Document Type

Journal Article

Department/Unit

Department of Mathematics

Language

English

Abstract

Meshless collocation methods are often seen as a flexible alternative to overcome difficulties that may occur with other methods. As various meshless collocation methods gain popularity, finding appropriate settings becomes an important open question. Previously, we proposed a series of sequential-greedy algorithms for selecting quasi-optimal meshless trial subspaces that guarantee stable solutions from meshless methods, all of which were designed to solve a more general problem: “Let $A$ be an $M \times N$ matrix with full rank $M$; choose a large $M \times K$ submatrix formed by $K\leq M$ columns of $A$ such that it is numerically of full rank.” In this paper, we propose a block-greedy algorithm based on a primal/dual residual criterion. Similar to all algorithms in the series, the block-greedy algorithm can be implemented in a matrix-free fashion to reduce the storage requirement. Most significantly, the proposed algorithm reduces the computational cost from the previous $\mathcal{O}(K^4+NK^2)$ to at most $\mathcal{O}(NK^2)$. Numerical examples are given to demonstrate how this efficient and ready-to-use approach can benefit the stability and applicability of meshless collocation methods.

Keywords

Kansa method, kernel collocation, radial basis function, adaptive greedy algorithm, basis selection

Publication Date

2016

Source Publication Title

SIAM Journal on Scientific Computing

Volume

38

Issue

2

Start Page

A1224

End Page

A1250

Publisher

Society for Industrial and Applied Mathematics

DOI

10.1137/15M1037627

Link to Publisher's Edition

https://doi.org/10.1137/15M1037627

ISSN (print)

10648275

ISSN (electronic)

10957197

Included in

Mathematics Commons

Share

COinS