### 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(n^{2})) 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 10^{108} x 10^{108}. 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 language | English (US) |
---|---|

Pages (from-to) | 547-587 |

Number of pages | 41 |

Journal | SIAM Review |

Volume | 59 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2017 |

### Fingerprint

### 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

*SIAM Review*,

*59*(3), 547-587. https://doi.org/10.1137/15M1040827

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

Research output: Contribution to journal › Review article

*SIAM Review*, vol. 59, no. 3, pp. 547-587. https://doi.org/10.1137/15M1040827

}

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 -