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

TL;DR

This paper develops an exponential sum approximation for fractional powers of operators, enabling efficient low-rank tensor computations and providing theoretical insights into the low-rank structure of solutions to fractional differential equations.

Contribution

It introduces a new exponential sum approximation for fractional operators that preserves low-rank tensor structures and predicts low-rank solution approximability.

Findings

Efficient approximation of fractional operator actions on tensors.

Theoretical prediction of low-rank solution structures.

Applicability to various low-rank tensor formats.

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 $x = \mathcal A^{-\alpha} c$, where $\mathcal A$ is a discretization of the classical Laplacian, and $\alpha$ a fractional exponent between $0$ and $1$. In this work, we derive an exponential sum approximation for $f(z) =z^{-\alpha}$ that is accurate over $[1, \infty)$ 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.