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

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...

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Bibliographic Details
Published in:Journal of computational physics 2009-08, Vol.228 (15), p.5574-5591
Main Authors: Acebrón, Juan A., Rodríguez-Rozas, Ángel, Spigler, Renato
Format: Article
Language:English
Subjects:
ALGORITHMS
APPROXIMATIONS
Computational techniques
Domain decomposition
Exact sciences and technology
Fault tolerant algorithms
FUNCTIONALS
INTERPOLATION
MATHEMATICAL METHODS AND COMPUTING
Mathematical methods in physics
MONTE CARLO METHOD
Monte Carlo methods
Nonlinear parabolic problems
NONLINEAR PROBLEMS
Parallel computing
PARTIAL DIFFERENTIAL EQUATIONS
Physics
PROBABILISTIC ESTIMATION
Random trees
RANDOMNESS
SPACE-TIME
STOCHASTIC PROCESSES
TWO-DIMENSIONAL CALCULATIONS
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Summary: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.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2009.04.034