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

Boian Alexandrov, Gianmarco Manzini (), Erik W. Skau, Phan Minh Duc Truong and Radoslav G. Vuchov
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Boian Alexandrov: Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Gianmarco Manzini: Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Erik W. Skau: Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Phan Minh Duc Truong: Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Radoslav G. Vuchov: Computational Mathematics Department, Sandia National Laboratories, Albuquerque, NM 87185, USA

Mathematics, 2023, vol. 11, issue 3, 1-24

Abstract: 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.

Keywords: multidimensional numerical integration; highly-dimensional cubatures; Gauss-Legendre integration rule; Clenshaw-Curtis integration rule (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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