Domain decomposition solution of elliptic boundary-value problems via Monte Carlo and quasi-Monte Carlo methods

Domain decomposition of two-dimensional domains on which boundary-value elliptic problems are formulated, is accomplished by probabilistic (Monte Carlo) as well as by quasi-Monte Carlo methods, generating only few interfacial values and interpolating on them. Continuous approximations for the trace of solution are thus obtained, to be used as boundary data for the sub-problems. The numerical treatment can then proceed by standard deterministic algorithms, separately in each of the so-obtained subdomains. Monte Carlo and quasi-Monte Carlo simulations may naturally exploit multiprocessor architectures, leading to parallel computing, as well as the ensuing domain decomposition does. The advantage such as scalability obtained increasing the number of processors is shown, both theoretically and experimentally, in a number of test examples, and the possibility of using clusters of computers (grid computing) emphasized.