ANALYSIS ON COMPUTATIONAL ISSUES WHEN APPROXIMATING FRACTIONAL POWERS OF SPARSE SPD MATRICES

Fractional diffusion has many applications in science and engineering as it models non-local processes and phenomena. However, numerically solving such problems involves systems of linear algebraic equations with dense matrices. For practical problems such systems can be extremely large and applying the usual LU factorization methods becomes an extremely expensive computational task. The Best Uniform Rational Approximation (BURA) and related methods have been developed in order to compute an approximation of the inverse (Formula presented) of a symmetric positive definite matrix (Formula presented) via an approximation of the scalar function tα, α ∈ (0, 1), t ∈ [0, 1]. Thus, the solution of a system of linear algebraic equations (Formula presented) can be computed approximately via computing several auxiliary systems with as sparse matrices as (Formula presented). This paper is devoted to the analysis of various numerical issues that arise in the process of the BURA computations when α ∈ (1, 2). Different reformulations of the classical BURA setting are considered in order to improve the stability of the computational process. Furthermore, since the direct BURA method does not preserve the symmetric positive definite property of (Formula presented), alternatives are proposed. They are based on a superposition of several BURA solvers with smaller αi ∈ (0, 1]. Theoretical and experimental analysis on their behavior is provided.