A direct prediction of the shape parameter—A purely scattered data approach. (September 2020)

• Record Type: Journal Article
• Authors: Luh, Lin-Tian
• Is Part Of: Engineering analysis with boundary elements. Volume 118(2020)
• Page Start: 277
• Page End: 284
• Publication Date: 2020-09
• Subjects: Radial basis function -- Multiquadric -- Inverse multiquadric -- Shape parameter -- Interpolation; 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.2020.05.018
• Languages: English
• ISSNs: 0955-7997

Abstract

Abstract: In this paper we present an approach which predicts directly without search the optimal choice of the shape parameter c contained in the multiquadrics ( − 1 ) ⌈ β 2 ⌉ ( c 2 + ∥ x ∥ 2 ) β 2, β > 0, and the inverse multiquadrics ( c 2 + ∥ x ∥ 2 ) β 2, β < 0 . Unlike the simplex scheme where the data points are required to be evenly spaced, as in a recent paper of the author, here we allow them to be arbitrarily scattered in the simplex, making it much more useful. Besides this, we aim at helping non-mathematicians use our approach and hence remove some complicated requirements for the domain. The drawback is that its theoretical ground is not so strong as in the evenly spaced data setting. However, experiments show that it works well. The experimentally optimal value of c coincides with the theoretically predicted one. Since the fill distance, which reflects the amount of data points needed, involved is always of reasonable size, this approach is supposed to be practically useful.