RBF approximation of three dimensional PDEs using Tensor Krylov subspace methods

TL;DR

This paper introduces tensor Krylov subspace algorithms for efficiently solving three-dimensional PDEs with radial basis functions, offering a computationally feasible alternative to traditional SVD methods.

Contribution

It develops novel tensor Krylov subspace algorithms using Einstein product for regularized solutions of tensor PDE problems, improving computational efficiency.

Findings

Algorithms effectively solve large-scale tensor PDE problems.

Regularized solutions obtained in few iterations.

Tensor methods outperform traditional approaches in computational cost.

Abstract

In this paper, we propose different algorithms for the solution of a tensor linear discrete ill-posed problem arising in the application of the meshless method for solving PDEs in three-dimensional space using multiquadric radial basis functions. It is well known that the truncated singular value decomposition (TSVD) is the most common effective solver for ill-conditioned systems, but unfortunately the operation count for solving a linear system with the TSVD is computationally expensive for large-scale matrices. In the present work, we propose algorithms based on the use of the well known Einstein product for two tensors to define the tensor global Arnoldi and the tensor Gloub Kahan bidiagonalization algorithms. Using the so-called Tikhonov regularization technique, we will be able to provide computable approximate regularized solutions in a few iterations.