Probabilistically induced domain decomposition methods for elliptic boundary-value problems

Authors:
Juan A. Acebrón;Maria Pia Busico;Piero Lanucara;Renato Spigler
Affiliations:
Departamento de Automática, Escuela Politécnica, Universidad de Alcalá, Crta. Madrid-Barcelona km. 31.600, 28871 Alcalá de Henares, Spain;CASPUR, Via dei Tizii 6b, 00185 Rome, Italy;CASPUR, Via dei Tizii 6b, 00185 Rome, Italy;Dipartimento di Matematica, Universití di "Roma Tre", Largo S.L. Murialdo 1, 00146 Rome, Italy
Venue:
Journal of Computational Physics
Year:
2005

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Abstract

Monte Carlo as well as quasi-Monte Carlo methods are used to generate only few interfacial values in two-dimensional domains where boundary-value elliptic problems are formulated. This allows for a domain decomposition of the domain. A continuous approximation of the solution is obtained interpolating on such interfaces, and then used as boundary data to split the original problem into fully decoupled subproblems. The numerical treatment can then be continued, implementing any deterministic algorithm on each subdomain. Both, Monte Carlo (or quasi-Monte Carlo) simulations and the domain decomposition strategy allow for exploiting parallel architectures. Scalability and natural fault tolerance are peculiarities of the present algorithm. Examples concern Helmholtz and Poisson equations, whose probabilistic treatment presents additional complications with respect to the case of homogeneous elliptic problems without any potential term and source.