In this book of the well-known specialist on Monte Carlo methods weighted methods for solving problems of mathematical physics, such as boundary value problems for elliptic equations, the Boltzmann equation, radiation transfer and diffusion equations, are presented and studied. The presented estimates make it possible to evaluate special linear functionals, for example, derivatives with respect to parameters of a problem.
The book contains 6 sections: 1. Estimation of integrals and solution of integral equations. 2. Estimation of derivatives. 3. Solution of the Helmholtz equation. 4. Solution of metaharmonic equations and elliptic systems. 5. Monte Carlo methods with calculating parametric derivatives in the radiation transport theory. 6. Solution of nonlinear problems. There is also an Appendix, where the problem of simulation of random fields and variables is considered.
In this book new weak conditions are presented under which the corresponding vector Monte Carlo estimates are unbiased and their variances are finite. The author has also constructed new Monte Carlo methods for solving the Helmholtz equation with a nonconstant parameter. New results for nonlinear problems are also presented. A new method of substantiating and optimizing the iterative Monte Carlo estimates without using the Neumann series is presented.
This book will be of interest to specialists in the field of computational mathematics and physics, probability theory and mathematical statistics.