Exploring Monte Carlo Methods

1st Edition

Authors: William Dunn J. Kenneth Shultis
Hardcover ISBN: 9780444515759
eBook ISBN: 9780080930619
Imprint: Elsevier Science
Published Date: 6th April 2011
Page Count: 398
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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

Table of Contents

Praise for Exploring Monte Carlo Methods



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


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


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


Chapter 4: Sampling, Scoring, and Precision

4.1 Sampling

4.2 Scoring

4.3 Accuracy and Precision

4.4 Summary


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


Chapter 6: Markov Chain Monte Carlo

6.1 Markov Chains to the Rescue

6.2 Brief Revi


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© Elsevier Science 2011
Elsevier Science
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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.


"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