Monte Carlo Methods for Particle Transport

Acknowledgments About the Author Introduction History of Monte Carlo Simulation Status of Monte Carlo Codes Motivation for Writing This Book Overview of the Book Recommendations to Instructors Author's Expectation References Random Variables and Sampling Introduction Random Variables Discrete Random Variable Continuous Random Variable Notes on pdf and cdf Characteristics Random Numbers Derivation of the Fundamental Formulation of Monte Carlo (FFMC) Sampling One-Dimensional Density Functions Analytical Inversion Numerical Inversion Probability Mixing Method Rejection Technique Numerical Evaluation Table Lookup Sampling Multidimensional Density Functions Example Procedures for Sampling a Few Commonly Used Distributions Normal Distribution Watt Spectrum Cosine and Sine Function Sampling Remarks References Problems Random Number Generation (RNG) Introduction Random Number Generation Approaches Pseudorandom Number Generators (PRNGs) Congruential Generators Multiple Recursive Generator Testing Randomness x2-Test Frequency Test Serial Test Gap Test Poker Test Moment Test Serial Correlation Test Serial Test via Plotting Examples for PRNG Tests Evaluation of PRNG Based on Period and Average Serial Test via Plotting Remarks References Problems Fundamentals of Probability and Statistics Introduction Expectation Value One-Dimensional Density Function Multidimensional Density Function Useful Theorems Associated with the "True Variance" Definition of Sample Expectation Values Used in Statistics Sample Mean Expected Value of the Sample Variance Precision and Accuracy of a Statistical Process Uniform Distribution Bernoulli and Binomial Distributions Geometric Distribution Poisson Distribution Normal ("Gaussian") Distribution Limit Theorems and Their Applications Corollary to the de Moivre-Laplace Limit Theorem Central Limit Theorem Formulations of Uncertainty and Relative Error for a Random Process General Random Process Special Case of Bernoulli Process Confidence Interval for Finite Sampling Introduction to Student's t-Distribution Determination of Confidence Interval and Application of the t-Distribution Test of Normality of Distribution Test of Skewness Coefficient Shapiro-Wilk Test for Normality References Problems Integrals and Associated Variance Reduction Techniques Introduction Estimation of Integrals Variance Reduction Techniques Associated with Integrals Importance Sampling Correlation Sampling Technique Stratified Sampling Technique Combined Sampling Remarks References Problems Fixed-Source Monte Carlo Particle Transport Introduction Introduction to the Linear Boltzmann Equation Introduction the Monte Carlo Method Determination of Free Flight, i.e., Path-Length Selection of Interaction Type Selection of Scattering Angle A Monte Carlo Algorithm for Estimation of Transmitted Particles Perturbation Calculations via Correlated Sampling Analysis of Monte Carlo Results Remarks References Problems Variance Reduction Techniques in Particle Transport Introduction Effectiveness of Variance Reduction Algorithms Biasing of Density Functions Implicit Capture (or Survival Biasing) Russian Roulette Biasing the Path-Length to the Next Collision Exponential Transformation Forced Collision Splitting Techniques Geometric Splitting with Russian Roulette Energy Splitting with Russian Roulette Angular Splitting with Russian Roulette Weight-Window Technique Application of Combination of Importance Sampling, pdf biasing, and Splitting Technique in Particle Transport Importance (Adjoint) Function Methodology in Deterministic Transport Theory Determination of Detector Response Use of Deterministic Importance (Adjoint) Function for Importance Sampling Remarks References Problems Tallying Introduction Major Quantities in a Particle Transport Simulation Tallying in a Steady-State System Collision Estimator Path-Length Estimator Surface-Crossing Estimator Analytical Estimator Tallying in a Time-Dependent System Tallies in Nonanalog Simulations Estimation of Relative Error Associated Physical Quantities Propagation of Error Remarks References Problems Geometry and Particle Tracking Introduction Discussion on a Combinatorial Geometry Approach Definition of Surfaces Definition of Cells Examples Description of Boundary Conditions Particle Tracking Remarks References Problems Eigenvalue or Criticality Monte Carlo Particle Transport Introduction Theory of Power-Iteration for Eigenvalue Problems Monte Carlo Eigenvalue Calculation Random Variables Associated with a Fission Process Direction of Fission Neutrons Monte Carlo Simulation of a Criticality Problem Estimators for Sampling Fission Neutrons Issues Associated with the Standard Eigenvalue Calculation Procedure Diagnostic Methods for Source Convergence Fission Matrix (FM) Methodology Issues Associated with the FM Method Remarks References Problems Vector and Parallel Processing of Monte Carlo Methods Introduction Vector Processing Vector Performance Parallel Processing Parallel Performance Vectorization of Monte Carlo Methods Parallelization of the Monte Carlo Methods Other Possible Parallel Monte Carlo Algorithms Development of a Parallel Algorithm Using MPI Remarks References Problems Appendices One to Six