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

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.