On the stochastic/deterministic numerical solution of composite deterministic elliptic PDE problems

Abstract

We consider stochastic numerical solvers for deterministic elliptic Partial Differential Equation (PDE) problems. We concentrate on those that are characterized by their multidomain or/and multi-physics nature. In particular we consider either plain random walk on spheres methods or synergies of conventional deterministic PDE solving methods and traditional probabilistic Monte Carlo approaches. Our main objectives are two. One is to clearly define the context and the practical approach concerning the use of deterministic components that lead to effective numerical solvers for linear deterministic PDEs. The other is the design and implementation of a proof-ofconcept computational framework that allows experimentation in order to elucidate the capabilities and identify the emerging computational characteristics of the proposed approaches. A class of model problems in two and three space dimensions are first considered and experimental results are presented and discussed. © 2015, North Atlantic University Union NAUN. All rights reserved.