MATH858D: Stochastic Methods with Applications

Lecture notes and HWs are for Spring 2019

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: Sampling and Monte Carlo integration

— Estimators, Estimates, and Sampling Distributions

Refs:

Lecture notes: prob_basic_concepts.pdf, sampling.pdf

Homework: HW1, HW2

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)

Lecture notes: markov_chains.pdf

Homework:

An introduction to data analysis

— Principal component analysis (PCA)

— Multidimensional scaling (MDS)

— Diffusion maps

— Multiscale geometric methods

— Basics of Data Assimilation

Refs:

Codes: DataAssimilation.zip --codes mimic those from 

Lecture notes:

Homework: HW7, SubjSim12countries.mat, MakeSpiral.m

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:

Lecture notes: SDEs.pdf

Homework:

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

Lecture notes:

Codes: Paths&Saddles2019.zip — MATLAB codes for finging transition paths and transition states

Copyright 2010, 2015 , 2017, 2018  by Maria Cameron