Chorin / Hald Stochastic Tools in Mathematics and Science

2. Auflage 2009
ISBN: 978-1-4419-1002-8
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 1, 162 Seiten

Reihe: Surveys and Tutorials in the Applied Mathematical Sciences

This introduction to probability-based modeling covers basic stochastic tools used in physics, chemistry, engineering and the life sciences. Topics covered include conditional expectations, stochastic processes, Langevin equations, and Markov chain Monte Carlo algorithms. The applications include data assimilation, prediction from partial data, spectral analysis and turbulence. A special feature is the systematic analysis of memory effects.

Autoren/Hrsg.

Chorin, Alexandre J.

Hald, Ole H

Inhaltsverzeichnis

1;Prefaces;6
1.1;Preface to the Second Edition;6
1.2;Preface to the First Edition;7
2;Contents;9
3;CHAPTER 1;11
3.1;1.1. Least Squares Approximation;11
3.2;1.2. Orthonormal Bases;16
3.3;1.3. Fourier Series;19
3.4;1.4. Fourier Transform;22
3.5;1.5. Exercises;26
3.6;1.6. Bibliography;27
4;CHAPTER 2;28
4.1;2.1. Definitions;28
4.2;2.2. Expected Values and Moments;31
4.3;2.3. Monte Carlo Methods;37
4.4;2.4. Parametric Estimation;41
4.5;2.5. The Central Limit Theorem;43
4.6;2.6. Conditional Probability and Conditional Expectation;46
4.7;2.7. Bayes' Theorem;50
4.8;2.8. Exercises;52
4.9;2.9. Bibliography;54
5;CHAPTER 3;56
5.1;3.1. De nition of Brownian Motion;56
5.2;3.2. Brownian Motion and the Heat Equation;58
5.3;3.3. Solution of the Heat Equation by Random Walks;59
5.4;3.4. The Wiener Measure;63
5.5;3.5. Heat Equation with Potential;65
5.6;3.6. Physicists' Notation for Wiener Measure;73
5.7;3.7. Another Connection Between Brownian Motion and theHeat Equation;75
5.8;3.8. First Discussion of the Langevin Equation;77
5.9;3.9. Solution of a Nonlinear Di erential Equation byBranching Brownian Motion;82
5.10;3.10. A Brief Introduction to Stochastic ODEs;84
5.11;3.11. Exercises;86
5.12;3.12. Bibliography;89
6;CHAPTER 4;91
6.1;4.1. Weak De nition of a Stochastic Process;91
6.2;4.2. Covariance and Spectrum;94
6.3;4.3. Scaling and the Inertial Spectrum of Turbulence;96
6.4;4.4. Random Measures and Random Fourier Transforms;99
6.5;4.5. Prediction for Stationary Stochastic Processes;104
6.6;4.6. Data Assimilation;109
6.7;4.7. Exercises;112
6.8;4.8. Bibliography;114
7;CHAPTER 5;116
7.1;5.1. Mechanics;116
7.2;5.2. Statistical Mechanics;119
7.3;5.3. Entropy and Equilibrium;122
7.4;5.4. The Ising Model;126
7.5;5.5. Markov Chain Monte Carlo;128
7.6;5.6. Renormalization;133
7.7;5.7. Exercises;138
7.8;5.8. Bibliography;141
8;CHAPTER 6;142
8.1;6.1. More on the Langevin Equation;142
8.2;6.2. A Coupled System of Harmonic Oscillators;145
8.3;6.3. Mathematical Addenda;147
8.3.1;6.3.1. How to write a nonlinear system of ordinary di erentialequations as a linear partial di erential equation.;148
8.3.2;6.3.2. More on the semigroup notation.;150
8.3.3;6.3.3. Hermite polynomials and projections.;151
8.4;6.4. The Mori-Zwanzig Formalism;152
8.5;6.5. More on Fluctuation-Dissipation Theorems;157
8.6;6.6. Scale Separation and Weak Coupling;159
8.7;6.7. Long Memory and the t-Model;160
8.8;6.8. Exercises;163
8.9;6.9. Bibliography;164
9;Index;167