Probabilistically induced domain decomposition methods for elliptic boundary-value problems

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.

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