# Graduate Courses

## Qualifying Graduate Courses

All courses are 3 credit courses

### ACMS 60650. Applied Partial Differential Equations

This course covers fundamental theory for partial differential equations as well as tools and methods for solving these equations, and the implication for the PDE models in applications. Topics include fundamental solutions, maximum principles, model derivations, solution behaviors and the implications, etc. Selected topics from current research will also be included.

### ACMS 60690. Numerical Analysis I

A solid theoretical introduction to numerical analysis. Least squares and the basic theory of orthogonal functions. Numerical integration in one variable. Numerical linear algebra. Methods to solve systems of nonlinear equations. Numerical solution of ordinary differential equations. Solution of some simple partial differential equations by difference methods.

### ACMS 60786. Applied Linear Models

The first half of this course introduces simple linear regression and multiple regression. The topics include least squares estimation for simple linear regression, matrix notation for multiple regression, least squares estimators, prediction intervals, expectations and distributions of quadratic Forms, analysis of variance, generalized linear hypothesis, simultaneous confidence intervals and joint confidence regions. The second half of this course focuses on advanced topics in linear regression, including the Gauss-Markov theorem, model selection and variable selection, nonlinear regression, Box-Cox transformation, generalized least squares, and techniques used when multicollinearity is a problem, such as principal components regression and ridge regression. Finally, an important generalization of linear regression model, mixed effect model, will be discussed.

### ACMS 60801. Statistical Inference

A first graduate course in the theory of statistics. Basic estimation including unbiased, maximum likelihood and moment estimation; testing hypotheses for standard distributions and contingency tables; confidence intervals and regions; introduction to nonparametric tests and linear regression.

### ACMS 60850. Applied Probability

A thorough introduction to probability theory. Elements of measure and integration theory. Basic setup of probability theory (sample spaces, independence). Random variables, the law of large numbers. Discrete random variables (including random walks); continuous random variables, the basic distributions and sums of random variables. Generating functions, branching processes, basic theory of characteristic functions, central limit theorems. Markov chains. Various stochastic processes, including Brownian motion, queues and applications. Martingales. Other topics as time permits.

## Elective Graduate Courses

All courses are 3 credit courses

### ACMS 50550. Functional Analysis

This one semester course will cover selected topics in Functional Analysis. The theory will be built on Banach and Hilbert spaces and will be applied to selected examples from application including Laplace equations, heat equations, and wave equations. Tools and methods such as fixed point theorems, Dirichlet principle, Semi-group, etc. will be covered in the course.

### ACMS 60212. Advanced Scientific Computing

This course covers fundamental material necessary for using high performance computing in science and engineering. There is a special emphasis on algorithm development, computer implementation, and the application of these methods to specific problems in science and engineering.

### ACMS 60470. Atmospheric Modeling and Data Analysis

This course focuses on the theory and application of modeling tools to investigate atmospheric dynamics and dispersion processes relevant for a range of environmental applications. Different numerical modeling frameworks, including Gaussian, Lagrangian and Eulerian models at different spatial scales will be presented. The students will learn how to perform model simulations and analyze simulated output using statistical techniques for model evaluation and uncertainty quantification. The course includes lectures, in-class practice of different modeling tools, homework, as well as the development of a research project focused on a real-world environmental case study.

### ACMS 60590. Finite Elements in Engineering

Fundamental aspects of the finite-element method are developed and applied to the solution of PDE’s encountered in science and engineering. Solution strategies for parabolic, elliptic, and hyperbolic equations are explored.

### ACMS 60630. Nonlinear Dynamical Systems

Theory of nonlinear dynamical systems has applications to a wide variety of fields, from physics, biology, and chemistry, to engineering, economics, and medicine. This is one of its most exciting aspects--that it brings researchers from many disciplines together with a common language. A dynamical system consists of an abstract phase space or state space, whose coordinates describe the dynamical state at any instant; and a dynamical rule which specifies the immediate future trend of all state variables, given only the present values of those same state variables. Dynamical systems are "deterministic" if there is a unique consequent to every state, and "stochastic" or "random" if there is more than one consequent chosen from some probability distribution. A dynamical system can have discrete or continuous time. The discrete case is defined by a map and the continuous case is defined by a "flow. Nonlinear dynamical systems have been shown to exhibit surprising and complex effects that would never be anticipated by a scientist trained only in linear techniques. Prominent examples of these include bifurcation, chaos, and solitons. This course will be self-contained.

### ACMS 60640. Introduction to Mathematical Biology

Theory of nonlinear dynamical systems has applications to a wide variety of fields, from mathematics, physics, biology, and chemistry, to engineering, economics, and medicine. This is one of its most exciting aspects--that it brings researchers from many disciplines together with a common language. Dynamical systems are "deterministic" if there is a unique consequent to every state, and "stochastic" or "random" if there is more than one consequent chosen from some probability distribution. A dynamical system can have discrete or continuous time. The discrete case is defined by a map and the continuous case is defined by a "flow." Nonlinear dynamical systems have been shown to exhibit surprising and complex effects. Prominent examples of these include bifurcation and chaos. Applications to population dynamics, cancer growth and spread of infection will be discussed amongst others. This course will be self-contained.

### ACMS 60790. Numerical Analysis II

A solid introduction to numerical partial differential equations with an emphasis on finite difference methods for time dependent equations and systems of equations. Interpolation. Stability and convergence of solutions in systems of PDE arising in science and engineering. High order accurate difference methods and Fourier methods. Well posed problems and general solutions for a variety of types of systems of equations with constant coefficients. Stability and convergence. Hyperbolic systems of equations.

### ACMS 60842. Time Series Analysis

This is an introductory and applied course in time series analysis. Popular time series models and computational techniques for model estimation, diagnostic and forecasting will be discussed. Although the book focuses on financial data sets, other data sets, such as climate data, earthquake data, will also be included and discussed within the same theoretical framework.

### ACMS 60852. Advanced Biostatistical Methods

This course introduces advanced statistical methods used in biological and biomedical research. Topics include study designs commonly used in health research including case-control, cross-sectional, prospective and retrospective studies; statistical analysis of different data types arising from biological and health research including Gaussian data, categorical data, count data, survival data, correlated/clustered data models, and diagnostic tests. All statistical methods are illustrated with examples from the biology and health sciences. Students are expected to have basic knowledge in R programming, probabilities and distribution theory, descriptive statistics, statistical inferences including hypothesis testing and estimation, and working knowledge of linear regression, before they can register for the course. Upon completion of the course, students are able to recognize and give examples of different types of data arising in biological and health studies and apply appropriate methods to analyze such data.

### ACMS 60855. Spatio-Temporal Statistics for Environmental Applications

The course aims at providing the foundations of methods for spatio-temporal models for environmental Statistics. The main topic covered will be Gaussian processes in space and time and related notions of stationarity, co-variance functions and optimal interpolation (kriging). Exploratory analysis and inference, with particular emphasis on approximation methods for very large data sets, will be covered in the second part of the course. The last part of the course. The last part of the course will be either dedicated to more methodical (e.g. asymptotics for spatial processes) or applied problems (e.g. climate model emulation, air pollution, visualization in Virtual Reality), depending on class interests.

### ACMS 60875. Statistical Methods in Data Mining

Data mining is widely used to discover useful patterns and relationships in data. We will emphasize on large complex datasets such as those in very large databases or web-based mining. The topics will include data visualization, decision trees, association rules, clustering, case based methods, etc.

### ACMS 60883. Applied Generalized Linear Models

Methods and applications of statistical techniques for analyzing categorical data in two and higher dimensions. Specific topics include tests of association for multiway contingency tables, logistic regression, loglinear models, and regression models for ordinal data. The theory of generalized linear models will be emphasized, as well as estimation methods such as the Newton-Raphson and Fisher scoring. Concepts will be illustrated using examples with real datasets from the social and biological sciences. Calculations will be performed in R.

### ACMS 60885. Applied Bayesian Statistics

A comprehensive treatment of the statistical analysis from a Bayesian perspective. Modern computational tools such as MCMC are emphasized. The principles of Bayesian analysis are described with an emphasis on practical rather than theoretical issues and illustrated with actual data. A variety models are considered, including linear/nonlinear regression, hierarchical models, generalized linear model, and mixed models. Issues on data collection, model formulation, computation, and model checking, and sensitivity analysis are also covered.

### ACMS 60886. Applied Bayesian Statistics II

Based on materials that we learned in applied Bayesian statistics (ACMS 60885), this course further introduces advanced Bayesian modeling and Bayesian nonparametrics, which are inevitable in modern Bayesian data analysis. Topics are organized into three major parts: advanced Bayesian modeling, advanced Bayesian computation, and Bayesian nonparamterics. In the advanced Bayesian modeling section, we will cover a hierarchial (linear) model, models for robust inference and missing data as well as generalized linear model. Efficient Gibbs and Metropolis samplers, hybrid Monte Carlo methods, and modal and distributional approximation will be studied in the advanced Bayesian computation section. Gaussian process models and various applications of the Dirichlet process models will be studied in the Bayesian nonparametric section.

### ACMS 60888. Statistical Computing and Monte Carlo Methods

This course introduces statistical computing and Monte Carlo methods. Topics are organized into two major parts: optimization and Monte Carlo methods. Optimization techniques are commonly used in statistics for finding maximum likelihood estimators, minimizing risks in a Bayesian decision problem, solving nonlinear least square problems, and a wide variety of other tasks all involving optimizations. Monte Carlo statistical methods, particularly those based on Markov chains, have been a part of standard computational techniques used in Bayesian data analysis, because a posterior distribution may not belong to a familiar distribution family. Prerequisite: Students are expected to have some basic knowledge in mathematical statistics and Bayesian Statistics. Students are expected to be familiar with R programming. Basic R implementation will not be dealt with in this class.

### ACMS 70860. Stochastic Analysis

This course is a sequel to ACMS 60850 (Applied Probability). It gives an introduction to stochastic modeling and stochastic differential equations, with application to models from biology and finance. Some topics covered will be: stochastic versus deterministic models; Brownian motion and related processes, e.g., the Ornstein-Uhlenbeck Process; diffusion processes and stochastic differential equations; discrete and continuous Markov chain models with applications; the long run behavior of Markov chains; the Poisson processes with applications; and numerical methods for stochastic processes.

### ACMS 80770. Topics in Applied Mathematics

The subject matter of this course will be an advanced topic in applied mathematics.

### ACMS 80870. Topics in Statistics

The subject matter of this course will be an advanced topic in statistics.

### ACMS 90620. Advanced Continuum Mechanics

This advanced continuum mechanics course is designed for graduate students in various branches of engineering and science, especially aerospace, mechanical, civil engineering, physics, chemistry, and applied mathematics. The course will cover fundamental mathematical concepts of tensor calculus including general tensors on an arbitrary curvilinear coordinate system. The continuum mechanics part focuses on deformation and motion of continua in both Lagrangian and Eulerian frames, the concept of stress, transport equations and objectivity under Euclidean and Galilean transformations, balance laws of mass, momentum and energy, and thermodynamics. The course also covers the principles of constitutive theory regardless of the type of matter including material frame indifference, equipresence, internal material constraints, material symmetry, smoothness and memory.