V. M. Ivanov
Russia, 195220, St.Petersburg, Grazhdansky 28,
St.Petersburg State Technical University (SPSTU),
Informatics faculty,
M. L. Korenevsky
Russia, 195220, St.Petersburg, Grazhdansky 28,
St.Petersburg State Technical University (SPSTU),
Informatics faculty
O. Yu. Kulchitsky
Russia, 195251, St.Petersburg, Polytechnicheskaya 29,
St.Petersburg State Technical University (SPSTU),
Faculty of Mechanics and Control Processes,
A new combined method for numerical solving regular second type Fredholm integral equations is presented. This method combines the Monte-Carlo method and traditional projection methods for solving integral equations. We seek the solution of an integral equation in the form of an expansion with respect to a given orthonormal basis taking into account an error of expansion. Integrals with unknown errors are approximated by formulas of the Monte-Carlo method. As a result, we arrive to a system of equations linear with respect to expansion coefficients and errors at nodes of a random integration grid. Once this system has been solved, the approximate solution is prolonged to a domain where the equation is defined. Investigation of the convergence of the method is carried out. We obtain estimates of the convergence rate that demonstrate the decrease of the root-mean-square deviation of the approximate solution from the exact one provided the number of random points used for approximation of integrals by the Monte-Carlo method increases. Recurrent formulas obtained for the inversion of the matrix of linear equations system make possible to increase gradually the size of a random grid with slight computational expenditures. We propose certain ways to adapt basis functions to properties of the solution that allows to increase significantly an accuracy of the method. Also, we propose a method for multiple solving the problem on independent equally distributed samples of a comparatively small size with subsequent averaging. The efficiency of the method is confirmed by studying a number of test problems.