# Exploring Monte Carlo Methods

1st Edition - April 6, 2011

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• Authors: William Dunn, J. Kenneth Shultis
• eBook ISBN: 9780080930619
• Hardcover ISBN: 9780444515759

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## 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

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

• 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

1.4 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

2.7 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

3.8 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

9.5 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

## Product details

• No. of pages: 398
• Language: English
• Published: April 6, 2011
• Imprint: Elsevier Science
• eBook ISBN: 9780080930619
• Hardcover ISBN: 9780444515759

### William Dunn

Dr. Bill Dunn graduated with a BS degree in Electrical Engineering from the University of Notre Dame and MS and PhD degrees in Nuclear Engineering from North Carolina State University (NCSU). He was employed by Carolina Power & Light Company for four years and then served on the faculty and staff of the Nuclear Engineering Department at NCSU for two years. From 1979 until 2002, Dr. Dunn was involved in contract research. From 1988 until 2002, he was President of Quantum Research Services. He is now Professor and former Department Head of the Mechanical and Nuclear Engineering Department at Kansas State University. He is an editor of the journal Radiation Physics and Chemistry and Treasurer of the International Radiation Physics Society. In 2015, Dr. Dunn was recognized with the Radiation Science and Technology Award by the American Nuclear Society.

#### Affiliations and Expertise

Professor and former Department Head of the Mechanical and Nuclear Engineering Department, Kansas State University, Department of Mechanical and Nuclear Engineering, Manhattan, USA

### J. Kenneth Shultis

J. Kenneth Shultis, born in Toronto, Canada, graduated from the University of Toronto with a BASc degree in Engineering Physics (1964). He gained his MS (1965) and PhD (1968) degrees in Nuclear Science and Engineering from the University of Michigan. After a postdoctoral year at the Mathematics Institute of the University of Groningen, the Netherlands, he joined the Nuclear Engineering faculty at Kansas State University in 1969. He teaches and conducts research in radiation transport, radiation shielding, Monte Carlo methods, reactor physics, optimization of new type of radiation detectors, numerical analysis, particle combustion, remote sensing, and utility energy and economic analyses. He is a Fellow of the American Nuclear Society, and has received many awards for his teaching and research. Dr. Shultis is the author or co-author of 6 text books on radiation shielding, radiological assessment, nuclear science and technology, and Monte Carlo methods. He has written over 200 research papers and reports, and served as a consultant to many private and governmental organizations.

#### Affiliations and Expertise

Faculty Member, Kansas State University, Department of Mechanical and Nuclear Engineering, Manhattan, USA

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