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Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator
Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of d...
Autores principales: | , , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
MDPI
2020
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7517260/ https://www.ncbi.nlm.nih.gov/pubmed/33286494 http://dx.doi.org/10.3390/e22070722 |
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author | Klus, Stefan Nüske, Feliks Hamzi, Boumediene |
author_facet | Klus, Stefan Nüske, Feliks Hamzi, Boumediene |
author_sort | Klus, Stefan |
collection | PubMed |
description | Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems. |
format | Online Article Text |
id | pubmed-7517260 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2020 |
publisher | MDPI |
record_format | MEDLINE/PubMed |
spelling | pubmed-75172602020-11-09 Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator Klus, Stefan Nüske, Feliks Hamzi, Boumediene Entropy (Basel) Article Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems. MDPI 2020-06-30 /pmc/articles/PMC7517260/ /pubmed/33286494 http://dx.doi.org/10.3390/e22070722 Text en © 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). |
spellingShingle | Article Klus, Stefan Nüske, Feliks Hamzi, Boumediene Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator |
title | Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator |
title_full | Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator |
title_fullStr | Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator |
title_full_unstemmed | Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator |
title_short | Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator |
title_sort | kernel-based approximation of the koopman generator and schrödinger operator |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7517260/ https://www.ncbi.nlm.nih.gov/pubmed/33286494 http://dx.doi.org/10.3390/e22070722 |
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