IJAM: Volume 37, No. 6 (2024)

DOI: 10.12732/ijam.v37i6.7

ANALYSIS ON COMPUTATIONAL ISSUES

WHEN APPROXIMATING FRACTIONAL POWERS

OF SPARSE SPD MATRICES

Dimitar Slavchev 1,*, Stanislav Harizanov 1,2,†,

and Nikola Kosturski 1,‡

1 Institute of Information and Communication Technologies

Bulgarian Academy of Sciences

Acad. G. Bonchev Str. Bl. 25A, Sofia - 1113, BULGARIA

2 Institute of Mathematics and Informatics

Bulgarian Academy of Sciences

Acad. G. Bonchev Str. Bl. 8, Sofia - 1113, BULGARIA

Abstract. 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 $\mathbb{A}^{-\alpha}$ of a symmetric positive definite matrix via an approximation of the scalar function $t^\alpha, \alpha \in (0,1), t \in [0,1]$. Thus, the solution of a system of linear algebraic equations $\mathbb{A}^{\alpha}\mathbf{u} = \mathbf{f}$ can be computed approximately via computing several auxiliary systems with as sparse matrices as $\mathbb{A}$.

This paper is devoted to the analysis of various numerical issues that arise in the process of the BURA computations when $\alpha \in (1,2)$. 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 alternatives are proposed. They are based on a superposition of several BURA solvers with smaller

$\alpha_i \in (0,1]$. Theoretical and experimental analysis on their behavior is provided.

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How to cite this paper?
DOI: 10.12732/ijam.v37i6.7
Source: International Journal of Applied Mathematics
ISSN printed version: 1311-1728
ISSN on-line version: 1314-8060
Year: 2024
Volume: 37
Issue: 6

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