Monte Carlo Methods For Integral Equations(systems)

With the rapid development of computer, the application of the Monte Carlo(MC) method has been extended to mathematics, engineering thermophysics, physics,mechanics, statistics, sociology, ?nance and other variety ?elds, since it was applied to Bu?on’s needle problem in 1777. And it tends to expand. The MC method is divided into static MC method and dynamic MC method(also called Markov Chain Monte Carlo method), based on the di?erence between the methods generating random samples to simulate. The MC method has a special advantage, which is the convergence rate wouldn’t be in?uenced by the dimension of the problem to be solved, so the method is almost the only e?ective algorithm dealing with the high-dimensional problems. However, there are shortcomings that the convergence rate is slow and the calculated precision is not high. In case of the given sample size, variance reduction techniques have been presented in order to improve the convergence rate and the calculated precision of the Monte Carlo method, such as the importance sampling technique and the control variate technique.Norwegian mathematician Niels Abel, Italian mathematician Vito Volterra, Swedish mathematician Erik Ivar Fredholm and German mathematician David Hilbert have devoted much to the process of integral equation to be an independent subject, the integral equation develops always with the application in the practical subjects, it also can transform each other with the di?erential equation and reduce the dimension of the problem to be solved, thus the integral equations are widely used in engineering system, social system and other ?elds.Although there are a lot of deterministic numerical methods to solve the numerical solutions of integral equations(system), such as the Nystr¨om method, the collocation method, the ?nite element method, the successive approximations method, the quadrature formula method and so on, however, some of the deterministic methods would produce a high-dimensional or high order system, in this case, the advantage of the Monte Carlo method just can counteract the inadequacy of the deterministic method.Therefore, combining the MC method with the deterministic methods is favored by many scholars. In this paper, using the stochastic simulation method to solve the in-tegral equation is studied to develop the scienti?c computing method of the integral equation, providing sci-tech workers with the experience of the numerical computation method.In this paper, applying the Monte Carlo methods to solve numerical solutions of the system of Volterra integral equations of the second kind, the Fredholm integral equation of the second kind and the Volterra integral equation of the second kind are studied, the Fredholm integral equations and the Volterra integral equations are the most basic kinds in the subject of integral equations and they have a wide range of backgrounds of the application in the practical problems. Therefore, developing the scienti?c computing method of the two equations would bring the important theoretical signi?cance and the practical applied value.In the ?rst chapter, all the four aspects are introduced brie?y, including how to accomplish the Monte Carlo method, its brilliant features and basic information, the development and the range of application of integral equations, some important literatures in the development of using Monte Carlo method to solve the integral equations(system), as well as the main research contents of this paper.In the second chapter, the quadrature formula method combined with the dynamic MC method based on the importance sampling technique is applied to solve the numerical solutions of the system of Volterra integral equations of the second kind, the initial probabilities and the transition probabilities generating dynamic random samples are showed. It is theoretically proved that the feasibility and e?ectiveness of the iterative Monte Carlo method base on the important sampling to solve the integral equations system. Finally, some classical examples are calculated to show that the calculated precision of the dynamic MC method is able to be reference.In the third chapter, the Gauss-Legendre quadrature formula combined with the dynamic MC method based on the control variate technique is applied to solve the numerical solutions of Fredholm integral equations of the second kind, the remainder of the compound Gauss-Legendre quadrature formula and the process of using Taylor expansion to build control variates are showed. It is proved in theory and examples that the iterative Monte Carlo control variate method to solve the Fredholm integral equations would improve the calculated precision.In the fourth chapter, the successive approximations method combined with thedynamic MC method based on the importance sampling technique is applied to solve the numerical solutions of Volterra integral equations of the second kind, how to generate the random samples of the continuous Markov Chain is showed. At last, the e?ciency of the method applied is acceptable illustrated by the numerical examples of literatures.In the last chapter, all the content in the whole paper is made a conclusion, then present some direction and problems which can be further studied.