Backward iterations for solving integral equations with polynomial nonlinearity

Abstract

The theory of adjoint operators is widely used in solving applied multidimensional problems with Monte Carlo method. Efficient algorithms are constructed using the duality principle for many problems described in linear integral equations of the second kind. On the other hand, important applications of adjoint equations for designing experiments were suggested by G.Marchuk and his colleagues in their respective works. Some results obtained in these fields are also generalized to the case of nonlinear operators. Linearization methods were mostly used for that purpose. Results for Lyapunov-Schmidt nonlinear polynomial equations were obtained in the theory of Monte Carlo methods. However, many interesting questions in this subject area are remained open. New results about dual processes used for solving polynomial equations with Monte Carlo method are presented. In particular, an adjoint Markov process for the branching process and the corresponding unbiased estimate of the functional of the solution to the equation are constructed in a general form. The possibility of constructing an adjoint operator to a nonlinear one is discussed.