Exploring Monte Carlo Methods
by William L. Dunn and J. Kenneth Shultis
Academic Press, Elsevier, Burlington, MD, 2012. ISBN 978-0-444-51575-9
Errata in Postscript or in
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TABLE OF CONTENTS
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
Chapter 2 The Basis of Monte Carlo
2.1 Single Continuous Variables
2.1.1 Probability Density Function
2.1.2 Cummulative Distribution Function
2.1.3 Some Example Distributions
2.1.4 Population Mean, Variance, and Standard deviation
2.1.5 Sample Mean, Variance, and Standard deviation
2.2 Discrete Random Variables
2.3 Multiple Random Variables
2.3.1 Two Random Variables
2.3.2 More Than Two Random Variables
2.3.3 Sums of Random Variables
2.3 The Law of Large Numbers
2.4 The Central Limit Theorem
2.5 Monte Carlo Quadrature
2.6 Monte Carlo Simulation
2.7 Summary
Chapter 3 Pseudorandom Numerber Generators
3.1 Binding Energy
3.2 Structure of the Generated Random Numbers
3.3 Characteristics of Good Random Number Generators
3.4 Tests for Congruential Generators
3.4.1 Spectral Test
3.4.2 Number of Hyperplanest
3.4.3 Distance between Points
3.4.4 Other Tests
3.5 Practical Multiplicative Congruential Generators
3.5.1 Generators with m=2^a
3.5.2 Prime Modulus Generators
3.5.3 A Minimal Standard Congruential Generator
3.5.4 Coding the Minimal Standard
3.5.5 Deficiencies of the Minimal Standard Generator
3.5.6 Optimum Multipliers for Prime Modulus Generators
3.6 Shuffling a Generator's Output
3.7 Skipping Ahead
3.8 Combining Generators
3.8.1 Bit Mixing
3.8.2 The Wichmann-Hill Generator
3.8.3 The L'Ecuyer Generator
3.9 Other Random Number Generators
3.9.1 Multiple Recursive Generators
3.9.2 Lagged Fibonacci Generators
3.9.3 Add-with-Carry Generators
3.9.4 Inversive Congruential Generators
3.9.5 Nonlinear Congruential Generators
3.10 Summary
Chapter 4 Sampling, Scoring and Precision
4.1 Sampling
4.1.1 Inverse CDF Method for Continuous Variables
4.1.2 Inverse CDF Method for Discrete Variables
4.1.3 Rejection Method
4.1.4 Composition Method
4.1.5 Rectangle-Wedge-Tail Decomposition Method
4.1.6 Sampling from a Nearly Linear PDF
4.1.7 Composition-Rejection Method
4.1.8 Ratio of Uniforms Method
4.1.9 Sampling from a Joint Distribution
4.1.10 Sampling from Specific Distributions
4.2 Scoring
4.2.1 Statistical Tests to Assess Results
4.2.2 Scoring for ``Successes-Over-Trials" Simulation84
4.2.3 Use of Weights in Scoring
4.2.4 Scoring for Multidimensional Integrals
4.3 Accuracy and Precision
4.3.1 Factors Affecting Accuracy
4.3.2 Factors Affecting Precision
4.4 Summary
Chapter 5 Variance Reduction Techniques
5.1 Use of Transformations
5.2 Importance Sampling
5.2.1 Application to Monte Carlo Integration
5.3 Systematic Sampling
5.3.1 Comparison to Straightforward Sampling
5.3.2 Systematic Sampling to Evaluate an Integral
5.3.3 Systematic Sampling as Importance Sampling
5.4 Stratified Sampling
5.4.1 Comparison to Straight Forward Sampling
5.4.2 Importance Sampling Versus Stratified Sampling
5.5 Correlated Sampling
5.5.1 Correlated Sampling With One Known Expected Value
5.5.2 Antithetic Variates
5.6 Partition of Integration Volume
5.7 Reduction of Dimensionality
5.8 Russian Roulette and Splitting
5.8.1 Application to Monte Carlo Simulation
5.9 Combinations of Different Variance Reduction Techniques
5.10 Biased Estimators
5.11 Improved Monte Carlo Integration Schemes
5.11.1 Weighted Monte Carlo Integration
5.11.2 Monte-Carlo Integration with Quasi-Random Numbers
5.12 Summary
Chapter 6 Markov Chain Monte Carlo
6.1 Markov Chains to the Rescue
6.1.1 Ergodic Markov Chains
6.1.2 The Metropolis-Hastings Algorithm
6.1.3 The Myth of Burn-in
6.1.4 The Proposal Distribution
6.1.5 Multidimensional Sampling
6.1.6 The Gibbs Sampler
6.2 Brief Review of Probability Concepts
6.3 Bayes Theorem
6.4 Inference and Decision Applications
6.4.1 Implementing MCMC with Data
6.5 Summary
Chapter 7 Inverse Monte Carlo
7.1 Formulation of the Inverse Problem
7.1.1 Integral Formulation
7.1.2 Practical Formulation
7.1.3 Optimization Formulation
7.1.4 Monte Carlo Approaches to Solving Inverse Problems
7.2 Inverse Monte Carlo by Iteration
7.3 Symbolic Monte Carlo
7.3.1 Uncertainties in Retrieved Values
7.3.2 The PDF is Fully Known
7.3.3 The PDF Is Unknown
7.3.4 Unknown Parameter in Domain of x
7.4 Inverse Monte Carlo by Simulation
7.5 General Applications of IMC
7.6 Summary
Chapter 8 Linear Operator Equations
8.1 Linear Algebraic Equations
8.1.1 Solution of Linear Equations by Random Walks
8.1.2 Solution of the Adjoint Linear Equations by Random Walks
8.1.3 Solution of Linear Equations by Finite Random Walks
8.2 Linear Integral Equations
8.2.1 Monte Carlo Solution of a Simple Integral Equation
8.2.2 A More General Procedure for Integral Equations
8.3 Linear Differential Equations
8.3.1 Monte Carlo Solution of Linear Differential Equations
8.3.2 Continuous Monte Carlo for Laplace's Equation
8.3.3 Generalization to Three Dimensions
8.3.4 Continuous Monte Carlo for Poisson's Equation
8.3.5 Continuous Monte Carlo for the 2-D Helmholtz Equation
8.3.6 Continuous Monte Carlo for the 3-D Helmholtz Equation
8.4 Eigenvalue Problems
8.5 Summary
Chapter 9 Fundamentals of Neutral Particle Transport
9.1 Description of the Radiation Field
9.1.1 Directions and Solid Angles
9.1.2 Particle Density
9.1.3 Flux Density
9.1.4 Fluence
9.1.5 Current Vector
9.2 Radiation Interactions
9.2.1 Interaction Coefficient / Macroscopic Cross Section
9.2.2 Attenuation of Uncollided Radiation
9.2.3 Average Travel Distance before an Interaction
9.2.4 Scattering Interaction Coefficients
9.2.5 Microscopic Cross Sections
9.2.6 Reaction Rate Density
9.3 Transport Equation
9.3.1 Integral Forms of the Transport Equation
9.4 Adjoint Transport Equation
9.4.1 Derivation of the Adjoint Transport Equation
9.4.2 Utility of the Adjoint Solution
9.5 Summary
Chapter 10 Particle Transport Simulation
10.1 Basic Approach for Monte Carlo Transport Simulations
10.2 Geometry
10.2.1 Combinatorial Geometry
10.3 Sources
10.4 Path Length Estimation
10.4.1 Travel Distance in Each Cell
10.4.2 Convex Versus Concave Cells
10.4.3 Effect of Computer Precision
10.5 Purely Absorbing Media
10.6 Type of Collision
10.6.1 Scattering Interactions
10.6.2 Photon Scattering from a Free Electron
10.6.3 Neutron Scattering
10.7 Time Dependence
10.8 Particle Weights
10.9 Scoring and Tallies
10.9.1 Fluence Averaged Over a Surface
10.9.2 Fluence in a Volume: Path Length Estimator
10.9.3 Fluence in a Volume: Reaction Density Estimator
10.9.4 Current through a Surface
10.9.5 Fluence at a Point: Next Event Estimator
10.9.6 Flow through a Surface: Leakage Estimator
10.10 An Example of One-Speed Particle Transport
10.11 Monte Carlo Based on the Integral Transport Equation
10.11.1 The Integral Transport Equation
10.11.2 The Integral Equation Method as Simulation
10.12 Variance Reduction
10.12.1 Importance Sampling
10.12.2 Truncation Methods
10.12.3 Splitting and Russian Roulette
10.12.4 Implicit Absorption
10.12.5 Interaction Forcing
10.12.6 Exponential Transformation
10.13 Summary
Appendix A: Some Common Probability Distributions
A.1 Continuous Distributions
A.1.1 Uniform Distribution
A.1.2 Exponential Distribution
A.1.3 Gamma Distribution
A.1.4 Beta Distribution
A.1.5 Weibull Distribution
A.1.6 Normal Distribution
A.1.7 Lognormal Distribution
A.1.8 Cauchy Distribution 7
A.1.9 Chi-Squared Distribution
A.1.10 Student's t Distribution
A.1.11 Pareto Distribution
A.2 Discrete Distributions
A.2.1 Bernoulli Distribution
A.2.2 Binomial Distribution
A.2.3 Geometric Distribution
A.2.4 Negative Binomial Distribution
A.2.5 Poisson Distribution
A.3 Joint Distributions
A.3.1 Bivariate Normal Distribution
A.3.2 Multinomial Distribution
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
B.2.1 Difference between the Weak and Strong Laws
B.2.2 Other Subtleties
Appendix C: Central Limit Theorem
C.1 Moment Generating Functions
C.1.1 Central Moments
C.1.2 Some Properties of the MGF
C.1.3 Uniqueness of the MGF
C.2 The Central Limit Theorem
Appendix D: Some Popular Monte Carlo Codes for Particle Transport
D.1 COG
D.2 EGSnrc
D.3 GEANT4
D.4 ITS
D.5 MCNP5\MCNPX
D.6 PENELOPE
D.7 SCALE
D.8 SRIM
D.9 TRIPOLI
Appendix E: Minimal Standard Pseudorandom Number Generator
E.1 FORTRAN77
E.2 FORTRAN90
E.3 Pascal
E.4 C and C++
E.5 Programming Considerations