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
1.3 Monte Carlo as Quadrature
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
2.6 Monte Carlo Quadrature
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
3.7 Skipping Ahead
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
9.4 Adjoint 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
