Monte Carlo Algorithms for the Solution of Quasi-Linear Dirichlet Boundary Value Problems of Elliptical Type

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 tr...

A Quasi-Monte Carlo Method for Elliptic Boundary Value Problems

In this paper we present and analyze a quasi-Monte Carlo method for solving elliptic boundary value problems. Our method transforms the given partial differential equation into an integral equation by employing a well known local integral representation. The kernel in this integral equation representation can be used as a transition density function to define a Markov process used in estimating...

An Efficient Algorithm for Accelerating Monte Carlo Approximations of the Solution to Boundary Value Problems

The numerical approximation of boundary value problems by means of a probabilistic representations often has the drawback that the Monte Carlo estimate of the solution is substantially biased due to the presence of the domain boundary. We introduce a scheme, which we have called the leading-term Monte Carlo regression, which seeks to remove that bias by replacing a ’cloud’ of Monte Carlo estima...

Quasi-Monte Carlo algorithms for unbounded, weighted integration problems

In this article we investigate Quasi-Monte Carlo methods for multidimensional improper integrals with respect to a measure other than the uniform distribution. Additionally, the integrand is allowed to be unbounded at the lower boundary of the integration domain. We establish convergence of the Quasi-Monte Carlo estimator to the value of the improper integral under conditions involving both the...

Quasi-linear elliptic boundary value problems