A discrete adapted hierarchical basis solver for radial basis function interpolation

被引：0
作者：
Julio E. Castrillón-Candás
Jun Li
Victor Eijkhout
机构：
[1] King Abdullah University of Science and Technology,Texas Advanced Computing Center
[2] Schlumberger,undefined
[3] University of Texas at Austin,undefined
来源：
BIT Numerical Mathematics | 2013年 / 53卷
关键词：
Radial basis function; Interpolation; Hierarchical basis; Integral equations; Fast summation methods; Stable completion; Lifting; Generalized least squares; Best linear unbiased estimator; 65D05; 65D07; 65F25; 65F10; 62J05; 41A15;
D O I：
暂无
中图分类号：
学科分类号：
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
收藏
页码：57 / 86
页数：29
相关论文
共 83 条
[1]  
Alpert B.(1993)Wavelet-like bases for the fast solution of second-kind integral equations SIAM J. Sci. Comput. 14 159-184
[2]  
Beylkin G.(1993)A class of bases in SIAM J. Math. Anal. 24 246-262
[3]  
Coifman R.(2001) for the sparse representation of integral operators Int. J. Numer. Methods Eng. 52 239-271
[4]  
Rokhlin V.(2001)Surface wavelets: a multiresolution signal processing tool for 3D computational modeling SIAM J. Math. Anal. 32 1272-1310
[5]  
Alpert B.K.(1999)Fast evaluation of radial basis functions: methods for four-dimensional polyharmonic splines Adv. Comput. Math. 11 253-270
[6]  
Amaratunga K.(2000)Fast fitting of radial basis functions: methods based on preconditioned GMRES iteration SIAM J. Sci. Comput. 22 1717-1740
[7]  
Castrillon-Candas J.(1991)Fast solution of the radial basis function interpolation equations: domain decomposition methods Commun. Pure Appl. Math. 44 141-183
[8]  
Beatson R.(1996)Fast wavelet transforms and numerical algorithms I Appl. Comput. Harmon. Anal. 3 127-153
[9]  
Cherrie J.(2006)Local decomposition of refinable spaces Comput. Math. Appl. 51 1185-1198
[10]  
Ragozin D.(2007)Shape preserving surface reconstruction using locally anisotropic RBF interpolants Appl. Math. Comput. 190 1050-1062
123456789