Exploring Monte Carlo Methods

Preface......Page 6
What Is Monte Carlo?......Page 8
A Brief History of Monte Carlo......Page 9
Monte Carlo as Quadrature......Page 15
Monte Carlo as Simulation......Page 19
Preview of Things to Come......Page 22
Bibliography......Page 23
Problems......Page 25
The Basis of Monte Carlo......Page 28
Probability Density Function......Page 29
Some Example Distributions......Page 30
Population Mean, Variance, and Standard Deviation......Page 32
Sample Mean, Variance, and Standard Deviation......Page 34
The general case......Page 35
Discrete Random Variables......Page 36
Two Random Variables......Page 38
More Than Two Random Variables......Page 40
Sums of Random Variables......Page 41
The Law of Large Numbers......Page 42
The Central Limit Theorem......Page 44
Monte Carlo Quadrature......Page 46
Monte Carlo Simulation......Page 48
Summary......Page 51
Problems......Page 52
Pseudorandom Number Generators......Page 54
Linear Congruential Generators......Page 55
Structure of the Generated Random Numbers......Page 56
Spectral Test......Page 59
Other Tests......Page 60
Generators with m = 2α......Page 61
Prime Modulus Generators......Page 63
Coding the Minimal Standard......Page 64
Optimum Multipliers for Prime Modulus Generators......Page 66
Shuffling a Generator's Output......Page 67
Skipping Ahead......Page 68
The L'Ecuyer Generator......Page 69
Multiple Recursive Generators......Page 70
Add-with-Carry Generators......Page 71
Summary......Page 72
Bibliography......Page 73
Problems......Page 74
Sampling......Page 76
Inverse CDF Method for Continuous Variables......Page 77
Inverse CDF Method for Discrete Variables......Page 79
Other Monte Carlo applications of table lookups......Page 80
Rejection Method......Page 81
Composition Method......Page 83
Rectangle-Wedge-Tail Decomposition Method......Page 85
Composition-Rejection Method......Page 86
Sampling from a Joint Distribution......Page 87
Sampling from Specific Distributions......Page 88
Scoring......Page 89
Relative error and figure of merit......Page 90
Variance of the variance......Page 91
Scoring for "Successes-Over-Trials'' Simulation......Page 92
Use of Weights in Scoring......Page 93
Scoring for Multidimensional Integrals......Page 94
Factors Affecting Accuracy......Page 96
Factors Affecting Precision......Page 97
Round-off errors......Page 98
Summary......Page 100
Bibliography......Page 101
Problems......Page 102
Variance Reduction Techniques......Page 104
Use of Transformations......Page 107
Importance Sampling......Page 108
Application to Monte Carlo Integration......Page 112
Systematic Sampling......Page 114
Comparison to Straightforward Sampling......Page 116
Systematic Sampling as Importance Sampling......Page 117
Stratified Sampling......Page 118
Comparison to Straightforward Sampling......Page 119
Correlated Sampling......Page 120
Correlated Sampling With One Known Expected Value......Page 121
Antithetic Variates......Page 124
Generalization......Page 126
Reduction of Dimensionality......Page 128
Russian Roulette and Splitting......Page 129
Application to Monte Carlo Simulation......Page 130
Combinations of Different Variance Reduction Techniques......Page 131
Biased Estimators......Page 132
Improved Monte Carlo Integration Schemes......Page 133
Weighted Monte Carlo Integration......Page 134
Bibliography......Page 136
Problems......Page 137
Markov Chains to the Rescue......Page 140
Ergodic Markov Chains......Page 143
The Metropolis-Hastings Algorithm......Page 144
Reversibility of the transition function......Page 146
Distribution of Metropolis-Hasting samples......Page 147
The Myth of Burn-in......Page 150
The Proposal Distribution......Page 151
Samples over infinite domains......Page 152
Multidimensional Sampling......Page 155
The Gibbs Sampler......Page 159
Brief Review of Probability Concepts......Page 160
Bayes Theorem......Page 161
Inference and Decision Applications......Page 166
Implementing MCMC with Data......Page 168
The likelihood function......Page 169
Summary......Page 173
Bibliography......Page 174
Problems......Page 175
Inverse Monte Carlo......Page 178
Formulation of the Inverse Problem......Page 180
Integral Formulation......Page 181
Practical Formulation......Page 182
Monte Carlo Approaches to Solving Inverse Problems......Page 183
Inverse Monte Carlo by Iteration......Page 184
Symbolic Monte Carlo......Page 185
Uncertainties in Retrieved Values......Page 186
A simple linear kernel......Page 188
Approximating an unknown PDF as a histogram......Page 192
The PDF is separable......Page 195
Exponential PDF......Page 196
The PDF is linear in the unknown parameters......Page 198
A one-dimensional linear PDF......Page 199
Unknown Parameter in Domain of x......Page 201
A Simple Two-Dimensional World......Page 203
Summary......Page 206
Bibliography......Page 207
Problems......Page 208
Linear Algebraic Equations......Page 210
Solution of Linear Equations by Random Walks......Page 211
Proof of Eq. (8.11)......Page 212
Limiting cases......Page 213
Solution of the Adjoint Linear Equations by Random Walks......Page 214
Solution of Linear Equations by Finite Random Walks......Page 216
Proof of Eq. (8.31)......Page 217
Linear Integral Equations......Page 218
Monte Carlo Solution of a Simple Integral Equation......Page 219
A More General Procedure for Integral Equations......Page 221
Linear Differential Equations......Page 223
Monte Carlo Solutions of Linear Differential Equations......Page 224
Form of solution......Page 225
Monte Carlo estimation of u(r0)......Page 226
Generalization to Three Dimensions......Page 227
Case: q is a constant......Page 229
Case: q(r) is variable......Page 230
Solution of the homogeneous equation......Page 232
Solution for a constant source term......Page 233
The Three-dimensional Helmholtz Equation......Page 236
Eigenvalue Problems......Page 237
Bibliography......Page 238
Problems......Page 239
The Fundamentals of Neutral Particle Transport......Page 242
Directions and Solid Angles......Page 243
Particle Density......Page 245
Flux Density......Page 246
Fluence......Page 247
Current Vector......Page 248
Interaction Coefficient Macroscopic Cross Section......Page 249
Attenuation of Uncollided Radiation......Page 251
Scattering Interaction Coefficients......Page 252
Microscopic Cross Sections......Page 254
Reaction Rate Density......Page 256
Transport Equation......Page 258
Integral equation for the angular flux density......Page 262
Integral equations for integrals of φ(r, E, Ω)......Page 265
Explicit form for the scalar flux density......Page 266
Alternate forms of the integral transport equation......Page 267
Adjoint Transport Equation......Page 268
Derivation of the Adjoint Transport Equation......Page 269
Utility of the Adjoint Solution......Page 270
Problems......Page 272
Monte Carlo Simulation of Neutral Particle Transport......Page 276
Geometry......Page 277
Combinatorial Geometry......Page 278
Isotropic Sources......Page 280
Travel Distance in Each Cell......Page 281
Effect of Computer Precision......Page 282
Purely Absorbing Media......Page 284
Type of Collision......Page 285
Direction cosines of the scattered particle......Page 286
Photon Scattering from a Free Electron......Page 290
Neutron Scattering......Page 291
Time Dependence......Page 292
Scoring and Tallies......Page 293
Fluence Averaged Over a Surface......Page 294
Average Fluence in a Volume: Path-Length Estimator......Page 295
Average Fluence in a Volume: Reaction-Density Estimator......Page 296
Fluence at a Point: Next-Event Estimator......Page 297
Flow Through a Surface: Leakage Estimator......Page 299
An Example of One-Speed Particle Transport......Page 300
Formal solution of the integral transport equation......Page 303
Evaluation of G by Monte Carlo......Page 304
Integral Equation Method as Simulation......Page 306
Importance Sampling......Page 307
Splitting and Russian Roulette......Page 309
Exponential Transformation......Page 310
Summary......Page 311
Problems......Page 312
Sampling and linear transformations......Page 314
Uniform Distribution......Page 315
Sampling from the rectangular distribution......Page 316
Exponential Distribution......Page 317
Translated exponential distribution......Page 318
Gamma Distribution......Page 319
Sampling from a gamma distribution......Page 320
Beta Distribution......Page 321
Sampling from a beta distribution......Page 322
Weibull Distribution......Page 323
Normal Distribution......Page 324
Sampling from a normal distribution......Page 325
Cauchy Distribution......Page 326
Sampling from the Cauchy distribution......Page 327
Chi-Squared Distribution......Page 328
Student's t Distribution......Page 329
Pareto Distribution......Page 330
Discrete Distributions......Page 331
Bernoulli Distribution......Page 332
Binomial Distribution......Page 333
Sampling from a binomial distribution......Page 335
Negative Binomial Distribution......Page 336
Poisson Distribution......Page 337
Sampling from the Poisson distribution......Page 339
Multivariate Normal Distribution......Page 340
Multinomial Distribution......Page 341
Bibliography......Page 342
The Weak Law of Large Numbers......Page 344
Difference Between the Weak and Strong Laws......Page 346
Bibliography......Page 347
Moment Generating Functions......Page 348
Some Properties of the MGF......Page 349
Central Limit Theorem......Page 351
Bibliography......Page 353
Availability......Page 354
Significant Features......Page 355
Applications......Page 356
Significant Features......Page 357
Availability......Page 358
Significant Features......Page 359
MCNP and MCNPX......Page 360
Applications......Page 361
Significant Features......Page 362
Current Version and Availability......Page 363
Applications......Page 364
Significant Features......Page 365
Applications......Page 366
Acronyms......Page 368
Applications......Page 369
Significant Features......Page 370
Applications......Page 371
Principal Authors......Page 372
Significant Features......Page 373
Bibliography......Page 374
FORTRAN77......Page 380
Pascal......Page 381
Bibliography......Page 382
C......Page 384
I......Page 385
M......Page 386
P......Page 387
S......Page 389
V......Page 390
W......Page 391