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

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.