Multilevel quasi-interpolation on a sparse grid with the Gaussian

作者：Usta Fuat^*; Levesley Jeremy

来源：Numerical Algorithms, 2018, 77(3): 793-808.

DOI：10.1007/s11075-017-0340-y

• Quasi-interpolation
• Multilevel
• Sparse grids
• Hyperbolic crosses
• Quadrature
• High dimension

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.

• 出版日期2018-3