It is well known that radial basis function interpolants suffer from bad conditioning if the basis of translates is used. In the recent work by Pazouki and Schaback (2011), [5], the authors gave a quite general way to build stable and orthonormal bases for the native space associated to a kernel on a domain . The method is simply based on the factorization of the corresponding kernel matrix. Starting from that setting, we describe a particular basis which turns out to be orthonormal in and in , where is a set of data sites of the domain . The basis arises from a weighted singular value decomposition of the kernel matrix. This basis is also related to a discretization of the compact operator , and provides a connection with the continuous basis that arises from an eigendecomposition of . Finally, using the eigenvalues of this operator, we provide convergence estimates and stability bounds for interpolation and discrete least-squares approximation.
A new stable basis for radial basis function interpolation
Santin, GabrieleMembro del Collaboration Group
2013-01-01
Abstract
It is well known that radial basis function interpolants suffer from bad conditioning if the basis of translates is used. In the recent work by Pazouki and Schaback (2011), [5], the authors gave a quite general way to build stable and orthonormal bases for the native space associated to a kernel on a domain . The method is simply based on the factorization of the corresponding kernel matrix. Starting from that setting, we describe a particular basis which turns out to be orthonormal in and in , where is a set of data sites of the domain . The basis arises from a weighted singular value decomposition of the kernel matrix. This basis is also related to a discretization of the compact operator , and provides a connection with the continuous basis that arises from an eigendecomposition of . Finally, using the eigenvalues of this operator, we provide convergence estimates and stability bounds for interpolation and discrete least-squares approximation.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.