Improving Greedy Algorithms for Rational Approximation

TL;DR

This paper introduces two improved greedy algorithms for rational approximation of fractional Laplace operators, enhancing preconditioning techniques for multiphysics problems with promising numerical results.

Contribution

The work develops two novel greedy algorithms for rational approximation, improving existing methods and demonstrating their effectiveness in preconditioning fractional operators.

Findings

Algorithms achieve non-increasing approximation error.

Numerical results confirm improved performance in preconditioning.

Methods are flexible and applicable to other approximation problems.

Abstract

When developing robust preconditioners for multiphysics problems, fractional functions of the Laplace operator often arise and need to be inverted. Rational approximation in the uniform norm can be used to convert inverting those fractional operators into inverting a series of shifted Laplace operators. Care must be taken in the approximation so that the shifted Laplace operators remain symmetric positive definite, making them better conditioned. In this work, we study two greedy algorithms for finding rational approximations to such fractional operators. The first algorithm improves the orthogonal greedy algorithm discussed in [Li et al., SISC, 2024] by adding one minimization step in the uniform norm to the procedure. The second approach employs the weak Chebyshev greedy algorithm in the uniform norm. Both methods yield non-increasing error. Numerical results confirm the effectiveness of our proposed algorithms, which are also flexible and applicable to other approximation problems. Moreover, with effective rational approximations to the fractional operator, the resulting algorithms show good performance in preconditioning a Darcy-Stokes coupled problem.