Parametric estimates by the Monte Carlo method
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Bibliographic Information
Parametric estimates by the Monte Carlo method
VSP, 1999
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Note
Includes bibliographical references (p. [184]-188)
Description and Table of Contents
Description
01/07 This title is now available from Walter de Gruyter. Please see www.degruyter.com for more information.
This monograph is devoted to the further development of parametric weight Monte Carlo estimates for solving linear and nonlinear integral equations, radiation transfer equations, and boundary value problems, including problems with random parameters. The use of these estimates leads to the construction of new, effective Monte Carlo methods for calculating parametric multiple derivatives of solutions and for the main eigenvalues.
The book opens with an introduction on the theory of weight Monte Carlo methods. The following chapters contain new material on solving boundary value problems with complex parameters, mixed problems to parabolic equations, boundary value problems of the second and third kind, and some improved techniques related to vector and nonlinear Helmholtz equations. Special attention is given to the foundation and optimization of the global 'walk on grid' method for solving the Helmholtz difference equation. Additionally, new Monte Carlo methods for solving stochastic radiation transfer problems are presented, including the estimation of probabilistic moments of corresponding critical parameters.
Table of Contents
1. INTRODUCTION: WEIGHT UNBIASED MONTE CARLO ESTIMATES
Integral equations, linear functionals
Terminating Markov chains
Standard weight estimates in the Monte Carlo method, biasedness
Variances of the standard estimates
The main approaches to variance reduction
The use of recurrent representations
Randomization
Vector estimates related to the triangular system of integral equations
Calculation of parametric derivatives and the main eigenvalues of integral operators
Test integral equations and problems
The extensions of unbiasedness conditions
Approximate confidence intervals
2. PARAMETRIC ESTIMATES FOR SOLVING PROBLEMS OF MATHEMATICAL PHYSICS
Introductory information
Solving the Helmholtz equation with a complex parameter
Solution of boundary value problems of the second and third kinds
Solution of the Dirichlet problem for the vector and nonlinear Helmholtz equations
Estimating the main eigenvalue of the Laplace operator
Global algorithms of the Monte Carlo method for solving n-dimensional difference equations
3. PARAMETRIC ESTIMATES FOR STUDYING THE RADIATION TRANSFER IN INHOMOGENEOUS MEDIA
Introductory information
Calculation of parametric derivatives and critical values of parameters
Use of the averaged estimates by the Monte Carlo method for the study of the effects of medium stochasticity
- Modelling the homogeneous stochastic fields
- Partially averaged weight estimates
- Finiteness conditions for the variance of a partially averaged weight estimate
- Asymptotic estimation of the passage probability
- Test problem
- Additional remarks
Critical parameters of the particle transport process with multiplication in a stochastic medium
- Averaging the constants and the solution of the transfer equation
- Use of the diffusion approximation
- Estimation by the Monte Carlo method
- Use of the simplest mathematical models
- Use of the second order parametric derivatives
New approach to path estimates in the Monte Carlo method
Monte Carlo estimates for derivatives of polarized radiation
A. THE IMPROVEMENT OF RANDOM NUMBER GENERATORS BY MODULO 1 SUMMATION
Estimates of the nonuniformity of distributions of the congruent sums of random quantities
Congruent sums of grid random quantities
Improvement in the random number generators by congruent summation
B. ON MODELLING CHEMICAL REACTIONS BY THE MONTE CARLO METHOD
Introduction
General scheme of chemical reaction modelling by the Monte Carlo method
Conditions of coexistence of steady states in chemical systems
Calculation of quasi-potentials of dynamic systems
Examples
C. ONE UNSOLVED MINIMAX PROBLEM
References
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