Maria K. Cameron

MATH858D: Stochastic Methods with Applications

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, Spring 2021

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

Homework: HW₁

Sampling

• Pseudorandom numbers
• Sampling random variables with given distribution
• Monte Carlo integration
• Estimators and estimates

Lecture notes: sampling.pdf

Homework: 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

• [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, HW₆, LJ7.zip

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

• [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: SDEs.pdf

Homework: HW₇, HW₈

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

Homework: HW₉, HW₁₀, LJ6in2Dsetup.m

An Introduction to data analysis

• Diffusion maps
• Approximating differential operators by means of diffusion maps

• [1] Ronald R. Coifman, Stéphane Lafon, Diffusion maps, Applied and Computational Harmonic Analysis, Volume 21, Issue 1, 2006, Pages 5-30, ISSN 1063-5203, https://doi.org/10.1016/j.acha.2006.04.006.

Lecture notes: DiffusionMapsMATH858T.pdf

Take-home final 2021: final2021.pdf, gmam.m

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Some additional course materials from Spring 2019

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 -- codes mimic those from [1]

Lecture notes: data_analysis.pdf

Homework: HW₇, SubjSim12countries.mat, MakeSprial.m

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

Take-home final 2019