by implementing the parallelized AG procedure including sliced
reduction following the algorithm described in Ref. 12. We then will explore the follow-ing
approaches for improving the starting point for the rank reduction of the integrals:
165
Object Description
| Title | Development of predictive electronic structure methods and their application to atmospheric chemistry, combustion, and biologically relevant systems |
| Author | Epifanovskiy, Evgeny |
| Author email | epifanov@usc.edu; epifanov@usc.edu |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Chemistry |
| School | College of Letters, Arts and Sciences |
| Date defended/completed | 2011-03-21 |
| Date submitted | 2011 |
| Restricted until | Unrestricted |
| Date published | 2011-04-28 |
| Advisor (committee chair) | Krylov, Anna I. |
| Advisor (committee member) |
Wittig, Curt Johnson, Clifford |
| Abstract | This work demonstrates electronic structure techniques that enable predictive modeling of the properties of biologically relevant species. Chapters 2 and 3 present studies of the electronically excited and detached states of the chromophore of the green fluorescent protein, the mechanism of its cis-trans isomerization, and the effect of oxidation. The bright excited ππ∗ state of the chromophore in the gas phase located at 2.6 eV is found to have an autoionizing resonance nature as it lies above the electron detachment level at 2.4 eV. The calculation of the barrier for the ground-state cis-trans isomerization of the chromophore yields 14.8 kcal/mol, which agrees with an experimental value of 15.4 kcal/mol; the electronic correlation and solvent stabilization are shown to have an important effect. In Chapter 3, a one-photon two-electron mechanism is proposed to explain the experimentally observed oxidative reddening of the chromophore. Chapter 4 considers the excited states of uracil. It demonstrates the role of the one-electron basis set and triples excitations in obtaining the converged values of the excitation energies of the nπ∗ and ππ∗ states. The effects of the solvent and protein environment are included in some of the models.; Chapter 5 describes an implementation of the algorithm for locating and exploring intersection seams between potential energy surfaces. The theory is illustrated with examples from atmospheric and combustion chemistry. |
| Keyword | electronic structure theory; coupled clusters theory; equation of motion theory; organic chromophore; green fluorescent protein; uracil |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Provenance | Electronically uploaded by the author |
| Type | texts |
| Legacy record ID | usctheses-m3801 |
| Contributing entity | University of Southern California |
| Rights | Epifanovskiy, Evgeny |
| Repository name | Libraries, University of Southern California |
| Repository address | Los Angeles, California |
| Repository email | cisadmin@lib.usc.edu |
| Filename | etd-Epifanovskiy-4557 |
| Archival file | uscthesesreloadpub_Volume14/etd-Epifanovskiy-4557.pdf |
Description
| Title | Page 175 |
| Contributing entity | University of Southern California |
| Repository email | cisadmin@lib.usc.edu |
| Full text | In sum, both the storage requirement and the cost of contractions between tensors represented in the low-rank CP format is dramatically reduced compared to the original tensors. However, the contractions (except when one of the tensors is a two-index tensor) lead to a rank increase. Thus, to preserve the low-rank representation, a rank reduction procedure should be performed after each contraction. Rank reduction. The rank-reduction procedure amounts to the minimization problem of the difference between an original tensor A and its lower-rank approximation, ˜A : jjA ˜A jj e (6.4) where e determines the accuracy. The accelerated gradient (AG) algorithm22 has been successfully employed in Ref. 12. The most time-consuming step in the AG algorithm is the calculation of the gradient. The complexity of this step is O(d ˜R n (˜R +R)) where d is the dimension of the tensor, and R and ˜R are the ranks of A and ˜A , respectively. Thus, the cost of this step depends critically on the initial rank. The authors of Ref. 12, who employed the trivial decomposition as the initial step for rank reduction, have suggested that a better starting point can be provided by using RI or Cholesky decomposed integrals. To further reduce the cost of the rank reduction algorithm, the sliced factorization procedure has been developed12, in which the initial reduction is performed for smaller blocks of data. This algorithm can be trivially parallelized. We plan to begin by implementing the parallelized AG procedure including sliced reduction following the algorithm described in Ref. 12. We then will explore the follow-ing approaches for improving the starting point for the rank reduction of the integrals: 165 |
