New Monte Carlo Methods with Estimating Derivatives - Tapa dura

It is possible to use weighted Monte Carlo methods for solving many problems of mathematical physics (boundary-value problems for elliptic equations, the Boltzmann equation, radiation transfer and diffusion equations). Weight estimates make it possible to evaluate special functionals, for example, derivatives with respect to parameters of a problems. This book presents new weak conditions under which the corresponding vector Monte Carlo estimates are unbiased and the variances are finite. The author has also constructued new Monte Carlo methods for solving the Helmhotz equation with a non-constant parameter, including the stationary Schrodinger equation. New results for linear and nonlinear problems are also presented. Some methods of random function simulation are considreed in the appendix. A new method of substantiating and optimzing the recurrent Monte Carlo estimates without using the Neumann series is presented in the introduction. This book should be of interest to specialists in the field of computational mathematics and physics, probability theory and mathematical statistics.