RBF approximation of three dimensional PDEs using tensor Krylov subspace methods. (June 2022)

• Record Type: Journal Article
• Authors: El Guide, M.; Jbilou, K.; Ratnani, A.
• Is Part Of: Engineering analysis with boundary elements. Volume 139(2022)
• Page Start: 77
• Page End: 85
• Publication Date: 2022-06
• Subjects: Krylov subspaces -- Linear tensor equations -- Tensors -- Global Arnoldi -- Global Golub–Kahan -- Einstein product; Boundary element methods -- Periodicals; Engineering mathematics -- Periodicals; Équations intégrales de frontière, Méthodes des -- Périodiques; Mathématiques de l'ingénieur -- Périodiques; Boundary element methods; Engineering mathematics; Periodicals; 620.00151
• DOI: 10.1016/j.enganabound.2022.02.019
• Languages: English
• ISSNs: 0955-7997

Abstract

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. The tensor formulation will allow us to develop an RBF approximation for high dimensional PDEs based on Hierarchical tensors and Adaptive Cross Approximation, which in turn reduces significantly the storage and computational costs, while the accuracy of the method is preserved. The performance of the proposed methods is illustrated with a variety of benchmark examples and large-scale industrial applications with high degrees of freedom.