Monte Carlo Methods for Particle Transport

1. Auflage 2014
ISBN: 978-1-4665-9253-7
Verlag: Taylor & Francis Inc

Buch, Englisch, 272 Seiten, Format (B × H): 156 mm x 242 mm, Gewicht: 528 g

The Monte Carlo method has become the de facto standard in radiation transport. Although powerful, if not understood and used appropriately, the method can give misleading results.

Monte Carlo Methods for Particle Transport teaches appropriate use of the Monte Carlo method, explaining the method’s fundamental concepts as well as its limitations. Concise yet comprehensive, this well-organized text:

Introduces the particle importance equation and its use for variance reduction
Describes general and particle-transport-specific variance reduction techniques
Presents particle transport eigenvalue issues and methodologies to address these issues
Explores advanced formulations based on the author’s research activities
Discusses parallel processing concepts and factors affecting parallel performance
Featuring illustrative examples, mathematical derivations, computer algorithms, and homework problems, Monte Carlo Methods for Particle Transport provides graduate students and nuclear engineers and scientists with a practical guide to the application of the Monte Carlo method.

Zielgruppe

Nuclear engineers and scientists.

Autoren/Hrsg.

Haghighat, Alireza

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein

Inhaltsverzeichnis

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

Über Autor(innen)

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.