Analysis of BURA and BURA-based approximations of fractional powers of sparse SPD matrices

Abstract

Numerical methods applicable to the approximation of spectral fractional diffusion operators in multidimensional domains with general geometry are analyzed. Over the past decade, several approaches have been proposed to approximate the inverse operator \(\mathcal {A}^{-\alpha }\), \(\alpha \in (0,1)\). Despite their different origins, they can all be written as a rational approximation. Let the matrix \(\mathbb {A}\) be obtained after finite difference or finite element discretization of \(\mathcal {A}\). The BURA (Best Uniform Rational Approximation) method was introduced to approximate the inverse matrix \({\mathbb A}^{-\alpha }\) based on an approximation of the scallar function \(z^\alpha \), \(\alpha \in (0,1)\), \(z\in [0,1]\). In this paper we study BURA and BURA-based methods for fractional powers of sparse symmetric and positive definite (SPD) matrices, presentiing the concept, general framework and error analysis. Our contributions concern approximations of \(\mathbb {A}^{-\alpha }\) and \(\mathbb {A}^\alpha \) for arbitrary \(\alpha > 0\), thus significantly expanding the range of available currently results. Assymptotically accurate error estimates are obtained. The rate of convergence is exponential with respect to the degree of BURA. Numerical results are presented to illustrate and better interpret the theoretical estimates.