Приближенное решение многомерных интегральных уравнений методами Монте Карло и квази-Монте Карло

Abstract

The article considers an approach based on the random cubature method for solving both single and multidimensional singular integral equations, Volterra and Fredholm equations of the 1st kind, for ill-posed problems in the theory of integral equations, etc. A variant of the quasi-Monte Carlo method is studied. The integral in an integral equation is approximated using the traditional Monte Carlo method for calculating integrals. Multidimensional interpolation is applied on an arbitrary set of points. Examples of applying the method to a one-dimensional integral equation with a smooth kernel using both random and low-dispersed pseudo-random nodes are considered. A multidimensional linear integral equation with a polynomial kernel and a multidimensional nonlinear problem – the Hammerstein integral equation – are solved using the Newton method. The existence of several solutions is shown. Multidimensional integral equations of the first kind and their solution using regularization are considered. The Monte Carlo and quasi-Monte Carlo methods have not been used to solve such problems in the studied literature. The Lavrentiev regularization method was used, as well as random and pseudo-random nodes obtained using the Halton sequence. The problem of eigenvalues is solved. It is established that one of the best methods considered is the Leverrier-Faddeev method. The results of solving the problem for a different number of quadrature nodes are presented in the table. An approach based on parametric regularization of the core, an interpolation-projection method, and averaged adaptive densities are studied. The considered methods can be successfully applied in solving spatial boundary value problems for areas of complex shape. These approaches allow us to expand the range of problems in the theory of integral equations solved by Monte Carlo and quasi-Monte Carlo methods, since there are no restrictions on the value of the norm of the integral operator. A series of examples demonstrating the effectiveness of the method under study is considered.