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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Numerical analysis for inchworm Monte Carlo method: Sign problem and error growth
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by Zhenning Cai, Jianfeng Lu and Siyao Yang HTML | PDF
Math. Comp. 92 (2023), 1141-1209 Request permission

Abstract:

We consider the numerical analysis of the inchworm Monte Carlo method, which is proposed recently to tackle the numerical sign problem for open quantum systems. We focus on the growth of the numerical error with respect to the simulation time, for which the inchworm Monte Carlo method shows a flatter curve than the direct application of Monte Carlo method to the classical Dyson series. To better understand the underlying mechanism of the inchworm Monte Carlo method, we distinguish two types of exponential error growth, which are known as the numerical sign problem and the error amplification. The former is due to the fast growth of variance in the stochastic method, which can be observed from the Dyson series, and the latter comes from the evolution of the numerical solution. Our analysis demonstrates that the technique of partial resummation can be considered as a tool to balance these two types of error, and the inchworm Monte Carlo method is a successful case where the numerical sign problem is effectively suppressed by such means. We first demonstrate our idea in the context of ordinary differential equations, and then provide complete analysis for the inchworm Monte Carlo method. Several numerical experiments are carried out to verify our theoretical results.
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Additional Information
  • Zhenning Cai
  • Affiliation: Department of Mathematics, National University of Singapore, Level 4, Block S17, 10 Lower Kent Ridge Road, Singapore 119076
  • MR Author ID: 892483
  • Email: matcz@nus.edu.sg
  • Jianfeng Lu
  • Affiliation: Department of Mathematics, Department of Physics, and Department of Chemistry, Duke University, Box 90320, Durham, North Carolina 27708
  • MR Author ID: 822782
  • ORCID: 0000-0001-6255-5165
  • Email: jianfeng@math.duke.edu
  • Siyao Yang
  • Affiliation: Department of Mathematics, National University of Singapore, Level 4, Block S17, 10 Lower Kent Ridge Road, Singapore 119076
  • MR Author ID: 1264228
  • ORCID: 0000-0002-6651-6224
  • Email: siyao_yang@u.nus.edu
  • Received by editor(s): July 28, 2022
  • Received by editor(s) in revised form: June 3, 2022, and July 23, 2022
  • Published electronically: November 21, 2022
  • Additional Notes: The first author was supported by the Academic Research Fund of the Ministry of Education of Singapore under grant No. R-146-000-291-114. The second author was supported in part by the National Science Foundation via grants DMS-1454939 and DMS-2012286.
  • © Copyright 2022 American Mathematical Society
  • Journal: Math. Comp. 92 (2023), 1141-1209
  • DOI: https://doi.org/10.1090/mcom/3785
  • MathSciNet review: 4550323