# Positive approximations of the inverse of fractional powers of SPD M-matrices

## TL;DR

This paper introduces a new rational approximation method for efficiently computing the inverse fractional powers of SPD M-matrices, preserving positivity and providing error bounds, with applications in fractional diffusion problems.

## Contribution

It develops and analyzes positive rational approximations of fractional matrix powers using BURA, ensuring preservation of key properties and offering sharp error estimates.

## Key findings

- The method preserves positivity of the approximation.
- Error bounds are established for the rational approximations.
- Numerical tests confirm theoretical properties and effectiveness.

## Abstract

This study is motivated by the recent development in the fractional calculus and its applications. During last few years, several different techniques are proposed to localize the nonlocal fractional diffusion operator. They are based on transformation of the original problem to a local elliptic or pseudoparabolic problem, or to an integral representation of the solution, thus increasing the dimension of the computational domain. More recently, an alternative approach aimed at reducing the computational complexity was developed. The linear algebraic system $\cal A^\alpha \bf u=\bf f$, $0< \alpha <1$ is considered, where $\cal A$ is a properly normalized (scalded) symmetric and positive definite matrix obtained from finite element or finite difference approximation of second order elliptic problems in $\Omega\subset\mathbb{R}^d$, $d=1,2,3$. The method is based on best uniform rational approximations (BURA) of the function $t^{\beta-\alpha}$ for $0 < t \le 1$ and natural $\beta$. The maximum principles are among the major qualitative properties of linear elliptic operators/PDEs. In many studies and applications, it is important that such properties are preserved by the selected numerical solution method. In this paper we present and analyze the properties of positive approximations of $\cal A^{-\alpha}$ obtained by the BURA technique. Sufficient conditions for positiveness are proven, complemented by sharp error estimates. The theoretical results are supported by representative numerical tests.