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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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.

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

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, 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