Challenging the Curse of Dimensionality in Multidimensional Numerical Integration by Using a Low-Rank Tensor-Train Format

Alexandrov, Boian; Manzini, Gianmarco; Skau, Erik W.; Truong, Phan Minh Duc; Vuchov, Radoslav Georgiev

doi:10.3390/math11030534

Challenging the Curse of Dimensionality in Multidimensional Numerical Integration by Using a Low-Rank Tensor-Train Format

Journal Article · Thu Jan 19 00:00:00 EST 2023 · Mathematics

DOI:https://doi.org/10.3390/math11030534· OSTI ID:2311692

^[1]; ^[1]; ^[1]; ^[1]; Vuchov, Radoslav Georgiev ^[2]

Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

Numerical integration is a basic step in the implementation of more complex numerical algorithms suitable, for example, to solve ordinary and partial differential equations. The straightforward extension of a one-dimensional integration rule to a multidimensional grid by the tensor product of the spatial directions is deemed to be practically infeasible beyond a relatively small number of dimensions, e.g., three or four. In fact, the computational burden in terms of storage and floating point operations scales exponentially with the number of dimensions. This phenomenon is known as the curse of dimensionality and motivated the development of alternative methods such as the Monte Carlo method. The tensor product approach can be very effective for high-dimensional numerical integration if we can resort to an accurate low-rank tensor-train representation of the integrand function. In this work, we discuss this approach and present numerical evidence showing that it is very competitive with the Monte Carlo method in terms of accuracy and computational costs up to several hundredths of dimensions if the integrand function is regular enough and a sufficiently accurate low-rank approximation is available.

View Accepted Manuscript (DOE)

Research Organization:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States); Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE Laboratory Directed Research and Development (LDRD) Program; USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: 89233218CNA000001; NA0003525

OSTI ID:: 2311692

Alternate ID(s):: OSTI ID: 2441387

Report Number(s):: LA-UR--22-29661; SAND--2023-00119J

Journal Information:: Mathematics, Journal Name: Mathematics Journal Issue: 3 Vol. 11; ISSN 2227-7390

Publisher:: MDPICopyright Statement

Country of Publication:: United States

Language:: English

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