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A model reduction method for solving the eigenvalue problem of semiclassical random Schrödinger operators

Panchi Li, Zhiwen Zhang

TL;DR

This work addresses the eigenvalue problem for semiclassical random Schrödinger operators with infinite-dimensional randomness. The authors develop a model-reduction pipeline combining stochastic-dimension truncation, multiscale finite element methods (MsFEM), proper orthogonal decomposition (POD), and quasi-Monte Carlo (qMC) integration to efficiently approximate eigenpairs and their statistics. They derive comprehensive error bounds that couple truncation, MsFEM discretization, POD approximation, and qMC integration, culminating in a total RMS error of $\mathcal{O}(H^6+\sqrt{\rho}+s^{-2/p+1}+N^{-\alpha})$ for the minimal eigenvalue and a parallel bound for mean functionals of eigenfunctions. Numerical experiments validate the predicted convergence, reveal superconvergence of the MsFEM, demonstrate substantial computational savings via the POD reduction, and illustrate eigenfunction localization under spatial randomness. The results indicate that the MsFEM-POD-qMC framework is a robust and efficient tool for simulating complex quantum systems governed by semiclassical random Schrödinger operators, including Anderson localization phenomena.

Abstract

In this paper, we compute the eigenvalue problem (EVP) for the semiclassical random Schrödinger operators, where the random potentials are parameterized by an infinite series of random variables. After truncating the series, we introduce the multiscale finite element method (MsFEM) to approximate the resulting parametric EVP. We then use the quasi-Monte Carlo (qMC) method to calculate empirical statistics within a finite-dimensional random space. Furthermore, using a set of low-dimensional proper orthogonal decomposition (POD) basis functions, the referred degrees of freedoms for constructing multiscale basis are independent of the spatial mesh. Given the bounded assumption on the random potentials, we then derive and prove an error estimate for the proposed method. Finally, we conduct numerical experiments to validate the error estimate. In addition, we investigate the localization of eigenfunctions for the Schrödinger operator with spatially random potentials. The results show that our method provides a practical and efficient solution for simulating complex quantum systems governed by semiclassical random Schrödinger operators.

A model reduction method for solving the eigenvalue problem of semiclassical random Schrödinger operators

TL;DR

This work addresses the eigenvalue problem for semiclassical random Schrödinger operators with infinite-dimensional randomness. The authors develop a model-reduction pipeline combining stochastic-dimension truncation, multiscale finite element methods (MsFEM), proper orthogonal decomposition (POD), and quasi-Monte Carlo (qMC) integration to efficiently approximate eigenpairs and their statistics. They derive comprehensive error bounds that couple truncation, MsFEM discretization, POD approximation, and qMC integration, culminating in a total RMS error of for the minimal eigenvalue and a parallel bound for mean functionals of eigenfunctions. Numerical experiments validate the predicted convergence, reveal superconvergence of the MsFEM, demonstrate substantial computational savings via the POD reduction, and illustrate eigenfunction localization under spatial randomness. The results indicate that the MsFEM-POD-qMC framework is a robust and efficient tool for simulating complex quantum systems governed by semiclassical random Schrödinger operators, including Anderson localization phenomena.

Abstract

In this paper, we compute the eigenvalue problem (EVP) for the semiclassical random Schrödinger operators, where the random potentials are parameterized by an infinite series of random variables. After truncating the series, we introduce the multiscale finite element method (MsFEM) to approximate the resulting parametric EVP. We then use the quasi-Monte Carlo (qMC) method to calculate empirical statistics within a finite-dimensional random space. Furthermore, using a set of low-dimensional proper orthogonal decomposition (POD) basis functions, the referred degrees of freedoms for constructing multiscale basis are independent of the spatial mesh. Given the bounded assumption on the random potentials, we then derive and prove an error estimate for the proposed method. Finally, we conduct numerical experiments to validate the error estimate. In addition, we investigate the localization of eigenfunctions for the Schrödinger operator with spatially random potentials. The results show that our method provides a practical and efficient solution for simulating complex quantum systems governed by semiclassical random Schrödinger operators.

Paper Structure

This paper contains 19 sections, 12 theorems, 156 equations, 10 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries on the semiclassical Schrödinger operator with random potentials
  3. Numerical approximations
  4. Stochastic dimension truncation
  5. MsFEM approximation
  6. A POD reduction method
  7. The qMC method
  8. Parametric regularity
  9. Convergence analysis
  10. Convergence analysis of the MsFEM for the EVP
  11. Dimension truncation error
  12. QMC error
  13. POD error
  14. Total error
  15. Numerical experiments
  16. ...and 4 more sections

Key Result

Lemma 2.1

For all $\boldsymbol{\omega} \in \Omega$ and $\lambda \in \mathds{R}$, define $\mathcal{A}_{\lambda}(\boldsymbol{\omega}; \cdot, \cdot): H_P^1(D)\times H_P^1(D) \rightarrow \mathds{R}$ to be the shifted bilinear form Restricted to the $L^2$-orthogonal complement of the eigenspace corresponding to $\lambda_1(\boldsymbol{\omega})$, denoted by $E(\boldsymbol{\omega}, \lambda_1(\boldsymbol{\omega}))^

Figures (10)

  • Figure 1: Numerical convergence rates of the FEM and MsFEM approximation for the EVP of the Schrödinger operator with the 1D double-well potential.
  • Figure 2: The checkboard potential and the numerical convergence rates of the FEM and MsFEM methods.
  • Figure 3: The basis functions and the error between the multiscale basis solved by \ref{['equ:optimal-problem-objective']}-\ref{['equ:optimal-problem']} and \ref{['equ:optimal-problem2']}. 1st column: sketches of basis functions. 2nd column: the case for $\omega \in \Omega_0$. 3rd column: the case for $\omega \in \Omega/\Omega_0$.
  • Figure 4: Numerical convergence rates with respect to $s$. The red and blue symbols denote the results corresponding to $q = 4/3$ and $q = 3$, respectively.
  • Figure 5: Numerical convergence rates of FEM and MsFEM-POD in physic space and the comparison of the CPU time.
  • ...and 5 more figures

Theorems & Definitions (27)

  • Lemma 2.1: gilbert2019analysis, Lemma 3.1
  • Proposition 3.1: benner2015surveyHolmes_Lumley_Berkooz_1996
  • Lemma 4.1
  • proof
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • proof
  • Theorem 5.1
  • Remark 5.1
  • ...and 17 more