Optimal Solvers for Linear Systems with Fractional Powers of Sparse SPD Matrices

TL;DR

This paper introduces efficient algorithms for solving fractional powers of symmetric positive definite matrices arising from elliptic PDE discretizations, utilizing rational approximation and fast solvers, with numerical validation in 1D and 3D.

Contribution

It develops novel algorithms based on rational approximation for fractional elliptic operators, leveraging existing fast solvers for shifted systems, and demonstrates their effectiveness through numerical experiments.

Findings

Algorithms are efficient for 1D and 3D problems.

Numerical experiments confirm the method's effectiveness.

Approach outperforms traditional methods in speed and accuracy.

Abstract

In this paper we consider efficient algorithms for solving the algebraic equation ${\mathcal A}^\alpha {\bf u}={\bf f}$, $0< \alpha <1$, where ${\mathcal A}$ is a symmetric and positive definite matrix obtained form finite difference or finite element approximations of second order elliptic problems in ${\mathbb R}^d$, $d=1,2,3$. The method is based on the best uniform rational approximation of the function $t^{\beta-\alpha}$ for $0 < t \le 1$ and natural $\beta$, and the assumption that one has at hand an efficient method (e.g. multigrid, multilevel, or other fast algorithm) for solving equations like $({\mathcal A} +c {\mathcal I}){\bf u}= {\bf f}$, $c \ge 0$. The provided numerical experiments on model problems with ${\mathcal A}$ obtained by finite element approximation of elliptic equations in one and three spacial dimensions confirm the efficiency of the proposed algorithms.