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Author Toral, Raúl.

Title Stochastic numerical methods : an introduction for students and scientists / Raúl Toral, and Pere Colet
Published Weinheim, Germany : Wiley-VCH, [2014]
©2014
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Description 1 online resource (xvi, 402 pages)
Contents 880-01 Stochastic Numerical Methods; Contents; Preface; Chapter 1 Review of probability concepts; 1.1 Random Variables; 1.2 Average Values, Moments; 1.3 Some Important Probability Distributions with a Given Name; 1.3.1 Bernoulli Distribution; 1.3.2 Binomial Distribution; 1.3.3 Geometric Distribution; 1.3.4 Uniform Distribution; 1.3.5 Poisson Distribution; 1.3.6 Exponential Distribution; 1.3.7 Gaussian Distribution; 1.3.8 Gamma Distribution; 1.3.9 Chi and Chi-Square Distributions; 1.4 Successions of Random Variables; 1.5 Jointly Gaussian Random Variables
880-01 Machine generated contents note: 1. Review of probability concepts -- 1.1. Random Variables -- 1.2. Average Values, Moments -- 1.3. Some Important Probability Distributions with a Given Name -- 1.3.1. Bernoulli Distribution -- 1.3.2. Binomial Distribution -- 1.3.3. Geometric Distribution -- 1.3.4. Uniform Distribution -- 1.3.5. Poisson Distribution -- 1.3.6. Exponential Distribution -- 1.3.7. Gaussian Distribution -- 1.3.8. Gamma Distribution -- 1.3.9. Chi and Chi-Square Distributions -- 1.4. Successions of Random Variables -- 1.5. Jointly Gaussian Random Variables -- 1.6. Interpretation of the Variance: Statistical Errors -- 1.7. Sums of Random Variables -- 1.8. Conditional Probabilities -- 1.9. Markov Chains -- Further Reading and References -- Exercises -- 2. Monte Carlo Integration -- 2.1. Hit and Miss -- 2.2. Uniform Sampling -- 2.3. General Sampling Methods -- 2.4. Generation of Nonuniform Random Numbers: Basic Concepts -- 2.5. Importance Sampling -- 2.6. Advantages of Monte Carlo Integration -- 2.7. Monte Carlo Importance Sampling for Sums -- 2.8. Efficiency of an Integration Method -- 2.9. Final Remarks -- Further Reading and References -- Exercises -- 3. Generation of Nonuniform Random Numbers: Noncorrelated Values -- 3.1. General Method -- 3.2. Change of Variables -- 3.3. Combination of Variables -- 3.3.1. Rejection Method -- 3.4. Multidimensional Distributions -- 3.5. Gaussian Distribution -- 3.6. Rejection Methods -- Further Reading and References -- Exercises -- 4. Dynamical Methods -- 4.1. Rejection with Repetition: a Simple Case -- 4.2. Statistical Errors -- 4.3. Dynamical Methods -- 4.4. Metropolis et al. Algorithm -- 4.4.1. Gaussian Distribution -- 4.4.2. Poisson Distribution -- 4.5. Multidimensional Distributions -- 4.6. Heat-Bath Method -- 4.7. Tuning the Algorithms -- 4.7.1. Parameter Tuning -- 4.7.2. How Often-- 4.7.3. Thermalization -- Further Reading and References -- Exercises -- 5. Applications to Statistical Mechanics -- 5.1. Introduction -- 5.2. Average Acceptance Probability -- 5.3. Interacting Particles -- 5.4. Ising Model -- 5.4.1. Metropolis Algorithm -- 5.4.2. Kawasaki Interpretation of the Ising Model -- 5.4.3. Heat-Bath Algorithm -- 5.5. Heisenberg Model -- 5.6. Lattice Φ4 Model -- 5.6.1. Monte Carlo Methods -- 5.7. Data Analysis: Problems around the Critical Region -- 5.7.1. Finite-Size Effects -- 5.7.2. Increase of Fluctuations -- 5.7.3. Critical Slowing Down -- 5.7.4. Thermalization -- Further Reading and References -- Exercises -- 6. Introduction to Stochastic Processes -- 6.1. Brownian Motion -- 6.2. Stochastic Processes -- 6.3. Stochastic Differential Equations -- 6.4. White Noise -- 6.5. Stochastic Integrals. Ito and Stratonovich Interpretations -- 6.6. Ornstein-Uhlenbeck Process -- 6.6.1. Colored Noise -- 6.7. Fokker--Planck Equation -- 6.7.1. Stationary Solution -- Further Reading and References -- Exercises -- 7. Numerical Simulation of Stochastic Differential Equations -- 7.1. Numerical Integration of Stochastic Differential Equations with Gaussian White Noise -- 7.1.1. Integration Error -- 7.2. Ornstein--Uhlenbeck Process: Exact Generation of Trajectories -- 7.3. Numerical Integration of Stochastic Differential Equations with Ornstein--Uhlenbeck Noise -- 7.3.1. Exact Generation of the Process gh(t) -- 7.4. Runge--Kutta-Type Methods -- 7.5. Numerical Integration of Stochastic Differential Equations with Several Variables -- 7.6. Rare Events: The Linear Equation with Linear Multiplicative Noise -- 7.7. First Passage Time Problems -- 7.8. Higher Order () Methods -- 7.8.1. Heun Method -- 7.8.2. Midpoint Runge--Kutta -- 7.8.3. Predictor--Corrector -- 7.8.4. Higher Order-- Further Reading and References -- Exercises -- 8. Introduction to Master Equations -- 8.1. Two-State System with Constant Rates -- 8.1.1. Particle Point of View -- 8.1.2. Occupation Numbers Point of View -- 8.2. General Case -- 8.3. Examples -- 8.3.1. Radioactive Decay -- 8.3.2. Birth (from a Reservoir) and Death Process -- 8.3.3. Chemical Reaction -- 8.3.4. Self-Annihilation -- 8.3.5. Prey--Predator Lotka--Volterra Model -- 8.4. Generating Function Method for Solving Master Equations -- 8.5. Mean-Field Theory -- 8.6. Fokker--Planck Equation -- Further Reading and References -- Exercises -- 9. Numerical Simulations of Master Equations -- 9.1. First Reaction Method -- 9.2. Residence Time Algorithm -- Further Reading and References -- Exercises -- 10. Hybrid Monte Carlo -- 10.1. Molecular Dynamics -- 10.2. Hybrid Steps -- 10.3. Tuning of Parameters -- 10.4. Relation to Langevin Dynamics -- 10.5. Generalized Hybrid Monte Carlo -- Further Reading and References -- Exercises -- 11. Stochastic Partial Differential Equations -- 11.1. Stochastic Partial Differential Equations -- 11.1.1. Kardar-Parisi-Zhang Equation -- 11.2. Coarse Graining -- 11.3. Finite Difference Methods for Stochastic Differential Equations -- 11.4. Time Discretization: von Neumann Stability Analysis -- 11.5. Pseudospectral Algorithms for Deterministic Partial Differential Equations -- 11.5.1. Evaluation of the Nonlinear Term -- 11.5.2. Storage of the Fourier Modes -- 11.5.3. Exact Integration of the Linear Terms -- 11.5.4. Change of Variables -- 11.5.5. Heun Method -- 11.5.6. Midpoint Runge--Kutta Method -- 11.5.7. Predictor--Corrector -- 11.5.8. Fourth-Order Runge--Kutta -- 11.6. Pseudospectral Algorithms for Stochastic Differential Equations -- 11.6.1. Heun Method -- 11.6.2. Predictor--Corrector -- 11.7. Errors in the Pseudospectral Methods -- Further Reading and References -- Exercises -- A. Generation of Uniform U(0, 1) Random Numbers -- A.1. Pseudorandom Numbers -- A.2. Congruential Generators -- A.3. Theorem by Marsaglia -- A.4. Feedback Shift Register Generators -- A.5. RCARRY and Lagged Fibonacci Generators -- A.6. Final Advice -- Exercises -- B. Generation of n-Dimensional Correlated Gaussian Variables -- B.1. Gaussian Free Model -- B.2. Translational Invariance -- Exercises -- C. Calculation of the Correlation Function of a Series -- Exercises -- D. Collective Algorithms for Spin Systems -- E. Histogram Extrapolation -- F. Multicanonical Simulations -- G. Discrete Fourier Transform -- G.1. Relation Between the Fourier Series and the Discrete Fourier Transform -- G.2. Evaluation of Spatial Derivatives -- G.3. Fast Fourier Transform -- Further Reading -- References
1.6 Interpretation of the Variance: Statistical Errors1.7 Sums of Random Variables; 1.8 Conditional Probabilities; 1.9 Markov Chains; Further Reading and References; Exercises; Chapter 2 Monte Carlo Integration; 2.1 Hit and Miss; 2.2 Uniform Sampling; 2.3 General Sampling Methods; 2.4 Generation of Nonuniform Random Numbers: Basic Concepts; 2.5 Importance Sampling; 2.6 Advantages of Monte Carlo Integration; 2.7 Monte Carlo Importance Sampling for Sums; 2.8 Efficiency of an Integration Method; 2.9 Final Remarks; Further Reading and References; Exercises
Chapter 3 Generation of Nonuniform Random Numbers: Noncorrelated Values3.1 General Method; 3.2 Change of Variables; 3.3 Combination of Variables; 3.3.1 A Rejection Method; 3.4 Multidimensional Distributions; 3.5 Gaussian Distribution; 3.6 Rejection Methods; Further Reading and References; Exercises; Chapter 4 Dynamical Methods; 4.1 Rejection with Repetition: a Simple Case; 4.2 Statistical Errors; 4.3 Dynamical Methods; 4.4 Metropolis et al. Algorithm; 4.4.1 Gaussian Distribution; 4.4.2 Poisson Distribution; 4.5 Multidimensional Distributions; 4.6 Heat-Bath Method; 4.7 Tuning the Algorithms
Further Reading and ReferencesExercises; Chapter 6 Introduction to Stochastic Processes; 6.1 Brownian Motion; 6.2 Stochastic Processes; 6.3 Stochastic Differential Equations; 6.4 White Noise; 6.5 Stochastic Integrals. Itô and Stratonovich Interpretations; 6.6 The Ornstein-Uhlenbeck Process; 6.6.1 Colored Noise; 6.7 The Fokker-Planck Equation; 6.7.1 Stationary Solution; Further Reading and References; Exercises; Chapter 7 Numerical Simulation of Stochastic Differential Equations; 7.1 Numerical Integration of Stochastic Differential Equations with Gaussian White Noise; 7.1.1 Integration Error
Summary Stochastic Numerical Methods introduces at Master level the numerical methods that use probability or stochastic concepts to analyze random processes. The book aims at being rather general and is addressed at students of natural sciences (Physics, Chemistry, Mathematics, Biology, etc.) and Engineering, but also social sciences (Economy, Sociology, etc.) where some of the techniques have been used recently to numerically simulate different agent-based models. Examples included in the book range from phase-transitions and critical phenomena, including details of data analysis
Bibliography Includes bibliographical references
Notes Online resource; title from PDF title page (Wiley, viewed June 18, 2014)
Subject Numerical analysis.
Stochastic analysis.
Stochastic control theory.
Form Electronic book
Author Colet, Pere.
ISBN 1306907977 (ebk)
3527411496
3527683119 (Mobi)
3527683127 (ePub)
3527683135 (electronic bk.)
3527683143 (ebk)
9781306907972 (ebk)
9783527411498
9783527683116 (Mobi)
9783527683123 (ePub)
9783527683130 (electronic bk.)
9783527683147 (ebk)
(Mobi)
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