Domain decomposition solution of nonlinear two-dimensional parabolic problems by random trees

Authors:
Juan A. Acebrón;Ángel Rodríguez-Rozas;Renato Spigler
Affiliations:
Center for Mathematics and its Applications, Department of Mathematics, Instituto Superior Técnico Av. Rovisco Pais 1049-001 Lisboa, Portugal;Center for Mathematics and its Applications, Department of Mathematics, Instituto Superior Técnico Av. Rovisco Pais 1049-001 Lisboa, Portugal;Dipartimento di Matematica, Universití "Roma Tre", Largo S.L. Murialdo 1, 00146 Rome, Italy
Venue:
Journal of Computational Physics
Year:
2009

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Abstract

A domain decomposition method is developed for the numerical solution of nonlinear parabolic partial differential equations in any space dimension, based on the probabilistic representation of solutions as an average of suitable multiplicative functionals. Such a direct probabilistic representation requires generating a number of random trees, whose role is that of the realizations of stochastic processes used in the linear problems. First, only few values of the sought solution inside the space-time domain are computed (by a Monte Carlo method on the trees). An interpolation is then carried out, in order to approximate interfacial values of the solution inside the domain. Thus, a fully decoupled set of sub-problems is obtained. The algorithm is suited to massively parallel implementation, enjoying arbitrary scalability and fault tolerance properties. Pruning the trees is shown to increase appreciably the efficiency of the algorithm. Numerical examples conducted in 2D, including some for the KPP equation, are given.