New Monte Carlo methods with estimating derivatives
type: Item
instance of: scholarly article
title: New Monte Carlo methods with estimating derivatives
zbMATH Open document ID: 0841.65003
author: G. A. Mikhaĭlov
published in: Inverse and Ill-Posed Problems Series
publication date: 1996-02-11T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 65-02
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 85A25
Mathematics Subject Classification ID: 76P05
Mathematics Subject Classification ID: 35K05
Mathematics Subject Classification ID: 80A20
Mathematics Subject Classification ID: 80A23
Mathematics Subject Classification ID: 65R20
Mathematics Subject Classification ID: 65C99
Mathematics Subject Classification ID: 60G60
zbMATH DE Number: 843453
zbMATH Keywords: textbook
zbMATH Keywords: estimation of integrals
zbMATH Keywords: estimation of derivatives
zbMATH Keywords: Monte Carlo methods
zbMATH Keywords: boundary value problems for elliptic equations
zbMATH Keywords: Boltzmann equation
zbMATH Keywords: radiation transfer
zbMATH Keywords: diffusion equations
zbMATH Keywords: integral equations
zbMATH Keywords: Helmholtz equation
zbMATH Keywords: metaharmonic equations
zbMATH Keywords: elliptic systems
zbMATH Keywords: nonlinear problems
zbMATH Keywords: random fields
reviewed by: I. Dimov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 843453