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Efficient Krylov-Regularization Solvers for Multiquadric RBF Discretizations of the 3D Helmholtz Equation

Mohamed El Guide, Khalide Jbilou, Kamal Lachhab, Driss Ouazar

TL;DR

This paper tackles the ill-conditioning of dense MQ-RBF discretizations for the 3D Helmholtz equation by embedding regularization directly into Krylov projections. It introduces three complementary strategies: Ine-TSVD, standard Tikhonov regularization, and a Hybrid Krylov–Tikhonov (HKT) scheme, with parameter choice via GCV or the L-curve, and efficient surrogate-based criteria on reduced spaces. Theoretical results establish existence, uniqueness, convergence, and smoothing under a discrete Picard condition, while extensive numerical tests on a unit cube, a unit sphere, and a pump-casing geometry demonstrate that HKT consistently offers the best accuracy-time balance, Ine-TSVD provides the fastest reconstructions for leading modes, and Tikhonov remains competitive when a full SVD is feasible. The work provides a robust, scalable framework for stable MQ-RBF Helmholtz solvers in complex 3D domains and lays groundwork for future extensions to high frequencies and more complex settings.

Abstract

Meshless collocation with multiquadric radial basis functions (MQ-RBFs) delivers high accuracy for the three-dimensional Helmholtz equation but produces dense, severely ill-conditioned linear systems. We develop and evaluate three complementary methods that embed regularization in Krylov projections to overcome this instability at scale: (i) an inexpensive TSVD that replaces the full SVD by a short Golub-Kahan bidiagonalization and a small projected SVD, retaining the dominant spectral content at greatly reduced cost; (ii) classical Tikhonov regularization with principled parameter choice (GCV/L-curve), expressed in SVD form for transparent filtering; and (iii) a hybrid Krylov-Tikhonov (HKT) scheme that first projects with Golub-Kahan and then selects the regularization parameter on the reduced problem, yielding stable solutions in few iterations. Extensive tests on canonical domains (cube and sphere) and a realistic industrial pump-casing geometry demonstrate that HKT consistently matches or surpasses the accuracy of full TSVD/Tikhonov at a fraction of the runtime and memory, while inexpensive TSVD provides the fastest viable reconstructions when only the leading modes are needed. These results show that coupling Krylov projection with TSVD/Tikhonov regularization provides a robust, scalable pathway for MQ-RBF Helmholtz methods in complex three-dimensional settings.

Efficient Krylov-Regularization Solvers for Multiquadric RBF Discretizations of the 3D Helmholtz Equation

TL;DR

This paper tackles the ill-conditioning of dense MQ-RBF discretizations for the 3D Helmholtz equation by embedding regularization directly into Krylov projections. It introduces three complementary strategies: Ine-TSVD, standard Tikhonov regularization, and a Hybrid Krylov–Tikhonov (HKT) scheme, with parameter choice via GCV or the L-curve, and efficient surrogate-based criteria on reduced spaces. Theoretical results establish existence, uniqueness, convergence, and smoothing under a discrete Picard condition, while extensive numerical tests on a unit cube, a unit sphere, and a pump-casing geometry demonstrate that HKT consistently offers the best accuracy-time balance, Ine-TSVD provides the fastest reconstructions for leading modes, and Tikhonov remains competitive when a full SVD is feasible. The work provides a robust, scalable framework for stable MQ-RBF Helmholtz solvers in complex 3D domains and lays groundwork for future extensions to high frequencies and more complex settings.

Abstract

Meshless collocation with multiquadric radial basis functions (MQ-RBFs) delivers high accuracy for the three-dimensional Helmholtz equation but produces dense, severely ill-conditioned linear systems. We develop and evaluate three complementary methods that embed regularization in Krylov projections to overcome this instability at scale: (i) an inexpensive TSVD that replaces the full SVD by a short Golub-Kahan bidiagonalization and a small projected SVD, retaining the dominant spectral content at greatly reduced cost; (ii) classical Tikhonov regularization with principled parameter choice (GCV/L-curve), expressed in SVD form for transparent filtering; and (iii) a hybrid Krylov-Tikhonov (HKT) scheme that first projects with Golub-Kahan and then selects the regularization parameter on the reduced problem, yielding stable solutions in few iterations. Extensive tests on canonical domains (cube and sphere) and a realistic industrial pump-casing geometry demonstrate that HKT consistently matches or surpasses the accuracy of full TSVD/Tikhonov at a fraction of the runtime and memory, while inexpensive TSVD provides the fastest viable reconstructions when only the leading modes are needed. These results show that coupling Krylov projection with TSVD/Tikhonov regularization provides a robust, scalable pathway for MQ-RBF Helmholtz methods in complex three-dimensional settings.

Paper Structure

This paper contains 24 sections, 8 theorems, 44 equations, 7 figures, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. The Multiquadric Radial Basis Function Method (MQ-RBF)
  3. The MQ-RBF Collocation Method
  4. Stabilizing the Solution: Truncated SVD and Tikhonov Regularization
  5. Krylov Subspace and Tikhonov Regularization Methods
  6. Inexpensive Truncated SVD (Ine-TSVD)
  7. Tikhonov Regularization and Parameter Choice
  8. Residual and solution norms in SVD variables
  9. Generalized cross-validation (GCV)
  10. L-curve criterion (detailed)
  11. Hybrid Krylov--Tikhonov via Golub--Kahan Projection
  12. Cheap GCV/L-curve on the projected problem
  13. Cost and storage
  14. Krylov Methods as Iterative Regularization: LSQR and GMRES
  15. Existence and Convergence Analysis
  16. ...and 9 more sections

Key Result

Theorem 4.1

Micchelli1986 \newlabelthm:micchelli-mq Let $\{X_j\}_{j=1}^N\subset\mathbb{R}^d$ be pairwise distinct. Define the MQ distance matrix $B\in\mathbb{R}^{N\times N}$ by $B_{ij}=\phi(\|X_i-X_j\|_2,\varepsilon)$. Then $B$ is conditionally positive definite of order $1$, i.e., Consequently, the augmented system is nonsingular and yields a unique interpolant.

Figures (7)

  • Figure 5.1: Collocation points in the unit cube for Example 1: (left) random distribution, (middle) uniform grid distribution, (right) Halton sequence.
  • Figure 5.2: Example 1 (Unit Cube): (Left) exact solution $u(x,y,z)$ given by \ref{['eq:exact_cube']}, and (Right) the approximate solution obtained by the HKT(140) method for $k=3$ with $N=2154$ uniformly distributed points. The approximate solution visually matches the exact solution, indicating the effectiveness of the regularized Krylov method.
  • Figure 5.3: Collocation points on the unit sphere for Example 2: (left) random distribution on the sphere, (middle) Halton sequence on the sphere, (right) uniform distribution.
  • Figure 5.4: Example 2 (Unit Sphere): (Left) geometry of the unit sphere domain with collocation points, and (Right) the approximate solution obtained by the HKT(180) method for $k=3$ with $N=2154$ uniformly distributed points (surface plot on the sphere). The solution is smooth and no spurious oscillations are visible, indicating effective regularization.
  • Figure 5.5: Geometry of the pump casing model (Example 3). This complex domain is used to test the proposed methods on an irregular three-dimensional geometry. Collocation points are distributed throughout the volume (surface not shown for clarity).
  • ...and 2 more figures

Theorems & Definitions (10)

  • Theorem 4.1: Micchelli's CPD(1) criterion specialized to MQ
  • Lemma 4.2: Complete monotonicity for MQ
  • Proposition 4.3: Generalized MQ and polynomial augmentation
  • Remark 4.4: Ill-conditioning vs. accuracy
  • Proposition 4.5: Existence and uniqueness for Tikhonov
  • Proposition 4.6: Existence and uniqueness for TSVD
  • Remark 4.7: Projected (hybrid) formulations and cheap parameter choice
  • Theorem 4.9: Convergence of Tikhonov under Assumption \ref{['ass:picard']}
  • Theorem 4.10: Convergence of TSVD (Ine-TSVD)
  • Theorem 4.11: Convergence and smoothing of the Hybrid Krylov--Tikhonov (via GKB)