Stochastic Numerical Methods
An Introduction for Students and Scientists
1. Edition July 2014
416 Pages, Softcover
100 Pictures (50 Colored Figures)
Textbook
Short Description
This unified and accessible textbook introduces numerical methods using probability and stochastic concepts and their applications for the analysis of random processes.
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The book introduces at a master's 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. The authors develop in detail examples from the phase-transitions field to explain the whole process from the numerical simulation (design of the convenient algorithm) to the data analysis (extraction of critical exponents, finite-size effects, etc). The core of the book covers Monte Carlo type methods with applications to statistical physics and phase transitions, numerical methods for stochastic differential equations - both ordinary and partial (including advanced pseudo-spectral methods-, Gillespie's method to simulate the dynamics of systems described by master equations (e.g. birth and death processes, and applications to Biology, such as protein expression and transcription). Finally, and in order to explain modern hybrid algorithms (combining Monte Carlo and stochastic differential equations), the authors explain the basics of molecular dynamics. Appendices with supplementary material for more advanced topics, end-of-chapter practical exercises, and useful codes for the core methods are included.
2. Monte Carlo Integration
3. Generation of Non-uniform Random Numbers: Non-correlated Values
4. Dynamical Methods
5. Applications to Statistical Mechanics
6. Introduction to Stochastic Processes
7. Numerical Simulation of Stochastic Differential
equations
8.Introduction to Master Equations
9. Numerical Simulations of Master Equations
10. Hybrid Monte Carlo
11. Stochastic Partial Differential Equations
A. Generation of Uniform ^U (0; 1) Random Numbers
B. Generation of n-dimensional Correlated Gaussian
Variables
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
Pere Colet is Research Professor at IFISC (CSIC-UIB). He obtained his M.Sc. degree in physics from Universitat de Barcelona (1987) and his Ph. D. also in Physics from Universitat de les Illes Balears (1991), Spain. He was a postdoctoral Fulbright fellow at the School of Physics of the Georgia Institute of Tecnology. In May 1995, he joined the Spanish Consejo Superior de Investigaciones Cientificas. He has co-authored over 100 papers in ISI journals as well as 35 other scientific publications. His research interests include fluctuations and nonlinear dynamics of semiconductor lasers, synchronization of chaotic lasers and encoded communications, synchronization of coupled nonlinear oscillators, pattern formation, and quantum fluctuations in nonlinear optical cavities and dynamics of dissipative solitons.
