Generating probability distributions on intervals and spheres with application to finite element method

Abstract

This work aims to build a bridge between probability methods and finite element methods. It starts with considering probability distributions supported in an interval [a,b], which incorporate the traditional probability distributions defined on the whole real space as limit cases on the one hand, lead to a type of spherical probability models with wide potential applications on the other hand. This type of probability has scaling and symmetry feature, and sufficient conditions under which a density function can be generated through discrete polynomial spectrum are given in this work followed by concrete examples. The density function ρ obtained in this way has the advantage of being positive definite. Computer based numerical simulation shows that the theoretically verified criteria for probability distribution are almost optimal with respect to our testing examples. After the establishment of an approximation theorem in L1 space, we propose a probabilistic Galerkin scheme that can be either continuous or discontinuous, which is potentially useful to asymptotically solve some PDEs on the sphere locally and globally.