Maria K. Cameron

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

• [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: 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)


• [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: markov_chains.pdf

Homework: HW₃, HW₄, HW₅, MakeRandomGraph.m, HW₆, LJ7.zip

An introduction to data analysis

• Principal component analysis (PCA)
• Multidimensional scaling (MDS)
• Diffusion maps
• Multiscale geometric methods
• Basics of Data Assimilation

Refs:

• [1] K. Law, A. Stuart, K. Zygalakis, Data Assimilation: a mathematical introduction, Springer, 2015

Codes: DataAssimilation.zip -- coes mimic those from [1]

Lecture notes: data_analysis.pdf

Homework: HW₇, SubjSim12countries.mat, MakeSprial.m

Brownian Motion

• Definition of Brownian Motion
• Brownian Motion and Heat Equation
• An Introduction to Stochastic Differential Equations (SDEs)
• Numberical integration of Stochastic ODEs: Euler-Maruyama, Milsteain's, MALA

Refs:

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


• [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: HW₈, HW₉, HW₁₀

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

• [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: LDT.pdf

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

Take-home final