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Quasi-Monte Carlo time-splitting methods for Schrödinger equation with Gaussian random potential

Zhizhang Wu, Zhiwen Zhang, Xiaofei Zhao

Abstract

In this paper, we study the Schrödinger equation with a Gaussian random potential (SE-GP) and develop an efficient numerical method to approximate the expectation of physical observables. The unboundedness of Gaussian random variables poses significant difficulties in both sampling and error analysis. Under time-splitting discretizations of SE-GP, we establish the regularity of the semi-discrete solution in the random space. Then, we introduce a non-standard weighted Sobolev space with properly chosen weight functions, and obtain a randomly shifted lattice-based quasi-Monte Carlo (QMC) quadrature rule for efficient sampling. This approach leads to a QMC time-splitting (QMC-TS) scheme for solving the SE-GP. We prove that the proposed QMC-TS method achieves a dimension-independent convergence rate that is almost linear with respect to the number of QMC samples. Numerical experiments illustrate the sharpness of the error estimate.

Quasi-Monte Carlo time-splitting methods for Schrödinger equation with Gaussian random potential

Abstract

In this paper, we study the Schrödinger equation with a Gaussian random potential (SE-GP) and develop an efficient numerical method to approximate the expectation of physical observables. The unboundedness of Gaussian random variables poses significant difficulties in both sampling and error analysis. Under time-splitting discretizations of SE-GP, we establish the regularity of the semi-discrete solution in the random space. Then, we introduce a non-standard weighted Sobolev space with properly chosen weight functions, and obtain a randomly shifted lattice-based quasi-Monte Carlo (QMC) quadrature rule for efficient sampling. This approach leads to a QMC time-splitting (QMC-TS) scheme for solving the SE-GP. We prove that the proposed QMC-TS method achieves a dimension-independent convergence rate that is almost linear with respect to the number of QMC samples. Numerical experiments illustrate the sharpness of the error estimate.

Paper Structure

This paper contains 16 sections, 18 theorems, 94 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Numerical method and main result
  4. Numerical method
  5. Dimension truncation
  6. Time discretization
  7. Quasi-Monte Carlo quadrature
  8. Main result
  9. Convergence analysis
  10. Preliminaries
  11. Dimension truncation error
  12. Temporal error
  13. QMC quadrature error
  14. Some discussions
  15. Numerical examples
  16. ...and 1 more sections

Key Result

Theorem 2.4

Let Assumptions assp: general--assp: convergence of V hold with $s \ge 3$, and we additionally assume eq: additional assumption if Assumption assp: summability of b holds with $p = 1$. If the Lie--Trotter splitting eq: Lie splitting is used in the QMC-TS method, then there exists a randomly shifted for all $t_n = n \tau \in [0, T]$ and some constant $C > 0$ independent of $m, \tau, N$, where $\ma

Figures (6)

  • Figure 1: Convergence in time for Example \ref{['expl: comparison of convergence rates']}
  • Figure 2: Convergence in the number of samples for Example \ref{['expl: comparison of convergence rates']}
  • Figure 3: Convergence in time for Example \ref{['expl: dimension independence in small D']}
  • Figure 4: Convergence in the number of samples for Example \ref{['expl: dimension independence in small D']}
  • Figure 5: Convergence of QMC in Example \ref{['expl: dimension independence in large D']}
  • ...and 1 more figures

Theorems & Definitions (41)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 31 more