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A new rational approximation algorithm via the empirical interpolation method

Aidi Li, Yuwen Li

TL;DR

The paper introduces the Rational Approximation via the Empirical Interpolation Method (rEIM), a dictionary-based approach that constructs partial-fraction rational approximants with shared, invariant poles to interpolate a family of parametrized functions. By leveraging entropy-number-based convergence theory, the authors establish sub-exponential convergence and demonstrate that the poles can be precomputed and reused across many target functions, yielding high efficiency for large sets of queries. The method is applied to space-fractional and evolution fractional PDEs, adaptive time-stepping, parameter-robust preconditioning, and matrix-exponential evaluations, with numerical experiments validating accuracy and efficiency. The work offers a principled framework for fast, scalable rational approximation in numerical analysis and scientific computing, with practical impact on fractional PDE solvers and related matrix-function computations.

Abstract

We present a new rational approximation algorithm based on the empirical interpolation method for interpolating a family of parametrized functions to rational polynomials with invariant poles, leading to efficient numerical algorithms for space-fractional differential equations, parameter-robust preconditioning, and evaluation of matrix functions. Compared to classical rational approximation algorithms, the proposed method is more efficient for approximating a large number of target functions. In addition, we derive a convergence estimate of our rational approximation algorithm using the metric entropy numbers. Numerical experiments are included to demonstrate the effectiveness of the proposed method.

A new rational approximation algorithm via the empirical interpolation method

TL;DR

The paper introduces the Rational Approximation via the Empirical Interpolation Method (rEIM), a dictionary-based approach that constructs partial-fraction rational approximants with shared, invariant poles to interpolate a family of parametrized functions. By leveraging entropy-number-based convergence theory, the authors establish sub-exponential convergence and demonstrate that the poles can be precomputed and reused across many target functions, yielding high efficiency for large sets of queries. The method is applied to space-fractional and evolution fractional PDEs, adaptive time-stepping, parameter-robust preconditioning, and matrix-exponential evaluations, with numerical experiments validating accuracy and efficiency. The work offers a principled framework for fast, scalable rational approximation in numerical analysis and scientific computing, with practical impact on fractional PDE solvers and related matrix-function computations.

Abstract

We present a new rational approximation algorithm based on the empirical interpolation method for interpolating a family of parametrized functions to rational polynomials with invariant poles, leading to efficient numerical algorithms for space-fractional differential equations, parameter-robust preconditioning, and evaluation of matrix functions. Compared to classical rational approximation algorithms, the proposed method is more efficient for approximating a large number of target functions. In addition, we derive a convergence estimate of our rational approximation algorithm using the metric entropy numbers. Numerical experiments are included to demonstrate the effectiveness of the proposed method.
Paper Structure (20 sections, 11 theorems, 86 equations, 7 figures, 1 table, 2 algorithms)

This paper contains 20 sections, 11 theorems, 86 equations, 7 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Rational Approximation via the Empirical Interpolation Method
  3. Convergence Estimate of the REIM
  4. Rational Orthogonal Greedy Algorithm
  5. Applications of the rEIM
  6. Fractional Order PDEs
  7. Evolution Fractional PDEs
  8. Adaptive Step-Size Control
  9. Preconditioning
  10. Approximation of Matrix Exponentials
  11. Convergence Analysis
  12. Variation Norm of Functions
  13. Convergence Rate of Entropy Numbers
  14. Numerical Experiments
  15. Approximation of Power Functions
  16. ...and 5 more sections

Key Result

Theorem 2.3

\newlabelthm:rEIMerror0 Let $L_n:=\sup _{0\neq g\in {\rm Span}\{\mathcal{D}\}} \frac{\left\|\Pi_n g\right\|_{{L^\infty(I)}}}{\|g\|_{{L^\infty(I)}}}$ and $S_n$ be the volume of the $n$-dimensional unit ball. For any $f\in \mathscr{L}_1(\mathcal{D})$, the rEIM (Algorithm alg:rEIM) with $\mathcal{B}=

Figures (7)

  • Figure 1: Opposite poles $b_i$ in the rEIM (left); rEIM interpolation points $x_i$ (middle); Lebesgue constant $L_n$ (right).
  • Figure 2: Maximum norm errors of rational approximation algorithms for $x^{-s}$ on $[10^{-6},1]$.
  • Figure 3: Graded grid $\mathcal{T}_1$ with 4225 vertices (left); graded grid $\mathcal{T}_2$ with 19585 vertices (right).
  • Figure 4: rEIM $L^\infty$ error for $x^{-s}$ on $[10^{-8},1]$ (left); FEM $L^2$ errors for $s=0.25$, $N$ is the number of grid vertices (right).
  • Figure 5: $L^\infty$ error of the rEIM for $F_{0.5}$ (left) and $F_{1}$ (right).
  • ...and 2 more figures

Theorems & Definitions (21)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Proof 1
  • Corollary 2.4
  • Corollary 2.5
  • Lemma 4.1
  • Proof 2
  • Corollary 4.2
  • Proof 3
  • ...and 11 more