Exploring Monte Carlo Methods
- by William L. Dunn and J. Kenneth Shultis
- Academic Press, Elsevier, Burlington, MD, 2012. ISBN 978-0-444-51575-9
![[book8]](smallMCbook.jpg)
Errata in Postscript or in Adobe pdf format
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
2.3 Multiple Random Variables
- 2.3.1 Two Random Variables
2.3.2 More Than Two Random Variables
2.3.3 Sums of Random Variables
2.4 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
- 3.5.1 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
3.7 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
- 3.9.1 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
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
- 4.2.1 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
- 4.3.1 Factors Affecting Accuracy
4.3.2 Factors Affecting Precision
Chapter 5 Variance Reduction Techniques
- 5.1 Use of Transformations
5.2 Importance Sampling
- 5.2.1 Application to Monte Carlo Integration
- 5.3.1 Comparison to Straightforward Sampling
5.3.2 Systematic Sampling to Evaluate an Integral
5.3.3 Systematic Sampling as Importance Sampling
- 5.4.1 Comparison to Straight Forward Sampling
5.4.2 Importance Sampling Versus Stratified Sampling
- 5.5.1 Correlated Sampling With One Known Expected Value
5.5.2 Antithetic Variates
5.7 Reduction of Dimensionality
5.8 Russian Roulette and Splitting
- 5.8.1 Application to Monte Carlo Simulation
5.10 Biased Estimators
5.11 Improved Monte Carlo Integration Schemes
- 5.11.1 Weighted Monte Carlo Integration
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
6.3 Bayes Theorem
6.4 Inference and Decision Applications
- 6.4.1 Implementing MCMC with Data
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
7.3 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
7.5 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
- 8.2.1 Monte Carlo Solution of a Simple Integral Equation
8.2.2 A More General Procedure for Integral 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
8.5 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
- 9.2.1 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
- 9.3.1 Integral Forms of the Transport Equation
- 9.4.1 Derivation of the Adjoint Transport Equation
9.4.2 Utility of the Adjoint Solution
Chapter 10 Particle Transport Simulation
10.1 Basic Approach for Monte Carlo Transport Simulations10.2 Geometry
- 10.2.1 Combinatorial Geometry
- 10.3.1 Isotropic Sources
- 10.4.1 Travel Distance in Each Cell
10.4.2 Convex Versus Concave Cells
10.4.3 Effect of Computer Precision
10.6 Type of Collision
- 10.6.1 Scattering Interactions
10.6.2 Photon Scattering from a Free Electron
10.6.3 Neutron Scattering
10.8 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
- 10.11 Monte Carlo Based on the Integral Transport Equation
10.11.1 The Integral Transport Equation
10.11.2 The Integral Equation Method as Simulation
- 10.12.1 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