Stochastic Algorithms with Hermite Cubic Spline Interpolation for Global Estimation of Solutions of Boundary Value Problems

Abstract

Here we construct two new functional Monte Carlo algorithms for the numerical solution of three-dimensional Dirichlet boundary value problems for the linear and nonlinear Helmholtz equations. These algorithms are based on estimating the solution and, if necessary, its partial derivatives at grid nodes using first Monte Carlo methods followed by an appropriate interpolation scheme. This allows us to obtain an approximation of the solution in the entire domain, which is not commonly done with Monte Carlo. The Monte Carlo methods used in this paper include the random walk on spheres method and the walk in balls process (with possible branching in the nonlinear case) and the stochastic application of Green's formula. For global approximation, cubic spline interpolation is used. One of the proposed approximation algorithms is based on Hermite cubic spline interpolation and utilizes estimates of the solution and its first partial derivatives. The other algorithm is based on Lagrange tricubic spline interpolation on a uniform grid and needs only estimates of the solution. An important problem is to find the optimal values of the interpolation algorithm parameters, such as the number of grid nodes and the sample volume. For this we use a stochastic optimization approach; i.e., for both of the proposed approximation algorithms we construct upper bounds of the approximation errors and minimize computational cost functions constrained by a fixed error criterion with a stochastic technique. To study the effectiveness of these proposed methods, we make a comparison between three functional algorithms, which are based on the use of the Hermite cubic splines, on the Lagrange tricubic splines, and on more common multilinear interpolation.

Some approaches to error analysis and optimization of the DSMC method

“…The main purpose of this theory is elaboration of approaches for the construction and optimization of Monte Carlo algorithms for solution approximation in the whole domain [11,13,17,20,21]. These approaches are related to the preliminary discretization of the problem (introduction of a grid), estimation of the solution at the grid nodes by the Monte Carlo method, and subsequent interpolation of the solution from thus obtained approximate values at the nodes.…”

“…To choose the optimal values of the numerical parameters of functional Monte Carlo algorithms, namely, the sample size K and the number of grid nodes M, two following steps have to be carried out, as in [11,17,20,21].…”

“…The investigation of the 'external' error is realized on the base of the theory of functional Monte Carlo algorithms [11,13,17,20,21]. In Section 1 analysis of the statistical part of the 'external' error is carried out.…”