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Improving Greedy Algorithms for Rational Approximation

James H. Adler, Xiaozhe Hu, Xue Wang, Zhongqin Xue

TL;DR

The paper tackles efficient inversion of fractional operators arising in multiphysics preconditioners by developing uniform-norm rational approximations constructed via two greedy strategies. The first is an Improved Orthogonal Greedy Algorithm (IOGA) that adds a post-projection uniform-norm minimization step, and the second is a direct Weak Chebyshev Greedy Algorithm (WCGA) in the uniform norm; both preserve negative poles to ensure shifted operators remain SPD and guarantee nonincreasing uniform error. The authors validate these methods through function-approximation experiments and a Darcy–Stokes preconditioning test, showing robust performance, monotone error reduction, and practical efficiency with modest pole counts. The work demonstrates the flexibility of the approach and points to broad applicability to other approximation tasks and multiphysics preconditioners.

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.

Improving Greedy Algorithms for Rational Approximation

TL;DR

The paper tackles efficient inversion of fractional operators arising in multiphysics preconditioners by developing uniform-norm rational approximations constructed via two greedy strategies. The first is an Improved Orthogonal Greedy Algorithm (IOGA) that adds a post-projection uniform-norm minimization step, and the second is a direct Weak Chebyshev Greedy Algorithm (WCGA) in the uniform norm; both preserve negative poles to ensure shifted operators remain SPD and guarantee nonincreasing uniform error. The authors validate these methods through function-approximation experiments and a Darcy–Stokes preconditioning test, showing robust performance, monotone error reduction, and practical efficiency with modest pole counts. The work demonstrates the flexibility of the approach and points to broad applicability to other approximation tasks and multiphysics preconditioners.

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.
Paper Structure (13 sections, 2 theorems, 32 equations, 2 figures, 6 tables, 4 algorithms)

This paper contains 13 sections, 2 theorems, 32 equations, 2 figures, 6 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rational approximation
  4. Greedy algorithms
  5. Orthogonal greedy algorithm
  6. Weak Chebyshev greedy algorithm
  7. Improved Greedy Algorithms
  8. Algorithm 1: Improved Orthogonal Greedy Algorithm
  9. Algorithm 2: Weak Chebyshev Greedy Algorithm in the Uniform Norm
  10. Numerical results
  11. Function approximation
  12. Preconditioning of the Darcy-Stokes problem
  13. Conclusions

Key Result

Theorem 3.2

\newlabelOGAth0 The uniform error for the rational approximation obtained by the improved OGA satisfies where $\varphi_n$ is the rational approxiation for $f$ using the OGA with input integer $n$, and $\varphi_n^*$ is the one computed using the improved OGA.

Figures (2)

  • Figure 1: Number of poles needed for the improved OGA applied to the Darcy-Stokes problem.
  • Figure 2: Number of poles needed for the WCGA applied to the Darcy-Stokes problem.

Theorems & Definitions (5)

  • Remark 3.1
  • Theorem 3.2
  • Proof 1
  • Theorem 3.3
  • Proof 2