# Exploring Monte Carlo Methods

## 1st Edition

**Authors:**William Dunn J. Kenneth Shultis

**eBook ISBN:**9780080930619

**Hardcover ISBN:**9780444515759

**Imprint:**Elsevier Science

**Published Date:**6th April 2011

**Page Count:**398

## Description

Exploring Monte Carlo Methods is a basic text that describes the numerical methods that have come to be known as "Monte Carlo." The book treats the subject generically through the first eight chapters and, thus, should be of use to anyone who wants to learn to use Monte Carlo. The next two chapters focus on applications in nuclear engineering, which are illustrative of uses in other fields. Five appendices are included, which provide useful information on probability distributions, general-purpose Monte Carlo codes for radiation transport, and other matters. The famous "Buffon’s needle problem" provides a unifying theme as it is repeatedly used to illustrate many features of Monte Carlo methods.

This book provides the basic detail necessary to learn how to apply Monte Carlo methods and thus should be useful as a text book for undergraduate or graduate courses in numerical methods. It is written so that interested readers with only an understanding of calculus and differential equations can learn Monte Carlo on their own. Coverage of topics such as variance reduction, pseudo-random number generation, Markov chain Monte Carlo, inverse Monte Carlo, and linear operator equations will make the book useful even to experienced Monte Carlo practitioners.

## Key Features

- Provides a concise treatment of generic Monte Carlo methods
- Proofs for each chapter
- Appendixes include Certain mathematical functions; Bose Einstein functions, Fermi Dirac functions, Watson functions

## Readership

For undergraduate or graduate courses in numerical methods and users of large, general-purpose Monte Carlo codes

## Table of Contents

Praise for Exploring Monte Carlo Methods

Dedication

Preface

Chapter 1: Introduction

1.1 What Is Monte Carlo?

1.2 A Brief History of Monte Carlo

1.3 Monte Carlo as Quadrature

1.4 Monte Carlo as Simulation

1.5 Preview of Things to Come

1.6 Summary

Problems

Chapter 2: The Basis of Monte Carlo

2.1 Single Continuous Random Variables

2.2 Discrete Random Variables

2.3 Multiple Random Variables

2.4 The Law of Large Numbers

2.5 The Central Limit Theorem

2.6 Monte Carlo Quadrature

2.7 Monte Carlo Simulation

2.8 Summary

Problems

Chapter 3: Pseudorandom Number Generators

3.1 Linear Congruential Generators

3.2 Structure of the Generated Random Numbers

3.3 Characteristics of Good Random Number Generators

3.4 Tests for Congruential Generators

3.5 Practical Multiplicative Congruential Generators

3.6 Shuffling a Generator’s Output

3.7 Skipping Ahead

3.8 Combining Generators

3.9 Other Random Number Generators

3.10 Summary

Problems

Chapter 4: Sampling, Scoring, and Precision

4.1 Sampling

4.2 Scoring

4.3 Accuracy and Precision

4.4 Summary

Problems

Chapter 5: Variance Reduction Techniques

5.1 Use of Transformations

5.2 Importance Sampling

5.3 Systematic Sampling

5.4 Stratified Sampling

5.5 Correlated Sampling

5.6 Partition of the Integration Volume

5.7 Reduction of Dimensionality

5.8 Russian Roulette and Splitting

5.9 Combinations of Different Variance Reduction Techniques

5.10 Biased Estimators

5.11 Improved Monte Carlo Integration Schemes

5.12 Summary

Problems

Chapter 6: Markov Chain Monte Carlo

6.1 Markov Chains to the Rescue

6.2 Brief Review of Probability Concepts

6.3 Bayes Theorem

6.4 Inference and Decision Applications

6.5 Summary

Problems

Chapter 7: Inverse Monte Carlo

7.1 Formulation of the Inverse Problem

7.2 Inverse Monte Carlo by Iteration

7.3 Symbolic Monte Carlo

7.4 Inverse Monte Carlo by Simulation

7.5 General Applications of IMC

7.6 Summary

Problems

Chapter 8: Linear Operator Equations

8.1 Linear Algebraic Equations

8.2 Linear Integral Equations

8.3 Linear Differential Equations

8.4 Eigenvalue Problems

8.5 Summary

Problems

Chapter 9: The Fundamentals of Neutral Particle Transport

9.1 Description of the Radiation Field

9.2 Radiation Interactions with the Medium

9.3 Transport Equation

9.4 Adjoint Transport Equation

9.5 Summary

Problems

Chapter 10: Monte Carlo Simulation of Neutral Particle Transport

10.1 Basic Approach for Monte Carlo Transport Simulations

10.2 Geometry

10.3 Sources

10.4 Path-Length Estimation

10.5 Purely Absorbing Media

10.6 Type of Collision

10.7 Time Dependence

10.8 Particle Weights

10.9 Scoring and Tallies

10.10 An Example of One-Speed Particle Transport

10.11 Monte Carlo Based on the Integral Transport Equation

10.12 Variance Reduction and Nonanalog Methods

10.13 Summary

Problems

Some Common Probability Distributions

The Weak and Strong Laws of Large Numbers

Central Limit Theorem

Some Popular Monte Carlo Codes for Particle Transport

Minimal Standard Pseudorandom Number Generator

Index

## Details

- No. of pages:
- 398

- Language:
- English

- Copyright:
- © Elsevier Science 2011

- Published:
- 6th April 2011

- Imprint:
- Elsevier Science

- eBook ISBN:
- 9780080930619

- Hardcover ISBN:
- 9780444515759

## About the Author

### William Dunn

Dr. Dunn is with the Department of Mechanical and Nuclear Engineering at Kansas State University and was President of Quantum Research Services from 1988-2002. His research activities are primarily in the area of radiation applications. He is an Editor of the journal Applied Radiation and Isotopes and serves as Vice President, North America, of the International Radiation Physics Society.

### Affiliations and Expertise

Kansas State University, Department of Mechanical & Nuclear Engineering, Manhattan, U.S.A.

### J. Kenneth Shultis

Dr. Shultis has served as a faculty member in Nuclear Engineering for 38 years. He has written five books, including two that are routinely used as text books, and over 170 technical publications. His research interests include transport theory and radiative transfer, radiation protection and shielding, detector design and analysis, numerical analysis, radiological assessment, remote sensing, and risk analysis.

### Affiliations and Expertise

Kansas State University, Department of Mechanical & Nuclear Engineering, Manhattan, U.S.A.

## Reviews

"Anyone interested in learning about the basics of the Monte Carlo method, and its potential applications, will find this an excellent book…ideal book for an undergraduate or graduate course in mathematics or statistics." **--IEEE Electrical Insulation Magazine**

"Emphasizing the burgeoning practical applications rather than strict mathematical rigor of methods that have been used for about a century, this text is intended as an introduction for undergraduate or graduate courses, as a self-teaching guide, and as a reference. The first eight chapters are generic and are relevant to applications in any field. They include discussion of history and definition; the basis; sampling, scoring, and precision; variance reduction techniques; Markov chain and inverse Monte Carlo; and linear operator equations. Following are two chapters on radiation transport, a field familiar to authors William L. Dunn and J. Kenneth Shultis (both are nuclear engineers affiliated with Kansas State U.), but the focus remains on general principles that can be applied in many fields. Each chapter includes examples and problems and exercises. Five appendices contain supporting material." **--Reference and Research Book News**

"Overall, the book is very well written, and the contents are concisely and logically introduced. It should be very useful as a textbook for undergraduate or graduate courses in numerical methods employing Monte Carlo techniques like molecular simulations." **--Contemporary Physics**