RBFCUB: A numerical package for near-optimal meshless cubature on general polygons

In this paper we improve the cubature rules discussed in [12] for the computation of integrals by radial basis functions (RBFs). More precisely, we introduce in the context of meshless cubature a leave-one-out cross validation criterion for the optimization of the RBF shape parameter. This choice allows us to get highly reliable and accurate results for any kind of both infinity and finite regularity RBF. The efficacy of this approximation scheme is tested by numerical experiments on complicated polygonal regions. The related Matlab software is provided to the scientific community in [5].

[1]  Shmuel Rippa,et al.  An algorithm for selecting a good value for the parameter c in radial basis function interpolation , 1999, Adv. Comput. Math..

[2]  Alvise Sommariva,et al.  Compressed algebraic cubature over polygons with applications to optical design , 2020, J. Comput. Appl. Math..

[3]  Gregory E. Fasshauer,et al.  On choosing “optimal” shape parameters for RBF approximation , 2007, Numerical Algorithms.

[4]  Roberto Cavoretto,et al.  An adaptive LOOCV-based refinement scheme for RBF collocation methods over irregular domains , 2020, Appl. Math. Lett..

[5]  R. Schaback.  Multivariate Interpolation and Approximation by Translates of a Basis Function the Space of Functions (1.1) Often , 1995 .

[6]  M. Vianello,et al.  RBF Moment computation and meshless cubature on general polygonal regions , 2021, Appl. Math. Comput..

[7]  M. Urner.  Scattered Data Approximation , 2016 .

[8]  Gregory E. Fasshauer,et al.  Meshfree Approximation Methods with Matlab , 2007, Interdisciplinary Mathematical Sciences.

[9]  R. Cavoretto,et al.  On the search of the shape parameter in radial basis functions using univariate global optimization methods , 2019, Journal of Global Optimization.

[10]  Gregory E. Fasshauer,et al.  Kernel-based Approximation Methods using MATLAB , 2015, Interdisciplinary Mathematical Sciences.

[11]  F. Marchetti.  The extension of Rippa's algorithm beyond LOOCV , 2021, Appl. Math. Lett..

[12]  R. Cavoretto,et al.  A two-stage adaptive scheme based on RBF collocation for solving elliptic PDEs , 2020, Comput. Math. Appl..

[13]  C. S. Chen,et al.  A novel RBF collocation method using fictitious centres , 2020, Appl. Math. Lett..

[14]  Robert Schaback,et al.  Error estimates and condition numbers for radial basis function interpolation , 1995, Adv. Comput. Math..