On a generalized Gaussian radial basis function: Analysis and applications. (March 2020)

• Record Type: Journal Article
• Authors: Karimi, N.; Kazem, S.; Ahmadian, D.; Adibi, H.; Ballestra, L.V.
• Is Part Of: Engineering analysis with boundary elements. Volume 112(2020)
• Page Start: 46
• Page End: 57
• Publication Date: 2020-03
• Subjects: Radial basis functions (RBFs) -- Gaussian basis -- Strictly positive definite functions -- Fourier transform; 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.2019.11.011
• Languages: English
• ISSNs: 0955-7997

Abstract

Abstract: We introduce a new infinitely smooth generalized Gaussian radial basis function (GGRBF) involving two shape parameters: ψ ( r ; ϵ ; ϵ 0 ) = φ ( r ; ϵ ) exp ( φ ( r ; ϵ 0 ) − 1 ), where φ( r ; ϵ) is the Gaussian basis associated with the shape parameter ϵ and ϵ0 is an auxiliary shape parameter. A thorough theoretical analysis is performed proving that the proposed radial basis function is strictly positive definite and, when it is used for function approximation, yields exponential convergence rate. In addition, we show, both analytically and numerically, that, by suitably choosing the second auxiliary parameter, the ill-conditioning problems that usually occur because of flat radial basis can be avoided. The reported numerical experiments highlight the very satisfactory performances of the novel radial basis function.