The kernel regularized learning algorithm for solving Laplace equation with Dirichlet boundary

Abstract

In this paper, we give an investigation on the problem of solving Laplace equation with the kernel regularized regression. We provide a Sobolev type space corresponding to the Dirichlet boundary value problem on a compact domain, and with which define a reproducing kernel space (RKS), which is used as the hypothesis space for constructing kernel regularized learning algorithm. We give theory analysis for the convergence of the learning algorithm, bound an upper bound for the error. The discussions show that the learning rate is controlled by a [Formula: see text]-functional corresponding to the RKS. As an application we give the learning rate in case that the domain is the unit ball. The simulations show that the algorithm has better fitting performance. The investigations show that the problem of solving an elliptic boundary problem can be attributed to constructing an orthonormal basis with respect to a bilinear form corresponding to the boundary value condition.