Abstract
Motivated by the recent multilevel sparse kernel-based interpolation (MuSIK) algorithm proposed in Georgoulis et al. (SIAM J. Sci. Comput. 35, 815–832, 2013), we introduce the new quasi-multilevel sparse interpolation with kernels (Q-MuSIK) via the combination technique. The Q-MuSIK scheme achieves better convergence and run time when compared with classical quasi-interpolation. Also, the Q-MuSIK algorithm is generally superior to the MuSIK methods in terms of run time in particular in high-dimensional interpolation problems, since there is no need to solve large algebraic systems. We subsequently propose a fast, low complexity, high-dimensional positive-weight quadrature formula based on Q-MuSIKSapproximation of the integrand. We present the results of numerical experimentation for both quasi-interpolation and quadrature in high dimensions.
Highlights
Over the last half century, numerical methods have obtained much attention, among mathematicians and in the scientific and engineering communities.High dimensionality usually causes some problems for mathematical modelling on gridded data
Radial basis function (RBF) approximation is one of the tools which is effective in highdimensions when the data is scattered in its domain; see e.g. [17]
As can be seen, the sparse grid treatment itself already requires the use of anisotropic basis functions since interpolation of data with anisotropic distribution of data sites in the domain requires special consideration
Summary
Over the last half century, numerical methods have obtained much attention, among mathematicians and in the scientific and engineering communities. Smoothness of the basis functions (such as Gaussians) means that RBF methods have been successfully applied for the solution of smooth partial differential equations; see [12] Another powerful tool for multidimensional problems is quasi-interpolation, which is widely used in scientific computation, mechanics and engineering. In this paper we use the same scheme with quasi-interpolation QSIK and Q-MuSIK respectively instead of SIK and MuSIK, and observe similar good results in high dimensions. More detailed this study showed that SIK and MuSIK scheme are interpolatory for the special case of scaled Gaussian kernels In addition to this a numerical integration algorithm is proposed in [7], based on interpolating the (high-dimensional) integrand.
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