Stochastic Numerical Methods : An Introduction for Students and Scientists.

Format:

Book

Author/Creator:

Toral, Ral.

Contributor:

Colet, Pere.

Toral, Raúl.

Toral, Ra L.

Language:

English

Subjects (All):

Algorithms.

Equations, Simultaneous--Numerical solutions.

Equations, Simultaneous -- Numerical solutions.

Mathematics.

Physical Description:

1 online resource (419 pages)

Edition:

1st ed.

Place of Publication:

Weinheim : John Wiley & Sons, Incorporated, 2014.

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 (extraction of critical exponents, finite-size effects, etc.), to population dynamics, interfacial growth, chemical reactions, etc. Program listings are integrated in the discussion of numerical algorithms to facilitate their understanding. From the contents: Review of Probability Concepts Monte Carlo Integration Generation of Uniform and Non-uniform Random Numbers: Non-correlated Values Dynamical Methods Applications to Statistical Mechanics Introduction to Stochastic Processes Numerical Simulation of Ordinary and Partial Stochastic Differential Equations Introduction to Master Equations Numerical Simulations of Master Equations Hybrid Monte Carlo Generation of n-Dimensional Correlated Gaussian Variables Collective Algorithms for Spin Systems Histogram Extrapolation Multicanonical Simulations.

Contents:

Intro

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

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

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

Chapter 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 A Rejection Method

3.4 Multidimensional Distributions

3.5 Gaussian Distribution

3.6 Rejection Methods

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

4.7.1 Parameter Tuning.

4.7.2 How Often?

4.7.3 Thermalization

Chapter 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

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

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

7.2 The 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?

Exercises.

Chapter 8 Introduction to Master Equations

8.1 A Two-State System with Constant Rates

8.1.1 The Particle Point of View

8.1.2 The Occupation Numbers Point of View

8.2 The General Case

8.3 Examples

8.3.1 Radioactive Decay

8.3.2 Birth (from a Reservoir) and Death Process

8.3.3 A Chemical Reaction

8.3.4 Self-Annihilation

8.3.5 The Prey-Predator Lotka-Volterra Model

8.4 The Generating Function Method for Solving Master Equations

8.5 The Mean-Field Theory

8.6 The Fokker-Planck Equation

Chapter 9 Numerical Simulations of Master Equations

9.1 The First Reaction Method

9.2 The Residence Time Algorithm

Chapter 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

Chapter 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

A Generation of Uniform Û(0,1) Random Numbers

A.1 Pseudorandom Numbers

A.2 Congruential Generators

A.3 A Theorem by Marsaglia

A.4 Feedback Shift Register Generators

A.5 RCARRY and Lagged Fibonacci Generators

A.6 Final Advice

B Generation of n-Dimensional Correlated Gaussian Variables

B.1 The Gaussian Free Model

B.2 Translational Invariance

C Calculation of the Correlation Function of a Series

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 The Fast Fourier Transform

Further Reading

References

Index

EULA.

Notes:

Description based on publisher supplied metadata and other sources.

Other Format:

Print version: Toral, Ral Stochastic Numerical Methods

ISBN:

9783527683123

OCLC:

882609092