A discrete adapted hierarchical basis solver for radial basis function interpolation

被引：6

作者：

Castrillon-Candas, Julio E. ^{[1
]}

Li, Jun ^{[2
]}

Eijkhout, Victor ^{[3
]}

机构：

[1] King Abdullah Univ Sci & Technol, Thuwal 239556900, Saudi Arabia

[2] Schlumberger, Houston, TX 77056 USA

[3] Univ Texas Austin, Texas Adv Comp Ctr, Austin, TX 78712 USA

来源：

BIT NUMERICAL MATHEMATICS | 2013年 / 53卷 / 01期

关键词：

Radial basis function; Interpolation; Hierarchical basis; Integral equations; Fast summation methods; Stable completion; Lifting; Generalized least squares; Best linear unbiased estimator; SCATTERED-DATA INTERPOLATION; SPARSE REPRESENTATION; BASES; RECONSTRUCTION; DECOMPOSITION; ALGORITHM; EQUATIONS; GMRES;

D O I：

10.1007/s10543-012-0397-x

中图分类号：

TP31 [计算机软件];

学科分类号：

081202 ; 0835 ;

摘要：

In this paper we develop a discrete Hierarchical Basis (HB) to efficiently solve the Radial Basis Function (RBF) interpolation problem with variable polynomial degree. The HB forms an orthogonal set and is adapted to the kernel seed function and the placement of the interpolation nodes. Moreover, this basis is orthogonal to a set of polynomials up to a given degree defined on the interpolating nodes. We are thus able to decouple the RBF interpolation problem for any degree of the polynomial interpolation and solve it in two steps: (1) The polynomial orthogonal RBF interpolation problem is efficiently solved in the transformed HB basis with a GMRES iteration and a diagonal (or block SSOR) preconditioner. (2) The residual is then projected onto an orthonormal polynomial basis. We apply our approach on several test cases to study its effectiveness.

引用

页码：57 / 86

页数：30

共 56 条

[21]

BRAMBLE JH, 1990, MATH COMPUT, V55, P1, DOI 10.1090/S0025-5718-1990-1023042-6

[22] Local decomposition of refinable spaces and wavelets [J].

Carnicer, JM ;

Dahmen, W ;

Pena, JM .

APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS, 1996, 3 (02) :127-153

[23]

Carr JC, 2001, COMP GRAPH, P67, DOI 10.1145/383259.383266

[24] Regularizing properties of anisotropic radial basis functions [J].

Casciola, G. ;

Montefusco, L. B. ;

Morigi, S. .

APPLIED MATHEMATICS AND COMPUTATION, 2007, 190 (02) :1050-1062

[25] Shape preserving surface reconstruction using locally anisotropic radial basis function interpolants [J].

Casciola, G. ;

Lazzaro, D. ;

Montefusco, L. B. ;

Morigi, S. .

COMPUTERS & MATHEMATICS WITH APPLICATIONS, 2006, 51 (08) :1185-1198

[26] Spatially adapted multiwavelets and sparse representation of integral equations on general geometries [J].

Castrillón-Candás, JE ;

Amaratunga, K .

SIAM JOURNAL ON SCIENTIFIC COMPUTING, 2003, 24 (05) :1530-1566

[27] Fast evaluation of radial basis functions:: Methods for generalized multiquadrics in Rn [J].

Cherrie, JB ;

Beatson, RK ;

Newsam, GN .

SIAM JOURNAL ON SCIENTIFIC COMPUTING, 2002, 23 (05) :1549-1571

[28] Generalized hierarchical bases:: A wavelet-Ritz-Galerkin framework for Lagrangian FEM [J].

D'Heedene, S ;

Amaratunga, K ;

Castrillón-Candás, J .

ENGINEERING COMPUTATIONS, 2005, 22 (1-2) :15-37

[29] A note on the meshless method using radial basis functions [J].

Duan, Yong .

COMPUTERS & MATHEMATICS WITH APPLICATIONS, 2008, 55 (01) :66-75

[30] A meshless Galerkin method for Dirichlet problems using radial basis functions [J].

Duan, Yong ;

Tan, Yong-Ji .

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, 2006, 196 (02) :394-401

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