Malvin H. Kalos, Paula A. Whitlock. Monte Carlo Methods. Second Revised and Enlarged Edition WILEY- BLACKWELL. WILEY-VCH Verlag GmbH & Co.

Transcription

1 Malvin H. Kalos, Paula A. Whitlock Monte Carlo Methods Second Revised and Enlarged Edition WILEY- BLACKWELL WILEY-VCH Verlag GmbH & Co. KGaA

2 v I Contents Preface to the Second Edition IX Preface to the First Edition XI 1 What is Monte Carlo? Introduction Topics to be Covered A Short History of Monte Carlo 4 References 5 2 A Bit of Probability Random Events Random Variables The Binomial Distribution The Geometric Distribution The Poisson Distribution Continuous Random Variables Expectations of Continuous Random Variables Bivariate Continuous Random Distributions Sums of Random Variables: Monte Carlo Quadrature Distribution of the Mean of a Random Variable: A Fundamental Theorem Distribution of Sums of Independent Random Variables Monte Carlo Integration Monte Carlo Estimators 31 References 34 Further Reading 34 Elementary 34 More Advanced 34 3 Sampling Random Variables Transformation of Random Variables Numerical Transformation Sampling Discrete Distributions 43 Monte Carlo Methods. Second Edition. M.H. Kalos and P.A. Whitlock Copyright 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN;

3 VI Contents I 3.4 Composition of Random Variables Sampling the Sum of Two Uniform Random Variables Sampling a Random Variable Raised to a Power Sampling the Distribution f(z) = z(1 z) Sampling the Sum of Several Arbitrary Distributions Rejection Techniques Sampling a Singular pdf Using Rejection Sampling the Sine and Cosine of an Angle Kahn's Rejection Technique for a Gaussian Marsaglia et al. Method for Sampling a Gaussian Multivariate Distributions Sampling a Brownian Bridge The M(RT) 2 Algorithm Application of M(RT) Testing Sampling Methods 74 References 75 Further Reading 76 4 Monte Carlo Evaluation of Finite-Dimensional Integrals Importance Sampling The Use of Expected Values to Reduce Variance Correlation Methods for Variance Reduction Antithetic Variates Stratification Methods Adaptive Monte Carlo Methods Quasi-Monte Carlo Low-Discrepancy Sequences Error Estimation for Quasi-Monte Carlo Quadrature Applications of Quasi-Monte Carlo Comparison of Monte Carlo Integration, Quasi-Monte Carlo and Numerical Quadrature 104 References 105 Further Reading Random Walks, Integral Equations, and Variance Reduction Properties of Discrete Markov Chains Estimators and Markov Processes Applications Using Markov Chains Simulated Annealing Genetic Algorithms Poisson Processes and Continuous Time Markov Chains Brownian Motion Integral Equations Radiation Transport and Random Walks The Boltzmann Equation 126

4 Contents I VII 5.4 Variance Reduction Importance Sampling of Integral Equations 127 References 129 Further Reading Simulations of Stochastic Systems: Radiation Transport Radiation Transport as a Stochastic Process Characterization of the Source Tracing a Path Modeling Collision Events The Boltzmann Equation and Zero Variance Calculations Radiation Impinging an a Slab 144 References 147 Further Reading Statistical Physics Classical Systems The Hard Sphere Liquid Molecular Dynamics Kinetic Monte Carlo The Ising Model 155 References 156 Further Reading Quantum Monte Carlo Variational Monte Carlo Green's Function Monte Carlo Monte Carlo Solution of Homogeneous Integral Equations The Schrödinger Equation in Integral Form Green's Functions from Random Walks The Importance Sampling Transformation Diffusion Monte Carlo Path Integral Monte Carlo Quantum Chromodynamics 175 References 176 Further Reading Pseudorandom Numbers Major Classes of prn Generators Linear Recurrence Methods Tausworthe or Feedback Shift Register Generators Nonlinear Recursive Generators Combination Generators Statistical Testing of prng's Theoretical Tests 185

5 VIII I Contents Empirical Tests Comparing Two Pseudorandom Number Generators A Multiplicative Congruential Generator Proposed for 32-bit Computers A Bad Random Number Generator Pseudorandom Number Generation an Parallel Computers Splitting and Leapfrogging Parallel Sequences from Combination Generators Reproducibility and Lehmer Trees SPRNG: A Library of Pseudorandom Number Generators Summary 195 References 196 Index 199