Fast randomized iteration: Diffusion Monte Carlo through the lens of numerical linear algebra

Lek Heng Lim, Jonathan Weare

Research output: Contribution to journalReview article

Abstract

We review the basic outline of the highly successful diffusion Monte Carlo technique commonly used in contexts ranging from electronic structure calculations to rare event simulation and data assimilation, and propose a new class of randomized iterative algorithms based on similar principles to address a variety of common tasks in numerical linear algebra. From the point of view of numerical linear algebra, the main novelty of the fast randomized iteration schemes described in this article is that they have dramatically reduced operations and storage cost per iteration (as low as constant under appropriate conditions) and are rather versatile: we will show how they apply to the solution of linear systems, eigenvalue problems, and matrix exponentiation, in dimensions far beyond the present limits of numerical linear algebra. While traditional iterative methods in numerical linear algebra were created in part to deal with instances where a matrix (of size O(n2)) is too big to store, the algorithms that we propose are effective even in instances where the solution vector itself (of size O(n)) may be too big to store or manipulate. In fact, our work is motivated by recent diffusion Monte Carlo based quantum Monte Carlo schemes that have been applied to matrices as large as 10108 x 10108. We provide basic convergence results, discuss the dependence of these results on the dimension of the system, and demonstrate dramatic cost savings on a range of test problems.

Original languageEnglish (US)
Pages (from-to)547-587
Number of pages41
JournalSIAM Review
Volume59
Issue number3
DOIs
StatePublished - Jan 1 2017

Fingerprint

Numerical Linear Algebra
Linear algebra
Lens
Lenses
Iteration
Rare Event Simulation
Quantum Monte Carlo
Data Assimilation
Exponentiation
Iteration Scheme
Monte Carlo Techniques
Costs
Electronic Structure
Randomized Algorithms
Iterative methods
Convergence Results
Iterative Algorithm
Test Problems
Eigenvalue Problem
Electronic structure

Keywords

  • Diffusion Monte Carlo
  • Dimension reduction
  • Eigenvalue problem
  • Matrix exponentiation
  • Quantum Monte Carlo
  • Randomized algorithm

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Mathematics
  • Applied Mathematics

Cite this

Fast randomized iteration : Diffusion Monte Carlo through the lens of numerical linear algebra. / Lim, Lek Heng; Weare, Jonathan.

In: SIAM Review, Vol. 59, No. 3, 01.01.2017, p. 547-587.

Research output: Contribution to journalReview article

Lim, Lek Heng ; Weare, Jonathan. / Fast randomized iteration : Diffusion Monte Carlo through the lens of numerical linear algebra. In: SIAM Review. 2017 ; Vol. 59, No. 3. pp. 547-587.
@article{f5ff9907b44c4808977edf73cd5ad9e1,
title = "Fast randomized iteration: Diffusion Monte Carlo through the lens of numerical linear algebra",
abstract = "We review the basic outline of the highly successful diffusion Monte Carlo technique commonly used in contexts ranging from electronic structure calculations to rare event simulation and data assimilation, and propose a new class of randomized iterative algorithms based on similar principles to address a variety of common tasks in numerical linear algebra. From the point of view of numerical linear algebra, the main novelty of the fast randomized iteration schemes described in this article is that they have dramatically reduced operations and storage cost per iteration (as low as constant under appropriate conditions) and are rather versatile: we will show how they apply to the solution of linear systems, eigenvalue problems, and matrix exponentiation, in dimensions far beyond the present limits of numerical linear algebra. While traditional iterative methods in numerical linear algebra were created in part to deal with instances where a matrix (of size O(n2)) is too big to store, the algorithms that we propose are effective even in instances where the solution vector itself (of size O(n)) may be too big to store or manipulate. In fact, our work is motivated by recent diffusion Monte Carlo based quantum Monte Carlo schemes that have been applied to matrices as large as 10108 x 10108. We provide basic convergence results, discuss the dependence of these results on the dimension of the system, and demonstrate dramatic cost savings on a range of test problems.",
keywords = "Diffusion Monte Carlo, Dimension reduction, Eigenvalue problem, Matrix exponentiation, Quantum Monte Carlo, Randomized algorithm",
author = "Lim, {Lek Heng} and Jonathan Weare",
year = "2017",
month = "1",
day = "1",
doi = "10.1137/15M1040827",
language = "English (US)",
volume = "59",
pages = "547--587",
journal = "SIAM Review",
issn = "0036-1445",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "3",

}

TY - JOUR

T1 - Fast randomized iteration

T2 - Diffusion Monte Carlo through the lens of numerical linear algebra

AU - Lim, Lek Heng

AU - Weare, Jonathan

PY - 2017/1/1

Y1 - 2017/1/1

N2 - We review the basic outline of the highly successful diffusion Monte Carlo technique commonly used in contexts ranging from electronic structure calculations to rare event simulation and data assimilation, and propose a new class of randomized iterative algorithms based on similar principles to address a variety of common tasks in numerical linear algebra. From the point of view of numerical linear algebra, the main novelty of the fast randomized iteration schemes described in this article is that they have dramatically reduced operations and storage cost per iteration (as low as constant under appropriate conditions) and are rather versatile: we will show how they apply to the solution of linear systems, eigenvalue problems, and matrix exponentiation, in dimensions far beyond the present limits of numerical linear algebra. While traditional iterative methods in numerical linear algebra were created in part to deal with instances where a matrix (of size O(n2)) is too big to store, the algorithms that we propose are effective even in instances where the solution vector itself (of size O(n)) may be too big to store or manipulate. In fact, our work is motivated by recent diffusion Monte Carlo based quantum Monte Carlo schemes that have been applied to matrices as large as 10108 x 10108. We provide basic convergence results, discuss the dependence of these results on the dimension of the system, and demonstrate dramatic cost savings on a range of test problems.

AB - We review the basic outline of the highly successful diffusion Monte Carlo technique commonly used in contexts ranging from electronic structure calculations to rare event simulation and data assimilation, and propose a new class of randomized iterative algorithms based on similar principles to address a variety of common tasks in numerical linear algebra. From the point of view of numerical linear algebra, the main novelty of the fast randomized iteration schemes described in this article is that they have dramatically reduced operations and storage cost per iteration (as low as constant under appropriate conditions) and are rather versatile: we will show how they apply to the solution of linear systems, eigenvalue problems, and matrix exponentiation, in dimensions far beyond the present limits of numerical linear algebra. While traditional iterative methods in numerical linear algebra were created in part to deal with instances where a matrix (of size O(n2)) is too big to store, the algorithms that we propose are effective even in instances where the solution vector itself (of size O(n)) may be too big to store or manipulate. In fact, our work is motivated by recent diffusion Monte Carlo based quantum Monte Carlo schemes that have been applied to matrices as large as 10108 x 10108. We provide basic convergence results, discuss the dependence of these results on the dimension of the system, and demonstrate dramatic cost savings on a range of test problems.

KW - Diffusion Monte Carlo

KW - Dimension reduction

KW - Eigenvalue problem

KW - Matrix exponentiation

KW - Quantum Monte Carlo

KW - Randomized algorithm

UR - http://www.scopus.com/inward/record.url?scp=85031905851&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85031905851&partnerID=8YFLogxK

U2 - 10.1137/15M1040827

DO - 10.1137/15M1040827

M3 - Review article

AN - SCOPUS:85031905851

VL - 59

SP - 547

EP - 587

JO - SIAM Review

JF - SIAM Review

SN - 0036-1445

IS - 3

ER -