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

TL;DR

This paper introduces scalable Krylov-regularization methods for solving dense, ill-conditioned linear systems from multiquadric RBF discretizations of the 3D Helmholtz equation, improving stability and efficiency.

Contribution

It develops and evaluates three regularization techniques embedded in Krylov projections, including a hybrid scheme that enhances stability and reduces computational costs.

Findings

HKT matches or surpasses full SVD/Tikhonov accuracy

Inexpensive TSVD offers fast reconstructions with leading modes

Methods are effective on complex 3D geometries

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.