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# Exploring Monte Carlo Methods

- 2nd Edition - June 7, 2022
- Authors: William L. Dunn, J. Kenneth Shultis
- Language: English
- Paperback ISBN:9 7 8 - 0 - 1 2 - 8 1 9 7 3 9 - 4
- eBook ISBN:9 7 8 - 0 - 1 2 - 8 1 9 7 4 5 - 5

Exploring Monte Carlo Methods, Second Edition provides a valuable introduction to the numerical methods that have come to be known as "Monte Carlo." This unique and trusted resour… Read more

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Request a sales quote*Exploring Monte Carlo Methods, Second Edition*provides a valuable introduction to the numerical methods that have come to be known as "Monte Carlo." This unique and trusted resource for course use, as well as researcher reference, offers accessible coverage, clear explanations and helpful examples throughout. Building from the basics, the text also includes applications in a variety of fields, such as physics, nuclear engineering, finance and investment, medical modeling and prediction, archaeology, geology and transportation planning.

- Provides a comprehensive yet concise treatment of Monte Carlo methods
- Uses the famous "Buffon’s needle problem" as a unifying theme to illustrate the many aspects of Monte Carlo methods
- Includes numerous exercises and useful appendices on: Certain mathematical functions, Bose Einstein functions, Fermi Dirac functions and Watson functions

Upper-level UG and graduate students (as well as researchers) in relevant courses across physics, math, engineering, and other areas

- Cover image
- Title page
- Table of Contents
- Copyright
- Dedication
- About the Authors
- Preface to the Second Edition
- Preface to the First Edition
- 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
- References
- Chapter 2: The Basis of Monte Carlo
- 2.1. Functions of a Single Continuous Random Variable
- 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
- References
- Chapter 3: Pseudorandom Number Generators
- 3.1. Pseudorandom Numbers
- 3.2. Linear Congruential Generators
- 3.3. Tests for Linear Congruential Generators
- 3.4. Practical Multiplicative Congruential Generators
- 3.5. A Minimal Standard Congruential Generator
- 3.6. Skipping Ahead
- 3.7. Combining Generators
- 3.8. Other Congruential Random Number Generators
- 3.9. RNGs Using Linear Feedback Shift Registers
- 3.10. What Is a Linear Feedback Shift Register?
- 3.11. RNGs Based on Cellular Automata
- 3.12. “Recent” Random Number Generators
- 3.13. Assessment Tests for RNGs
- 3.14. Summary
- Problems
- References
- Chapter 4: Sampling, Scoring, and Precision
- 4.1. Sampling
- 4.2. Scoring
- 4.3. Accuracy and Precision
- 4.4. Summary
- Problems
- References
- 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 Integration Volume
- 5.7. Reduction of Dimensionality
- 5.8. Russian Roulette and Splitting
- 5.9. Combinations of Different Variance Reduction Methods
- 5.10. Biased Estimators
- 5.11. Improved Monte Carlo Integration Schemes
- 5.12. Summary
- Problems
- References
- Chapter 6: Markov Chain Monte Carlo
- 6.1. Review of the Ordinary Monte Carlo Method
- 6.2. Markov Chains to the Rescue
- 6.3. Brief Review of Probability Concepts
- 6.4. Use of MCMC in Bayesian Analysis
- 6.5. Inference and Decision Applications
- 6.6. Summary
- Problems
- References
- 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
- References
- Chapter 8: Linear Operator Equations
- 8.1. Linear Algebraic Equations
- 8.2. Linear Integral Equations
- 8.3. Ordinary Differential Equations
- 8.4. Transient Partial Differential Equations
- 8.5. Eigenvalue Problems
- 8.6. Summary
- Problems
- References
- 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. Integral Forms of the Transport Equation
- 9.5. Adjoint Transport Equation
- 9.6. Summary
- Problems
- References
- 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
- References
- Appendix A: Some Common Probability Distributions
- A.1. Discrete Distributions
- A.2. Continuous Distributions
- A.3. Joint Distributions
- References
- Appendix B: The Weak and Strong Laws of Large Numbers
- B.1. The Weak Law of Large Numbers
- B.2. The Strong Law of Large Numbers
- References
- Appendix C: Central Limit Theorem
- C.1. Moment-Generating Functions
- C.2. The Central Limit Theorem
- References
- Appendix D: Linear Operators
- D.1. Linear Operators
- D.2. Inner Product
- D.3. Adjoint of a Linear Operator
- D.4. Uses of the Adjoint Operator
- D.5. Eigenfunctions and Eigenvalues of an Operator
- D.6. Eigenfunctions of Real, Linear, Self-Adjoint Operators
- D.7. The Sturm–Liouville Operator
- D.8. Generalized Fourier Series
- D.9. Solving Inhomogeneous Ordinary Differential Equations
- References
- Appendix E: Some Popular Monte Carlo Codes for Particle Transport
- E.1. COG
- E.2. EGSnrc
- E.3. GEANT4
- E.4. MCSHAPE
- E.5. MCNP6
- E.6. PENELOPE
- E.7. SCALE
- E.8. SRIM
- E.9. TRIPOLI
- Appendix F: Minimal Standard Pseudorandom Number Generator
- F.1. FORTRAN77
- F.2. FORTRAN90
- F.3. Pascal
- F.4. C and C++
- F.5. Programming Considerations
- References
- Index

- No. of pages: 592
- Language: English
- Edition: 2
- Published: June 7, 2022
- Imprint: Elsevier Science
- Paperback ISBN: 9780128197394
- eBook ISBN: 9780128197455

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### William L. 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, USAJS

### 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, USARead

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