Monte Carlo Methods for Particle Transport

Alireza Haghighat is a professor at Virginia Tech. He has served as the director of the Nuclear Science and Engineering Lab in Arlington, Virginia, and led the Virginia Tech Theory Transport Group. He previously worked at Penn State and the University of Florida. He holds a Ph.D from the University of Washington. He has published numerous papers, received several best paper awards, and presented many invited workshops, seminars, and papers nationally and internationally. He is a recipient of the 2011 Radiation Protection Shielding Division’s Professional Excellence Award, and a recognition award from the Office of Global Threat Reduction. An ANS fellow, he has served in various ANS leadership positions.

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