Kernel-based approximation of the Koopman generator and Schr\"odinger
operator
- URL: http://arxiv.org/abs/2005.13231v3
- Date: Fri, 25 Dec 2020 18:23:49 GMT
- Title: Kernel-based approximation of the Koopman generator and Schr\"odinger
operator
- Authors: Stefan Klus, Feliks N\"uske, Boumediene Hamzi
- Abstract summary: We show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems.
The resulting algorithms are applied to molecular dynamics and quantum chemistry examples.
- Score: 0.3093890460224435
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: 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
Schr\"odinger 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 Schr\"odinger 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.
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