This introduction to Monte Carlo methods seeks to identify and study the unifying elements that underlie their effective application. Initial chapters provide a short treatment of the probability and statistics needed as background, enabling those without experience in Monte Carlo techniques to apply these ideas to their research. The book focuses on two basic themes: The first is the importance of random walks as they occur both in natural stochastic systems and in their relationship to integral and differential equations. The second theme is that of variance reduction in general and importance sampling in particular as a technique for efficient use of the methods. Random walks are introduced with an elementary example in which the modeling of radiation transport arises directly from a schematic probabilistic description of the interaction of radiation with matter. Building on this example, the relationship between random walks and integral equations is outlined. The applicability of these ideas to other problems is shown by a clear and elementary introduction to the solution of the Schr?dinger equation by random walks. The text includes sample problems that readers can solve by themselves to illustrate the content of each chapter. This is the second, completely revised and extended edition of the successful monograph, which brings the treatment up to date and incorporates the many advances in Monte Carlo techniques and their applications, while retaining the original elementary but general approach.
Author(s): Malvin H. Kalos, Paula A. Whitlock
Edition: 2
Publisher: Wiley-VCH
Year: 2008
Language: English
Pages: 217
Cover Page......Page 1
Related Titles......Page 3
Title Page......Page 4
ISBN: 978-3527407606......Page 5
Contents......Page 6
Preface to the Second Edition......Page 10
Preface to the First Edition......Page 12
1.1 Introduction......Page 14
1.2 Topics to be Covered......Page 16
1.3 A Short History of Monte Carlo......Page 17
References......Page 18
2.1 Random Events......Page 20
2.2 Random Variables......Page 22
The Binomial Distribution......Page 25
The Geometric Distribution......Page 26
2.3 Continuous Random Variables......Page 27
2.4 Expectations of Continuous Random Variables......Page 29
2.5 Bivariate Continuous Random Distributions......Page 32
2.6 Sums of Random Variables: Monte Carlo Quadrature......Page 34
2.7 Distribution of the Mean of a Random Variable: A Fundamental Theorem......Page 35
2.8 Distribution of Sums of Independent Random Variables......Page 38
2.9 Monte Carlo Integration......Page 41
2.10 Monte Carlo Estimators......Page 44
Further Reading......Page 47
3 Sampling Random Variables......Page 48
3.1 Transformation of Random Variables......Page 49
Sampling (2/pi)(1/(1+y^2))......Page 53
The Box–Muller Method for Sampling a Gaussian or Normal Distribution......Page 54
3.2 Numerical Transformation......Page 55
3.3 Sampling Discrete Distributions......Page 56
The Geometric Distribution......Page 58
Mixed Distributions......Page 59
Sampling the Sum of Two Uniform Random Variables......Page 60
Sampling a Random Variable Raised to a Power......Page 61
Sampling the Sum of Several Arbitrary Distributions......Page 63
3.5 Rejection Techniques......Page 66
Sampling the Sine and Cosine of an Angle......Page 70
Kahn’s Rejection Technique for a Gaussian......Page 72
Marsaglia et al. Method for Sampling a Gaussian......Page 73
3.6 Multivariate Distributions......Page 74
Sampling a Brownian Bridge......Page 75
3.7 The M(RT)^2 Algorithm......Page 77
3.8 Application of M(RT)......Page 85
3.9 Testing Sampling Methods......Page 87
References......Page 88
Further Reading......Page 89
4 Monte Carlo Evaluation of Finite-Dimensional Integrals......Page 90
4.1 Importance Sampling......Page 92
Singular Integrals......Page 97
Importance Sampling with Correlated Sampling......Page 100
4.2 The Use of Expected Values to Reduce Variance......Page 101
Expected Values in M(RT)^2......Page 103
4.3 Correlation Methods for Variance Reduction......Page 104
Antithetic Variates......Page 106
Stratification Methods......Page 108
4.4 Adaptive Monte Carlo Methods......Page 111
4.5 Quasi-Monte Carlo......Page 113
Low-Discrepancy Sequences......Page 114
Error Estimation for Quasi-Monte Carlo Quadrature......Page 116
4.6 Comparison of Monte Carlo Integration, Quasi-Monte Carlo and Numerical Quadrature......Page 117
References......Page 118
Further Reading......Page 119
5.1 Properties of Discrete Markov Chains......Page 120
Estimators and Markov Processes......Page 122
5.2 Applications Using Markov Chains......Page 123
Simulated Annealing......Page 124
Genetic Algorithms......Page 125
Poisson Processes and Continuous Time Markov Chains......Page 127
Birth Processes......Page 129
Birth/Death Processes......Page 130
Absorbing States......Page 133
Brownian Motion......Page 135
Radiation Transport and Random Walks......Page 137
The Boltzmann Equation......Page 139
Importance Sampling of Integral Equations......Page 140
References......Page 142
Further Reading......Page 143
6.1 Radiation Transport as a Stochastic Process......Page 144
6.2 Characterization of the Source......Page 148
6.3 Tracing a Path......Page 149
6.4 Modeling Collision Events......Page 153
6.5 The Boltzmann Equation and Zero Variance Calculations......Page 155
Radiation Impinging on a Slab......Page 157
Further Reading......Page 160
7.1 Classical Systems......Page 162
The Hard Sphere Liquid......Page 164
Molecular Dynamics......Page 166
Kinetic Monte Carlo......Page 167
The Ising Model......Page 168
References......Page 169
Further Reading......Page 170
8 Quantum Monte Carlo......Page 172
8.1 Variational Monte Carlo......Page 173
8.2 Green’s Function Monte Carlo......Page 174
Monte Carlo Solution of Homogeneous Integral Equations......Page 175
The Schroedinger Equation in Integral Form......Page 176
Green’s Functions from Random Walks......Page 178
The Importance Sampling Transformation......Page 180
8.3 Diffusion Monte Carlo......Page 183
8.4 Path Integral Monte Carlo......Page 185
Path Integral Ground State Calculations......Page 187
8.5 Quantum Chromodynamics......Page 188
References......Page 189
Further Reading......Page 191
9 Pseudorandom Numbers......Page 192
Multiplicative Congruential Generators......Page 193
Lagged Fibonacci Congruential Generators......Page 194
Tausworthe or Feedback Shift Register Generators......Page 195
Nonlinear Recursive Generators......Page 196
Combination Generators......Page 197
Theoretical Tests......Page 198
Runs-up and Runs-down Test......Page 199
A Multiplicative Congruential Generator Proposed for 32-bit Computers......Page 200
A Bad Random Number Generator......Page 202
9.4 Pseudorandom Number Generation on Parallel Computers......Page 205
Parallel Sequences from Combination Generators......Page 206
Reproducibility and Lehmer Trees......Page 207
9.5 Summary......Page 208
References......Page 209
d......Page 212
h......Page 213
m......Page 214
s......Page 215
z......Page 216
Back Page......Page 217
