Hermite collocation method for solving high dimensional problems

Abstract

In this paper, Hermite collocation method with Radial Basis Functions is applied to solve a high dimensional Poisson equation and an elliptic equation up to degree six. The resultant matrix generated from the Hermite method is positive definite, which guarantees the invertibility of the matrix. The numerical results indicate that the method provides an efficient algorithm for solving high dimensional problems. We solved the parabolic equation, ∆u = -dc²u, where d is the dimension of the space, Rd, and a point xεRd is the d-tuple, (E1, E2, …,Ed). Dirichlet boundary condition of the space are assumed. This has an exact solution given by u = Пdk=1sin[cEk] on the hyper-cube, [0, 1]d. The dimension of the space was varied from one to six. The numerical solution was generated by the radial basis functions. Key words: Hermite Collocation, radial basis functions, high dimensional problems.