Numerical Experiments for Some Markov Models for Solving Boundary Value Problems
The main purpose of this work is the analysis of some stochastic algorithms to determine values of harmonic functions at points of a bounded domain of Euclidean space. To solve the Dirichlet problem we use a Random Walk on Spheres algorithm. The Neumann problem is solved by means of integral equations of potential theory.
We compare Monte Carlo and quasi-Monte Carlo versions of these algorithms numerically. The desired value of the harmonic function is represented as the sum of a series of integrals on hypercubes whose dimension grows. Therefore, the asymptotic formulas for discrepancy cannot be used for estimation of the error of quasi-Monte Carlo algorithm. New results are obtained about the influence of the smoothness of the domain boundary on the accuracy of calculations.
KeywordsBoundary value problem quasi-Monte Carlo method Monte Carlo method
Supported by the Russian Foundation for Basic Research, projects No. 17-01-00267, 18-47-350002.
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