Low-rank tensor structure preservation in fractional operators by means of exponential sums

2022/08/10 by Angelo A. Casulli, Casulli, Angelo A., Leonardo Robol +1
Engineering · Mathematics · Physics and Astronomy · #Electromagnetic Scattering and Analysis #FOS: Mathematics #Numerical Analysis (math.NA) #Sparse and Compressive Sensing Techniques #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.2208.05189

openalex publication_date 2022/08/10 · openalex created_date 2022/08/12 · openalex updated_date 2026/07/28

Abstract

The use of fractional differential equations is a key tool in modeling non-local phenomena. Often, an efficient scheme for solving a linear system involving the discretization of a fractional operator is evaluating the matrix function x = \mathcal A^-α c, where \mathcal A is a discretization of the classical Laplacian, and α a fractional exponent between 0 and 1. In this work, we derive an exponential sum approximation for f(z) =z^-α that is accurate over [1, ∞) and allows to efficiently approximate the action of bounded and unbounded operators of this kind on tensors stored in a variety of low-rank formats (CP, TT, Tucker). The results are relevant from a theoretical perspective as well, as they predict the low-rank approximability of the solutions of these linear systems in low-rank tensor formats.

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