Monte Carlo Algorithms for the Solution of Quasi-Linear Dirichlet Boundary Value Problems of Elliptical Type
Аннотация
The application of Monte Carlo methods in various fields is constantly growing due to increases in computer capabilities. Increasing speed and memory, and the wide availability of multiprocessor computers, allow us to solve many problems using the "method of statistical sampling", better known as the Monte Carlo method. Monte Carlo methods are known to have particular strengths. These include: Algorithmic simplicity with a strong analogy to the underlying physical processes, solve complex realistic problems that include sophisticated geometry and many physical processes, solve problems with high dimensions, the ability to obtain point solutions or evaluation linear functional of the solution, error estimates can be empirically obtained for all types of problems in parallel way, and ease of efficient parallel implementation. A shortcoming of the method is slow rate of convergence of the error, namely <img src=image/13492367_01.gif>) where <img src=image/13492367_02.gif> is the number of numerical experiments or realizations of the random variable. In this paper, we will propose Monte Carlo algorithms for the solution of the interior Dirichlet boundary value problem (BVP) for the Helmholtz operator with a polynomial nonlinearity on the right-hand side. The statistical algorithm is justified and complexity of the proposed algorithms is investigated, also the ways of decreasing the computational work are considered.