Asymptotic spectral properties and preconditioning of an approximated nonlocal Helmholtz equation with Caputo fractional Laplacian and variable coefficient wave number $μ$

TL;DR

This work develops a rigorous GLT-based framework to analyze the asymptotic spectral properties of a finite-difference discretization of a 2D nonlocal Helmholtz equation with a Caputo fractional Laplacian and a variable wave number $\mu$. It characterizes singular value and eigenvalue distributions for both nonpreconditioned and preconditioned systems, proving that properly designed $\tau$-based preconditioners yield tight clustering at 1 and enable fast Krylov convergence. The paper provides detailed results for bounded and unbounded $\mu$, using truncation and $a.c.s.$ arguments to extend GLT conclusions, and validates them through extensive numerical experiments. The findings support robust, scalable solvers for fractional nonlocal problems and point to extensions to higher-order discretizations and non-Cartesian domains.

Abstract

The current study investigates the asymptotic spectral properties of a finite difference approximation of nonlocal Helmholtz equations with a Caputo fractional Laplacian and a variable coefficient wave number $μ$, as it occurs when considering a wave propagation in complex media, characterized by nonlocal interactions and spatially varying wave speeds. More specifically, by using tools from Toeplitz and generalized locally Toeplitz theory, the present research delves into the spectral analysis of nonpreconditioned and preconditioned matrix-sequences. We report numerical evidences supporting the theoretical findings. Finally, open problems and potential extensions in various directions are presented and briefly discussed.

Figures (10)

• Figure 1: Singular/Eigenvalues of the matrix $\hat{A}_{\mathbf{n}}$ for $\mu(x,y)=\exp(\hat{\imath}(x+4y))$, $\alpha=1.2$ and $n^2=2^{12}$. The first (second) panel reports in blue the singular values (real part of the eigenvalues) and in red the equispaced sampling of $t_\alpha$ in nondecreasing order. The third panel reports the eigenvalues in the complex plane.
• Figure 2: Singular/Eigenvalues of the matrix $\hat{A}_{\mathbf{n}}$ for $\mu(x,y)=\exp(\hat{\imath}(x+4y))$, $\alpha=1.4$ and $n^2=2^{12}$. The first (second) panel reports in blue the singular values (real part of the eigenvalues) and in red the equispaced sampling of $t_\alpha$ in nondecreasing order. The third panel reports the eigenvalues in the complex plane.
• Figure 3: Singular/Eigenvalues of the matrix $\hat{A}_{\mathbf{n}}$ for $\mu(x,y)=\exp(\hat{\imath}(x+4y))$, $\alpha=1.6$ and $n^2=2^{12}$. The first (second) panel reports in blue the singular values (real part of the eigenvalues) and in red the equispaced sampling of $t_\alpha$ in nondecreasing order. The third panel reports the eigenvalues in the complex plane.
• Figure 4: Singular/Eigenvalues of the matrix $\hat{A}_{\mathbf{n}}$ for $\mu(x,y)=\exp(\hat{\imath}(x+4y))$, $\alpha=1.8$ and $n^2=2^{12}$. The first (second) panel reports in blue the singular values (real part of the eigenvalues) and in red the equispaced sampling of $t_\alpha$ in nondecreasing order. The third panel reports the eigenvalues in the complex plane.
• Figure 5: Singular/Eigenvalues of the matrix $\hat{A}_{\mathbf{n}}$ for $\mu(x,y)=-2+\exp(\hat{\imath}(3x+2y))$, $\alpha=1.2$ and $n^2=2^{12}$. The first (second) panel reports in blue the singular values (real part of the eigenvalues) and in red the equispaced sampling of $t_\alpha$ in nondecreasing order. The third panel reports the eigenvalues in the complex plane.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Remark 1
• Definition 3
• Theorem 1
• Theorem 2
• Theorem 3
• Remark 2
• Remark 3
• Lemma 1