Wavelet kernel function based multiscale LSSVM for elliptic boundary value problems

Abstract

Recently, Mehrkanoon and Suykens designed a least square support vector machine (LSSVM) for learning solutions to partial differential equations in [1], where the Gaussian radial basis function is used as the kernel of the LSSVM. The purpose of the present paper is twofold: firstly, we extend the Gaussian kernel to wavelet kernel; and secondly, we propose a multiscale scheme, by noticing the multiscale nature of the wavelet kernel functions. The multiscale algorithm consists of a sequence of residual corrections to the solution of the partial differential equation, in which different scale parameters are employed to accommodate information at different scales. However, the direct computation of the multiscale problem would be difficult due to the mixing of different scales. To resolve this problem, we further introduce a multilevel algorithm, which decompose the multiscale algorithm into multiple levels: on the first level, a coarse data set and a large scale parameter are chosen and the target function is interpolated in this data set to capture the large-scale variations of the target function; next, on the second level, a smaller scale parameter is used to interpolate the residuals on a finer data set, capturing the finer details. The sum of both interpolants obviously better approximates the target function at the data sites on the finer data set. This process can be further applied to finer and finer scales till the anticipated accuracy is achieved. Note that sometimes the correction on the finer scale is necessary only at some local sites, thus an adaptive algorithm with only local corrections on finer scale is introduced. The numerical tests on some linear second order elliptic boundary value problems show the efficiency of the multilevel algorithm and the adaptive algorithm.