# Maria K. Cameron

### University of Maryland, Department of Mathematics

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### 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

#### Sampling

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

#### 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

#### 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

#### 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

#### An Introduction to data analysis

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

### 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