A low-rank approach to the computation of path integrals

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...

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Bibliographic Details
Published in:Journal of computational physics 2016-01, Vol.305, p.557-574
Main Authors: Litsarev, Mikhail S., Oseledets, Ivan V.
Format: Article
Language:English
Subjects:
Algorithms
Approximation
Computation
Convolution
Diffusion
Feynman–Kac formula
Floating point arithmetic
Integrals
Low-rank approximation
Mathematical analysis
Multidimensional integration
Path integral
Skeleton approximation
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Summary: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(nrMlog⁡M+nr2M) 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.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2015.11.009