Monte Carlo Method for Solving ODE Systems

Abstract

The Monte Carlo method is applied to solve Cauchy problems for a system of linear and nonlinear ordinary differential equations. The Monte Carlo method is relevant for the solution of large systems of equations and in the case of small smoothness of initial functions. In this case, the system is reduced to an equivalent system of integral equations of the Volterra type. For linear systems, this transformation allows removing constraints connected with a convergence of a majorizing process. Examples of estimates of solution functionals are provided, and a behavior of their variances are discussed. In the general case, a solution interval is divided into finite subintervals, on which the nonlinear function is approximated by a polynomial. The obtained integral equation is solved by using branched Markov chains with absorption. Algorithm parallelization problems arising in this case are discussed in this paper. A one-dimensional cubic equation is considered as an example. A choice of transition densities of branching is discussed. A method of generations is described in detail. Numerical results are compared with a solution obtained by the Runge–Kutta method.