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Numerical analysis and efficient implementation of fast collocation methods for fractional Laplacian model on nonuniform grids

Meijie Kong, Hongfei Fu

TL;DR

This work develops fast, matrix-free collocation schemes for the 1D fractional Laplacian on nonuniform grids by leveraging sum-of-exponentials (SOE) to approximate the hypersingular kernel. For $α∈(0,1)$, the original fast scheme is proven uniquely solvable on general nonuniform grids, while a modified scheme extends robust solvability to the uniform-grid case for $α∈(0,2)$. Efficient Krylov-subspace solvers (BiCGSTAB) with a banded preconditioner and fast $O(NN_e)$ (or $O(N\tilde{N}_e)$) matrix-vector multiplications enable large-scale, memory-efficient computation, with $N_e,\tilde{N}_e=O(\log^2 N)$. The authors provide a rigorous max-norm error analysis on symmetric graded grids, showing convergence rates depending on the grading parameter $\kappa$ and regularity index $\sigma$, and validate the theory numerically, highlighting the modified scheme’s superior performance on uniform grids and the original scheme’s strength on nonuniform grids. Overall, the paper delivers robust, scalable methods for nonlocal fractional problems and offers practical guidance for achieving optimal convergence on graded grids.

Abstract

We propose a fast collocation method based on Krylov subspace iterative solver on general nonuniform grids for the fractional Laplacian problem, in which the fractional operator is presented in a singular integral formulation. The method is proved to be uniquely solvable on general nonuniform grids for $α\in(0,1)$, provided that the sum-of-exponentials (SOE) approximation is sufficiently accurate. In addition, a modified scheme is developed and proved to be uniquely solvable on uniform grids for $α\in(0,2)$. Efficient implementation of the proposed fast collocation schemes based on fast matrix-vector multiplication is carefully discussed, in terms of computational complexity and memory requirement. To further improve computational efficiency, a banded preconditioner is incorporated into the Krylov subspace iterative solver. A rigorous maximum-norm error analysis for $α\in(0,1)$ is presented on specific graded grids, which shows that the convergence order depends on the grading parameter. Numerical experiments validate the predicted convergence and demonstrate the efficiency of the fast collocation schemes.

Numerical analysis and efficient implementation of fast collocation methods for fractional Laplacian model on nonuniform grids

TL;DR

This work develops fast, matrix-free collocation schemes for the 1D fractional Laplacian on nonuniform grids by leveraging sum-of-exponentials (SOE) to approximate the hypersingular kernel. For , the original fast scheme is proven uniquely solvable on general nonuniform grids, while a modified scheme extends robust solvability to the uniform-grid case for . Efficient Krylov-subspace solvers (BiCGSTAB) with a banded preconditioner and fast (or ) matrix-vector multiplications enable large-scale, memory-efficient computation, with . The authors provide a rigorous max-norm error analysis on symmetric graded grids, showing convergence rates depending on the grading parameter and regularity index , and validate the theory numerically, highlighting the modified scheme’s superior performance on uniform grids and the original scheme’s strength on nonuniform grids. Overall, the paper delivers robust, scalable methods for nonlocal fractional problems and offers practical guidance for achieving optimal convergence on graded grids.

Abstract

We propose a fast collocation method based on Krylov subspace iterative solver on general nonuniform grids for the fractional Laplacian problem, in which the fractional operator is presented in a singular integral formulation. The method is proved to be uniquely solvable on general nonuniform grids for , provided that the sum-of-exponentials (SOE) approximation is sufficiently accurate. In addition, a modified scheme is developed and proved to be uniquely solvable on uniform grids for . Efficient implementation of the proposed fast collocation schemes based on fast matrix-vector multiplication is carefully discussed, in terms of computational complexity and memory requirement. To further improve computational efficiency, a banded preconditioner is incorporated into the Krylov subspace iterative solver. A rigorous maximum-norm error analysis for is presented on specific graded grids, which shows that the convergence order depends on the grading parameter. Numerical experiments validate the predicted convergence and demonstrate the efficiency of the fast collocation schemes.

Paper Structure

This paper contains 9 sections, 14 theorems, 120 equations, 2 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Fast collocation methods and efficient implementations
  3. Numerical scheme
  4. Unique solvability
  5. A modified scheme
  6. Efficient implementation
  7. Error analysis
  8. Numerical experiments
  9. Conclusions

Key Result

Lemma 2.1

For given $\beta \in(0,2)$, an absolute tolerance error $\epsilon$, a cut-off restriction $\Delta x>0$ and a given position $X>0$, there exists a positive integer $N_{e}$, positive quadrature points $\{\lambda_s\}_{s=1}^{N_{e}}$ and corresponding positive weights $\{\theta_s\}_{s=1}^{N_{e}}$ satisfy where the number of exponentials satisfies

Figures (2)

  • Figure 1: Values of $g_{\alpha}(N)$ with respect to $\alpha\in(0.001,0.999)$
  • Figure 2: Comparisons of the two fast collocation schemes. From top to bottom: $\alpha=1.0, 1.5, 1.7$

Theorems & Definitions (33)

  • Lemma 2.1: JZZZ17
  • Remark 1
  • Remark 2
  • Theorem 2.2
  • proof
  • Remark 3
  • Remark 4
  • Lemma 2.3
  • proof
  • Theorem 2.4
  • ...and 23 more