MATH858D: Stochastic Methods with Applications (Spring 2015, 2017)

The goal of this course is to give an introduction to stochastic methods for the analysis and the study of complex physical, chemical, and biological systems, and their mathematical foundations. 

Syllabus 

Basic concepts of Probability
— Random Variables, Distributions, and Densities
— Expected Values and Moments
— The Law of Large Numbers
— The Central Limit Theorem
— Conditional Probability and Conditional Expectation
— Monte Carlo Methods

Refs:  

1. A. Chorin and O. Hald, “Stochastic Tools for Mathematics and Science”, 3rd edition, Springer, 2013 

2. L. Koralov and Ya. Sinai, Theory of probability and stochastic processes, 2nd edition, Springer, 2007 

Lecture notes       HW1     HW2


Basic Concepts of Statistics
— Estimators, Estimates, and Sampling Distributions
— Ordinary Least Squares and Sampling Distributions
— Bayes’s Theorem

Refs:  

1. A. Chorin and O. Hald, “Stochastic Tools for Mathematics and Science”,  3rd edition, Springer, 2013 

2. Ralph C. Smith, “Uncertainty Quantification, Theory, Implementation, and Applications”, SIAM 2014

Lecture notes     HW3

 

Markov Chains

— Discrete time Markov Chains
— Continuous time Markov Chains
— Representation of Energy Landscapes

— Markov Chain Monte Carlo Algorithms (Metropolis and Metropolis-Hastings)

— Transition Path Theory and Path Sampling Techniques

— Metastability and Spectral Theory

 Refs:

1. J. R. Norris, "Markov Chains", Cambridge University Press, 1998

2. Metzner, P., Schuette, Ch., Vanden-Eijnden, E.: Transition path theory for Markov jump processes. SIAM Multiscale Model. Simul. 7, 1192 – 1219 (2009) 

3. A. Bovier, Metastability, in “Methods of Contemporary Statistical Mechanics”, (ed. R. Kotecky), LNM 1970, Springer, 2009

4.  A. Chorin and O. Hald, “Stochastic Tools for Mathematics and Science”, 3rd edition, Springer, 2013 

Lecture notes      HW4         HW5        HW6

 

Brownian Motion
— Definition of Brownian Motion
— Brownian Motion and Heat Equation
— An Introduction to Stochastic Differential Equations (SDEs)

— Numerical integration of Stochastic ODEs: Euler-Maruyama, Milstein's, MALA

Refs:  

1A. Chorin and O. Hald, “Stochastic Tools for Mathematics and Science”, 3rd edition, Springer, 2013 (Links to an external site.)

2. Zeev Schuss, Theory and Applications of Stochastic Processes,  An analytical approach, Springer, 2010 

3. Grigorios Pavliotis, Stochastic processes and Applications, Diffusion Processes, the Fokker-Planck, and Langevin Equations, Springer, 2014 

4. Desmond J. Higham, An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Review,  43, 3, (2001) 525-546 

Lecture notes       HW7        HW8        HW9


An Introduction into the Large Deviation Theory
— The Freidlin-Wentzell Action Functional
— The Minimum Action Paths and the Minimum Energy Paths
— Methods for computing Minimum Energy Paths and saddle points

 Refs: 

1. Freidlin, M. I. and Wentzell, A. D., Random Perturbations of Dynamical Systems, 2nd edition, Springer, New York, 1998, 3rd Edition, Springer, New York, 2013

2. Heymann, M. and Vanden-Eijnden, E., The Geometric Minimum Action Method: A Least Action Principle on the Space of Curves, Comm. Pure Appl. Math. 61, 8, 1052-1117 (2008)

3. M. K. Cameron, Finding the Quasipotential for Nongradient SDE’s, Physica D: Nonlinear Phenomena, 241 (2012), pp. 1532-155

Lecture notes    HW10

  

Statistical Mechanics
— The Hamilton Equations
— The Liouville Equation

— The Hamilton-Jacobi Equation

— Entropy and Equilibrium
Refs:  

1. A. Chorin and O. Hald, “Stochastic Tools for Mathematics and Science”, 3rd edition, Springer, 2013 (Links to an external site.)

2. V. I. Arnold, Mathematical Methods of Classical Mechanics, Second Edition, Springer, 1989

Lecture notes    HW11

 

Time-Dependent Statistical Mechanics
— A Coupled System of Harmonic Oscillators

— The Zwanzig-Mori Formalism 

Refs:  

1.  A. Chorin and O. Hald, Stochastic Tools in Mathematics and Science, 3rd edition, Springer 2013

2.  Alina Chertock, David Gottlieb, Alex Solomonoff, Modified Optimal Prediction and its Application to a Particle-Method Problem, J Sci Comput (2008) 37: 189201 

Lecture notes

 Copyright 2010, 2015 , 2017 by Maria Cameron