A low-rank approach to the computation of path integrals

被引：5

作者：

Litsarev, Mikhail S. ^{[1
]}

Oseledets, Ivan V. ^{[1
,2
]}

机构：

[1] Skolkovo Innovat Ctr, Skolkovo Inst Sci & Technol, Moscow 143026, Russia

[2] Russian Acad Sci, Inst Numer Math, Moscow 119333, Russia

来源：

JOURNAL OF COMPUTATIONAL PHYSICS | 2016年 / 305卷

基金：

俄罗斯科学基金会;

关键词：

Low-rank approximation; Feynman-Kac formula; Path integral; Multidimensional integration; Skeleton approximation; Convolution; MULTIDIMENSIONAL NONLOCAL OPERATORS; KRONECKER-PRODUCT APPROXIMATION; TENSOR APPROXIMATION; EQUATION; DIMENSIONALITY; REPRESENTATION; DECOMPOSITION; ALGORITHMS; MATRICES;

D O I：

10.1016/j.jcp.2015.11.009

中图分类号：

TP39 [计算机的应用];

学科分类号：

081203 ; 0835 ;

摘要：

We present a method for solving the reaction-diffusion equation with general potential in free space. It is based on the approximation of the Feynman-Kac formula by a sequence of convolutions on sequentially diminishing grids. For computation of the convolutions we propose a fast algorithm based on the low-rank approximation of the Hankel matrices. The algorithm has complexity of O(nrMlogM + nr(2)M) flops and requires O(Mr) floating-point numbers in memory, where n is the dimension of the integral, r << n, and M is the mesh size in one dimension. The presented technique can be generalized to the higher-order diffusion processes. (C) 2015 Elsevier Inc. All rights reserved.

引用

页码：557 / 574

页数：18

共 108 条

[91] TUCKER DIMENSIONALITY REDUCTION OF THREE-DIMENSIONAL ARRAYS IN LINEAR TIME
Oseledets, I. V.
Savostianov, D. V.
Tyrtyshnikov, E. E.
[J]. SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS, 2008, 30 (03) : 939 - 956
[92] Constructive Representation of Functions in Low-Rank Tensor Formats
Oseledets, I. V.
[J]. CONSTRUCTIVE APPROXIMATION, 2013, 37 (01) : 1 - 18
[93] TENSOR-TRAIN DECOMPOSITION
Oseledets, I. V.
[J]. SIAM JOURNAL ON SCIENTIFIC COMPUTING, 2011, 33 (05) : 2295 - 2317
[94] APPROXIMATION OF 2d x 2d MATRICES USING TENSOR DECOMPOSITION
Oseledets, I. V.
[J]. SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS, 2010, 31 (04) : 2130 - 2145
[95] BREAKING THE CURSE OF DIMENSIONALITY, OR HOW TO USE SVD IN MANY DIMENSIONS
Oseledets, I. V.
Tyrtyshnikov, E. E.
[J]. SIAM JOURNAL ON SCIENTIFIC COMPUTING, 2009, 31 (05) : 3744 - 3759
[96] Fast orthogonalization to the kernel of the discrete gradient operator with application to Stokes problem
Oseledets, Ivan
Muravleva, Ekaterina
[J]. LINEAR ALGEBRA AND ITS APPLICATIONS, 2010, 432 (06) : 1492 - 1500
[97] TT-cross approximation for multidimensional arrays
Oseledets, Ivan
Tyrtyshnikov, Eugene
[J]. LINEAR ALGEBRA AND ITS APPLICATIONS, 2010, 432 (01) : 70 - 88
[98] Lectures on the functional renormalization group method
Polonyi, J
[J]. CENTRAL EUROPEAN JOURNAL OF PHYSICS, 2003, 1 (01): : 1 - 71
[99] FAST MULTIDIMENSIONAL CONVOLUTION IN LOW-RANK TENSOR FORMATS VIA CROSS APPROXIMATION
Rakhuba, M. V.
Oseledets, I. V.
[J]. SIAM JOURNAL ON SCIENTIFIC COMPUTING, 2015, 37 (02) : A565 - A582
[100] Schneider R, 2009, J COMPUT MATH, V27, P360

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