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E-raamat: Exploring Monte Carlo Methods

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William L. Dunn (Professor and former Department Head of the Mechanical and Nuclear Engineering Department, Kansas State University, Department of Mechanical and Nuclear Engineering, Manhattan, USA), J. Kenneth Shultis (Faculty Member, Kansas State University, Department )

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Ilmumisaeg: 07-Jun-2022
Kirjastus: Elsevier Science Publishing Co Inc
Keel: eng
ISBN-13: 9780128197455

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Exploring Monte Carlo Methods, Second Edition provides a valuable introduction to the numerical methods that have come to be known as "Monte Carlo." This unique and trusted resource for course use, as well as researcher reference, offers accessible coverage, clear explanations and helpful examples throughout. Building from the basics, the text also includes applications in a variety of fields, such as physics, nuclear engineering, finance and investment, medical modeling and prediction, archaeology, geology and transportation planning.

Provides a comprehensive yet concise treatment of Monte Carlo methods
Uses the famous "Buffon’s needle problem" as a unifying theme to illustrate the many aspects of Monte Carlo methods
Includes numerous exercises and useful appendices on: Certain mathematical functions, Bose Einstein functions, Fermi Dirac functions and Watson functions

About the Authors

xvii

Preface to the Second Edition

xix

Preface to the First Edition

xxi

1 Introduction

1.1 What Is Monte Carlo?

(1)

1.2 A Brief History of Monte Carlo

(8)

1.3 Monte Carlo as Quadrature

(4)

1.4 Monte Carlo as Simulation

(3)

1.5 Preview of Things to Come

(1)

1.6 Summary

(7)

Problems

(4)

References

(3)

2 The Basis of Monte Carlo

2.1 Functions of a Single Continuous Random Variable

(8)

2.1.1 Probability Density Function

(1)

2.1.2 Cumulative Distribution Function

(1)

2.1.3 Some Example Distributions

(3)

2.1.4 Population Mean, Variance, and Standard Deviation

(1)

2.1.5 Sample Mean, Variance, and Standard Deviation

(3)

2.2 Discrete Random Variables

(3)

2.2.1 Special Cases of Discrete Distributions

(1)

2.3 Multiple Random Variables

(5)

2.3.1 Two Random Variables

(2)

2.3.2 More Than Two Random Variables

(1)

2.3.3 Sums of Random Variables

(2)

2.4 The Law of Large Numbers

(1)

2.5 The Central Limit Theorem

(3)

2.6 Monte Carlo Quadrature

(2)

2.7 Monte Carlo Simulation

(3)

2.8 Summary

(4)

Problems

(1)

References

(2)

3 Pseudorandom Number Generators

3.1 Pseudorandom Numbers

(5)

3.1.1 Origins of Random Number Generators

(1)

3.1.2 Properties of Good RNCs

(2)

3.1.3 Classification of RNGs

(1)

3.2 Linear Congruential Generators

(3)

3.2.1 Structure of the Generated Random Numbers

(2)

3.3 Tests for Linear Congruential Generators

(2)

3.3.1 Spectral Test

(1)

3.3.2 Number of Hyperplanes

(1)

3.3.3 Distance Between Points

(1)

3.3.4 Other Tests

(1)

3.4 Practical Multiplicative Congruential Generators

(3)

3.4.1 Generators with m = 2"

(1)

3.4.2 Prime Modulus Generators

(1)

3.5 A Minimal Standard Congruential Generator

(6)

3.5.1 Coding the Minimal Standard

(2)

3.5.2 Deficiencies of the Minimal Standard Generator

(1)

3.5.3 Optimum Multipliers for Prime Modulus Generators

(1)

3.5.4 Shuffling a Generator's Output

(1)

3.5.5 Refinement of the Minimal Standard

(1)

3.6 Skipping Ahead

(1)

3.7 Combining Generators

(2)

3.7.1 Bit Mixing

(1)

3.7.2 The Wichmann--Hill Generator

(1)

3.7.3 The L'Ecuyer Generator

(1)

3.8 Other Congruential Random Number Generators

(3)

3.8.1 Multiple Recursive Generators

(1)

3.8.2 Lagged Fibonacci Generators

(1)

3.8.3 Add-with-Carry Generators

(1)

3.8.4 Inversive Congruential Generators

(1)

3.8.5 Nonlinear Congruential Generators

(1)

3.9 RNGs Using Linear Feedback Shift Registers

(4)

3.9.1 The Tausworthe Bit-Level RNG

(4)

3.10 What Is a Linear Feedback Shift Register?

(7)

3.10.1 Random Numbers From Random Bits

(2)

3.10.2 LFSR Extensions

(1)

3.10.3 Mersenne Twister

(2)

3.11 RNGs Based on Cellular Automata

(8)

3.11.1 One-Dimensional Cellular Automata

(3)

3.11.2 Random Number Generation From Cellular Automata

(2)

3.11.3 Calculating Elementary Cellular Automata

(1)

3.11.4 Some CA-Based RNGs

(1)

3.12 "Recent" Random Number Generators

(3)

3.12.1 Some RNG Developments in the Last 30 Years

(2)

3.13 Assessment Tests for RNGs

101

(2)

3.13.1 Basic Statistical Tests

101

(1)

3.13.2 The Diehard Test Suite

101

(1)

3.13.3 The Dieharder Test Suite

102

(1)

3.13.4 Other Test Libraries for RNCs

103

(1)

3.14 Summary

103

(8)

Problems

105

(2)

References

107

(4)

4 Sampling, Scoring, and Precision

4.1 Sampling

111

(19)

4.1.1 Inverse CDF Method for Continuous Variables

112

(3)

4.1.2 Inverse CDF Method for Discrete Variables

115

(1)

4.1.3 Sampling by Table Lookup

116

(5)

4.1.4 Rejection Method

121

(3)

4.1.5 Composition Method

124

(1)

4.1.6 Rectangle-Wedge-Tail Decomposition Method

125

(1)

4.1.7 Sampling From a Nearly Linear PDF

126

(1)

4.1.8 Composition-Rejection Method

127

(1)

4.1.9 Ratio of Uniforms Method

127

(1)

4.1.10 Sampling From a Joint Distribution

128

(1)

4.1.11 Sampling From Specific Distributions

128

(2)

4.2 Scoring

130

(7)

4.2.1 Use of Weights in Scoring

131

(2)

4.2.2 Scoring for "Successes-Over-Trials" Simulation

133

(1)

4.2.3 Scoring for Multidimensional Integrals

134

(1)

4.2.4 Statistical Tests to Assess Results

135

(2)

4.3 Accuracy and Precision

137

(6)

4.3.1 Factors Affecting Accuracy

138

(1)

4.3.2 Factors Affecting Precision

139

(2)

4.3.3 The Use of Batches

141

(2)

4.4 Summary

143

(11)

Problems

145

(3)

References

148

(6)

5 Variance Reduction Techniques

5.1 Use of Transformations

154

(2)

5.2 Importance Sampling

156

(5)

5.2.1 Application to Monte Carlo Integration

160

(1)

5.3 Systematic Sampling

161

(5)

5.3.1 Comparison to Straightforward Sampling

164

(1)

5.3.2 Systematic Sampling to Evaluate an Integral

165

(1)

5.3.3 Systematic Sampling as Importance Sampling

166

(1)

5.4 Stratified Sampling

166

(3)

5.4.1 Comparison to Straightforward Sampling

167

(1)

5.4.2 Importance Sampling Versus Stratified Sampling

168

(1)

5.5 Correlated Sampling

169

(7)

5.5.1 Correlated Sampling with One Known Expected Value

170

(2)

5.5.2 Antithetic Variates

172

(4)

5.6 Partition of Integration Volume

176

(1)

5.7 Reduction of Dimensionality

177

(1)

5.8 Russian Roulette and Splitting

178

(2)

5.8.1 Application to Monte Carlo Simulation

179

(1)

5.9 Combinations of Different Variance Reduction Methods

180

(1)

5.10 Biased Estimators

181

(1)

5.11 Improved Monte Carlo Integration Schemes

182

(3)

5.11.1 Weighted Monte Carlo Integration

183

(1)

5.11.2 Monte Carlo Integration with Quasi-Random Numbers

184

(1)

5.12 Summary

185

(4)

Problems

185

(2)

References

187

(2)

6 Markov Chain Monte Carlo

6.1 Review of the Ordinary Monte Carlo Method

189

(2)

6.2 Markov Chains to the Rescue

191

(29)

6.2.1 A Discrete Random Variable

192

(3)

6.2.2 A Continuous Random Variable

195

(1)

6.2.3 MCMC Versus OMC

196

(3)

6.2.4 The Metropolis-Hastings Algorithm

199

(5)

6.2.5 The Myth of Burn-in

204

(2)

6.2.6 The M-H Independence Sampler

206

(5)

6.2.7 Selecting a Proposal Distribution

211

(1)

6.2.8 Multidimensional Sampling

212

(3)

6.2.9 The Gibbs Sampler

215

(4)

6.2.10 The Problem of High Dimensionality

219

(1)

6.3 Brief Review of Probability Concepts

220

(12)

6.3.1 Bayes' Theorem

222

(3)

6.3.2 Extending the Application of Bayes' Theorem

225

(2)

6.3.3 Probability and Confidence Intervals

227

(3)

6.3.4 Conjugate Prior Distributions

230

(2)

6.4 Use of MCMC in Bayesian Analysis

232

(9)

6.4.1 Using MCMC to Calculate the Posterior PDF

232

(4)

6.4.2 MCMC and Nonconjugate Prior PDFs

236

(2)

6.4.3 Estimating the Prior Distribution

238

(2)

6.4.4 Failure Rate Analysis

240

(1)

6.5 Inference and Decision Applications

241

(7)

6.5.1 Implementing MCMC with Data

243

(1)

6.5.2 The Likelihood Function

244

(4)

6.6 Summary

248

(10)

Problems

250

(2)

References

252

(6)

7 Inverse Monte Carlo

7.1 Formulation of the Inverse Problem

258

(3)

7.1.1 Integral Formulation

258

(1)

7.1.2 Practical Formulation

259

(1)

7.1.3 Optimization Formulation

260

(1)

7.1.4 Monte Carlo Approaches to Solving Inverse Problems

261

(1)

7.2 Inverse Monte Carlo by Iteration

261

(2)

7.3 Symbolic Monte Carlo

263

(18)

7.3.1 Uncertainties in Retrieved Values

264

(2)

7.3.2 The PDF Is Fully Known

266

(4)

7.3.3 The PDF Is Unknown

270

(8)

7.3.4 Unknown Parameter in Domain of x

278

(3)

7.4 Inverse Monte Carlo by Simulation

281

(2)

7.4.1 A Simple Two-Dimensional World

281

(2)

7.5 General Applications of IMC

283

(4)

7.5.1 Radiative Transfer

284

(2)

7.5.2 Photon Beam Modifier Design

286

(1)

7.5.3 Energy-Dispersive X-Ray Fluorescence

286

(1)

7.6 Summary

287

(4)

Problems

288

(1)

References

289

(2)

8 Linear Operator Equations

8.1 Linear Algebraic Equations

291

(9)

8.1.1 Solution of Linear Equations by Random Walks

293

(4)

8.1.2 Solving the Adjoint Linear Equations by Random Walks

297

(1)

8.1.3 Solution of Linear Equations by Finite Random Walks

298

(2)

8.1.4 Recent Interest in Monte Carlo Linear Equation Solvers

300

(1)

8.2 Linear Integral Equations

300

(7)

8.2.1 Frequently Encountered Integral Equations

301

(1)

8.2.2 Approximating Integral Equations by Algebraic Equations

302

(1)

8.2.3 Monte Carlo Solution of a Simple Integral Equation

303

(2)

8.2.4 A More General Procedure for Integral Equations

305

(2)

8.3 Ordinary Differential Equations

307

(31)

8.3.1 Reduction to First-Order Equations

310

(1)

8.3.2 Monte Carlo Solution of Initial Value Problems

310

(3)

8.3.3 Monte Carlo Solution of Partial Differential Equations

313

(2)

8.3.4 Discretization of Poisson's Equation

315

(2)

8.3.5 Discrete Random Walks for Poisson's Equation

317

(5)

8.3.6 Continuous Random Walks for the 2-D Laplace's Equation

322

(3)

8.3.7 Continuous Monte Carlo for 2-D Poisson Equation

325

(1)

8.3.8 Other Boundary Conditions

326

(3)

8.3.9 Extension to Three Dimensions

329

(1)

8.3.10 Continuous Monte Carlo for the 3-D Poisson Equation

330

(3)

8.3.11 Continuous Monte Carlo for the 2-D Helmholtz Equation

333

(4)

8.3.12 The 3-D Helmholtz Equation

337

(1)

8.4 Transient Partial Differential Equations

338

(15)

8.4.1 Reverse-Time Monte Carlo

338

(8)

8.4.2 Green's Function Approach

346

(7)

8.5 Eigenvalue Problems

353

(9)

8.5.1 Matrix Eigenvalue Problem

353

(3)

8.5.2 Eigenvalues of Integral Operators

356

(5)

8.5.3 Eigenvalues of Differential Operators

361

(1)

8.6 Summary

362

(8)

Problems

362

(5)

References

367

(3)

9 The Fundamentals of Neutral Particle Transport

9.1 Description of the Radiation Field

370

(6)

9.1.1 Directions and Solid Angles

370

(2)

9.1.2 Particle Density

372

(1)

9.1.3 Flux Density

373

(1)

9.1.4 Fluence

374

(1)

9.1.5 Current Vector

375

(1)

9.2 Radiation Interactions with the Medium

376

(10)

9.2.1 Interaction Coefficient/Macroscopic Cross Section

376

(2)

9.2.2 Attenuation of Uncollided Radiation

378

(1)

9.2.3 Average Travel Distance Before an Interaction

379

(1)

9.2.4 Scattering Interaction Coefficients

380

(1)

9.2.5 Microscopic Cross Sections

381

(3)

9.2.6 Reaction Rate Density

384

(2)

9.3 Transport Equation

386

(5)

9.3.1 One-Speed Transport Equation in Plane Geometry

390

(1)

9.4 Integral Forms of the Transport Equation

391

(6)

9.4.1 Integral Equation for the Angular Flux Density

391

(4)

9.4.2 Integral Equations for Integrals of φ(r, E, Ω)

395

(1)

9.4.3 Explicit Form for the Scalar Flux Density

396

(1)

9.4.4 Alternate Forms of the Integral Transport Equation

396

(1)

9.5 Adjoint Transport Equation

397

(4)

9.5.1 Derivation of the Adjoint Transport Equation

398

(2)

9.5.2 Utility of the Adjoint Solution

400

(1)

9.6 Summary

401

(5)

Problems

402

(1)

References

403

(3)

10 Monte Carlo Simulation of Neutral Particle Transport

10.1 Basic Approach for Monte Carlo Transport Simulations

406

(1)

10.2 Geometry

406

(3)

10.2.1 Combinatorial Geometry

407

(2)

10.3 Sources

409

(1)

10.3.1 Isotropic Sources

410

(1)

10.4 Path Length Estimation

410

(4)

10.4.1 Travel Distance in Each Cell

410

(2)

10.4.2 Convex Versus Concave Cells

412

(1)

10.4.3 Effect of Computer Precision

412

(2)

10.5 Purely Absorbing Media

414

(1)

10.6 Type of Collision

415

(7)

10.6.1 Scattering Interactions

416

(4)

10.6.2 Photon Scattering From a Free Electron

420

(1)

10.6.3 Neutron Scattering

421

(1)

10.7 Time Dependence

422

(1)

10.8 Particle Weights

423

(1)

10.9 Scoring and Tallies

423

(8)

10.9.1 Fluence Averaged Over a Surface

424

(1)

10.9.2 Fluence in a Volume: Path Length Estimator

425

(1)

10.9.3 Fluence in a Volume: Reaction Density Estimator

426

(1)

10.9.4 Average Current Through a Surface

427

(1)

10.9.5 Fluence at a Point: Next-Event Estimator

428

(2)

10.9.6 Flow Through a Surface: Leakage Estimator

430

(1)

10.10 An Example of One-Speed Particle Transport

431

(3)

10.11 Monte Carlo Based on the Integral Transport Equation

434

(5)

10.11.1 The Integral Transport Equation

434

(4)

10.11.2 The Integral Equation Method as Simulation

438

(1)

10.12 Variance Reduction and Nonanalog Methods

439

(4)

10.12.1 Importance Sampling

439

(2)

10.12.2 Truncation Methods

441

(1)

10.12.3 Splitting and Russian Roulette

441

(1)

10.12.4 Implicit Absorption

441

(1)

10.12.5 Interaction Forcing

442

(1)

10.12.6 Exponential Transformation

442

(1)

10.13 Summary

443

(4)

Problems

443

(2)

References

445

(2)

A Some Common Probability Distributions

A.1 Discrete Distributions

447

(17)

A.I.1 Bernoulli Distribution

449

(1)

A.1.2 Binomial Distribution

450

(4)

A.1.3 Geometric Distribution

454

(1)

A.1.4 Negative Binomial Distribution

454

(3)

A.1.5 Hypergeometric Distribution

457

(2)

A.1.6 Negative Hypergeometric Distribution

459

(2)

A.1.7 Poisson Distribution

461

(3)

A.2 Continuous Distributions

464

(23)

A.2.1 Uniform Distribution

465

(2)

A.2.2 Exponential Distribution

467

(2)

A.2.3 Gamma Distribution

469

(3)

A.2.4 Beta Distribution

472

(3)

A.2.5 Weibull Distribution

475

(2)

A.2.6 Normal Distribution

477

(1)

A.2.7 Lognormal Distribution

478

(2)

A.2.8 Cauchy Distribution

480

(1)

A.2.9 Logbeta Distribution

481

(2)

A.2.10 Chi-Squared Distribution

483

(1)

A.2.11 Student's t Distribution

484

(2)

A.2.12 Pareto Distribution

486

(1)

A.3 Joint Distributions

487

(10)

A.3.1 Multivariate Normal Distribution

488

(1)

A.3.2 Multinomial Distribution

489

(2)

A.3.3 Dirichlet Distribution

491

(2)

References

493

(4)

B The Weak and Strong Laws of Large Numbers

B.1 The Weak Law of Large Numbers

497

(2)

B.2 The Strong Law of Large Numbers

499

(2)

B.2.1 Difference Between the Weak and Strong Laws

499

(1)

B.2.2 Other Subtleties

500

(1)

References

500

(1)

C Central Limit Theorem

C.1 Moment-Generating Functions

501

(3)

C.1.1 Central Moments

502

(1)

C.1.2 Some Properties of the Moment-Generating Function

502

(2)

C.1.3 Uniqueness of the Moment-Generating Function

504

(1)

C.2 The Central Limit Theorem

504

(3)

References

506

(1)

D Linear Operators

D.1 Linear Operators

507

(1)

D.2 Inner Product

507

(1)

D.3 Adjoint of a Linear Operator

508

(3)

D.4 Uses of the Adjoint Operator

511

(6)

D.4.1 The Forward and Adjoint Problems

511

(1)

D.4.2 Solution Using the Green's Function

512

(2)

D.4.3 Relation Between Forward and Adjoint Green's Functions

514

(1)

D.4.4 Time-Dependent Green's Functions

514

(1)

D.4.5 Finding Averages of the Solution

515

(2)

D.5 Eigenfunctions and Eigenvalues of an Operator

517

(5)

D.5.1 Properties of Eigenfunctions

520

(2)

D.6 Eigenfunctions of Real, Linear, Self-Adjoint Operators

522

(5)

D.6.1 Eigenvalues Are Real

523

(1)

D.6.2 Eigenfunctions Are Orthogonal

524

(1)

D.6.3 Real Eigenfunctions

525

(1)

D.6.4 Eigenfunctions Form a Complete Basis Set

525

(2)

D.7 The Sturm--Liouville Operator

527

(4)

D.7.1 Boundary Conditions to Make Hermitian

528

(2)

D.7.2 Some Eigenfunction Properties

530

(1)

D.7.3 Important Applications of Sturm-Liouville Theory

531

(1)

D.8 Generalized Fourier Series

531

(3)

D.8.1 Fourier--Legendre Series

532

(1)

D.8.2 Fourier--Bessel Series

532

(2)

D.9 Solving Inhomogeneous Ordinary Differential Equations

534

(5)

D.9.1 Method 1: Find a Particular Solution

534

(1)

D.9.2 Method 2: Use the Green's Function

535

(1)

D.9.3 Method 3: The Eigenfunction Expansion Technique

535

(2)

References

537

(2)

E Some Popular Monte Carlo Codes for Particle Transport

E.1 COG

539

(1)

E.2 EGSnrc

540

(1)

E.3 CEANT4

541

(2)

E.4 MCSHAPE

543

(1)

E.5 MCNP6

544

(2)

E.6 PENELOPE

546

(2)

E.7 SCALE

548

(2)

E.8 SRIM

550

(1)

E.9 TRIPOLI

551

(2)

F Minimal Standard Pseudorandom Number Generator

F.1 FORTRAN77

553

(1)

F.2 FORTRAN90

554

(1)

F.3 Pascal

554

(1)

F.4 C and C++

555

(1)

F.5 Programming Considerations

555

(2)

References

556

(1)

Index

557

Dr. Bill Dunn graduated with a BS degree in Electrical Engineering from the University of Notre Dame and MS and PhD degrees in Nuclear Engineering from North Carolina State University (NCSU). He was employed by Carolina Power & Light Company for four years and then served on the faculty and staff of the Nuclear Engineering Department at NCSU for two years. From 1979 until 2002, Dr. Dunn was involved in contract research. From 1988 until 2002, he was President of Quantum Research Services. He is now Professor and former Department Head of the Mechanical and Nuclear Engineering Department at Kansas State University. He is an editor of the journal Radiation Physics and Chemistry and Treasurer of the International Radiation Physics Society. In 2015, Dr. Dunn was recognized with the Radiation Science and Technology Award by the American Nuclear Society. J. Kenneth Shultis, born in Toronto, Canada, graduated from the University of Toronto with a BASc degree in Engineering Physics (1964). He gained his MS (1965) and PhD (1968) degrees in Nuclear Science and Engineering from the University of Michigan. After a postdoctoral year at the Mathematics Institute of the University of Groningen, the Netherlands, he joined the Nuclear Engineering faculty at Kansas State University in 1969. He teaches and conducts research in radiation transport, radiation shielding, Monte Carlo methods, reactor physics, optimization of new type of radiation detectors, numerical analysis, particle combustion, remote sensing, and utility energy and economic analyses. He is a Fellow of the American Nuclear Society, and has received many awards for his teaching and research. Dr. Shultis is the author or co-author of 6 text books on radiation shielding, radiological assessment, nuclear science and technology, and Monte Carlo methods. He has written over 200 research papers and reports, and served as a consultant to many private and governmental organizations.

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