# Applied Mathematics and Statistics

2017-18 General Catalog

Baskin School of Engineering

(831) 459-2158

http://www.soe.ucsc.edu

## Lower-Division Courses

**3.** **Precalculus for the Social Sciences.** **F,S**

Introduces mathematical functions and their uses for modeling real-life problems in the social sciences. Includes inequalities, linear and quadratic equations, functions (linear, quadratic, polynomial, rational, power, exponential, logarithmic, trigonometric), inverses, and the composition of functions. Students cannot receive credit for both this course and Mathematics 3. Mathematics 3 can substitute for this course. (Formerly Precalculus for Science and Engineering.) Prerequisite(s): score of 200 or higher on the mathematics placement examination (MPE), or Mathematics 2. (General Education Code(s): MF.) *The Staff,* *P. Garaud,* *B. Mendes*

**5.** **Statistics.** **F,W,S**

Introduction to statistical methods/reasoning, including descriptive methods, data-gathering (experimental design and sample surveys), probability, interval estimation, significance tests, one- and two-sample problems, categorical data analysis, correlation and regression. Emphasis on applications to the natural and social sciences. Students cannot receive credit for this course if they have already received credit for course 7. (General Education Code(s): SR.) *The Staff,* *H. Lee,* *A. Kottas,* *B. Sanso,* *R. Morris,* *J. Katznelson,* *D. Draper,* *A. Rodriguez,* *B. Mendes*

**6.** **Precalculus for Statistics.** *****

Reviews and introduces mathematical methods useful in the elementary study of statistics, including logic, real numbers, inequalities, linear and quadratic equations, functions, graphs, exponential and logarithmic functions, and summation notation. (Formerly course 2, Pre-Statistics.) Prerequisite(s): Mathematics 2 or mathematics placement examination (MPE) score of 200 or higher or higher. (General Education Code(s): MF.) *B. Mendes,* *The Staff*

**7.** **Statistical Methods for the Biological, Environmental, and Health Sciences.** **F,W,S**

Case-study-based introduction to statistical methods as practiced in the biological, environmental, and health sciences. Descriptive methods, experimental design, probability, interval estimation, hypothesis testing, one- and two-sample problems, power and sample size calculations, simple correlation and simple linear regression, one-way analysis of variance, categorical data analysis. Prerequisite(s): score of 300 or higher on the mathematics placement examination (MPE), or course 2 or 3 or 6 or 11A or 15A or Mathematics 3 or 11A or 19A. Concurrent enrollment in course 7L is required. (General Education Code(s): SR.) *H. Lee,* *R. Prado,* *D. Draper,* *A. Rodriguez,* *J. Lee,* *R. Guhaniyogi,* *(F)* *The Staff*

**7L.** **Statistical Methods for the Biological, Environmental, and Health Sciences Laboratory** **(2 credits).** **F,W,S**

Computer-based laboratory course in which students gain hands-on experience in analysis of data sets arising from statistical problem-solving in the biological, environmental, and health sciences. Descriptive methods, interval estimation, hypothesis testing, one-and two-sample problems, correlation and regression, one-way analysis of variance, categorical data analysis. Prerequisite(s): score of 300 or higher on the mathematics placement examination (MPE), course 2 or 3 or 6 or 11A or 15A or Mathematics 3 or 11A or 19A. Concurrent enrollment in course 7 is required. *The Staff,* *H. Lee,* *R. Prado,* *R. Guhaniyogi,* *A. Rodriguez,* *J. Lee,* *D. Draper*

**10.** **Mathematical Methods for Engineers I.** **F,W,S**

Applications-oriented course on complex numbers and linear algebra integrating Matlab as a computational support tool. Introduction to complex algebra. Vectors, bases and transformations, matrix algebra, solutions of linear systems, inverses and determinants, eigenvalues and eigenvectors, and geometric transformations. Students cannot receive credit for this course and for courses 10A or Mathematics 21. Prerequisite(s): score of 400 or higher on the mathematics placement examination (MPE) or Mathematics 3. (General Education Code(s): MF.) *The Staff,* *H. Wang,* *B. Mendes,* *D. Venturi,* *N. Brummell,* *Q. Gong,* *J. Katznelson*

**11A.** **Mathematical Methods for Economists I.** **F,W,S**

Introduction to mathematical tools and reasoning, with applications to economics. Topics are drawn from differential calculus in one variable and include limits, continuity, differentiation, elasticity, Taylor polynomials, and optimization. Students cannot receive credit for both this course and Mathematics 11A or 19A or Applied Mathematics and Statistics 15A. (Also offered as Applied Mathematics and Statistics 11A. Students cannot receive credit for both courses.) (Also offered as Economics 11A. Students cannot receive credit for both courses.) Students who have already taken Mathematics 11A or 19A should not take this course. Prerequisite(s): score of 300 or higher on the mathematics placement examination (MPE), Applied Math and Statistics 2, 3, or 6, or Mathematics 3. (General Education Code(s): MF.) *The Staff,* *J. Katznelson,* *B. Mendes*

**11B.** **Mathematical Methods for Economists II.** **F,W,S**

Mathematical tools and reasoning, with applications to economics. Topics are drawn from multivariable differential calculus and single variable integral calculus, and include partial derivatives, linear and quadratic approximation, optimization with and without constraints, Lagrange multipliers, definite and indefinite integrals, and elementary differential equations. Students cannot receive credit for both this course and Mathematics 11B or 19B or Applied Math and Statistics 15B. (Also offered as Economics 11B. Students cannot receive credit for both courses.) Prerequisite(s): course 11A , Economics 11A, Mathematics 11A, or Mathematics 19A. (General Education Code(s): MF.) *The Staff,* *J. Katznelson,* *B. Mendes*

**15A.** **Case-Study Calculus I.** *****

Case-study-based, first-quarter introduction to single-variable calculus, with computing labs/discussion sections featuring contemporary symbolic, numerical, and graphical computing tools. Case studies drawn from biology, environmental sciences, health sciences, and psychology. Includes functions, mathematical modeling, limits, continuity, tangents, velocity, derivatives, the chain rule, implicit differentiation, higher derivatives, exponential and logarithmic functions and their derivatives, differentiating inverse functions, the mean value theorem, concavity, inflection points, function optimization, and curve-sketching. Students cannot receive credit for this course and course 11A or Economics 11A or Mathematics 11A or 19A. Prerequisite(s): course 3 or Mathematics 3 or score of 300 or higher on the mathematics placement examination (MPE) or by permission of instructor. (General Education Code(s): MF.) *The Staff,* *P. Garaud,* *B. Mendes*

**15B.** **Case-Study Calculus II.** *****

Case-study based, second-quarter introduction to single-variable calculus, with computing labs/discussion sections featuring symbolic numerical, and graphical computing tools. Case studies are drawn from biology, environmental science, health science, and psychology. Includes indefinite and definite integrals of functions of a single variable; the fundamental theorem of calculus; integration by parts and other techniques for evaluating integrals; infinite series; Taylor series, polynomial approximations. Students cannot receive credit for this course and course 11B or Economics 11B or Mathematics 11B of 19B. Prerequisite(s): course 15A or 11A or Economics 11A or Mathematics 11A or 19A. (General Education Code(s): MF.) *The Staff,* *P. Garaud,* *B. Mendes*

**20.** **Mathematical Methods for Engineers II.** **W,S**

Applications-oriented class on ordinary differential equations (ODEs) and systems of ODEs using Matlab as a computational support tool. Covers linear ODEs and systems of linear ODEs; nonlinear ODEs using substitution and Laplace transforms; phase-plane analysis; introduction to numerical methods. Students cannot receive credit for this course and for courses 20A or Mathematics 24. Prerequisite(s): Mathematics 19B, and course 10 or 10A or Mathematics 21. (General Education Code(s): MF.) *The Staff,* *D. Lee,* *Q. Gong,* *J. Katznelson*

**80A.** **Gambling and Gaming.** **F,S**

Games of chance and strategy motivated early developments in probability, statistics, and decision theory. Course uses popular games to introduce students to these concepts, which underpin recent scientific developments in economics, genetics, ecology, and physics. (General Education Code(s): SR.) *H. Lee,* *A. Kottas,* *B. Mendes,* *A. Rodriguez,* *R. Guhaniyogi,* *(F)* *The Staff*

**80B.** **Data Visualization.** **W**

Introduces the use of complex-data graphical representations to extract information from data. Topics include: summary statistics, boxplots, histograms, dotplots, scatterplots, bubble plots, and map-creation, as well as visualization of trees and hierarchies, networks and graphs, and text. (General Education Code(s): SR.) *A. Rodriguez,* *The Staff*

## Upper-Division Courses

**100.** **Mathematical Methods for Engineers III.** *****

Applications-oriented course on complex analysis and partial differential equations using Maple as symbolic math software support. In addition, introduces Fourier analysis, special functions, and asymptotic methods. Students cannot receive credit for this course and Physics 116B or Physics 116C. Prerequisite(s): course 20, or by permission of instructor. Enrollment limited to 25. *The Staff*

**107.** **Introduction to Fluid Dynamics.** **F**

Covers fundamental topics in fluid dynamics: Euler and Lagrange descriptions of continuum dynamics; conservation laws for inviscid and viscous flows; potential flows; exact solutions of the Navier-Stokes equation; boundary layer theory; gravity waves. Students cannot receive credit for this course and Applied Mathematics and Statistics 217. (Also offered as Physics 107. Students cannot receive credit for both courses.) Prerequisite(s): Mathematics 107 or Physics 116C or Earth and Planetary Sciences 111. *N. Brummell,* *The Staff*

**114.** **Introduction to Dynamical Systems.** **F**

Introduces continuous and discrete dynamical systems. Topics include: fixed points; stability; limit cycles; bifurcations; transition to and characterization of chaos; fractals. Examples are drawn from sciences and engineering. Students cannot receive credit for this course and course 214. (Formerly course 146.) Prerequisite(s): course 20 or 20A, or Mathematics 21 and Mathematics 24, or Physics 116B. Enrollment restricted to sophomores, juniors and seniors. (General Education Code(s): MF.) *P. Garaud,* *D. Venturi,* *D. Milutinovic,* *Q. Gong*

**115.** **Stochastic Modeling in Biology.** *****

Application of differential equations, probability, and stochastic processes to problems in cell, organismal, and population biology. Topics include life-history theory, behavioral ecology, and population biology. Students may not receive credit for this course and course 215. Prerequisite(s): course 131, a university-level course in biology, and operational knowledge of a programming language; or consent of instructor. *The Staff*

**131.** **Introduction to Probability Theory.** **F,W,S**

Introduction to probability theory and its applications. Combinatorial analysis, axioms of probability and independence, random variables (discrete and continuous), joint probability distributions, properties of expectation, Central Limit Theorem, Law of Large Numbers, Markov chains. Students cannot receive credit for this course and course 203 and Computer Engineering 107. Prerequisite(s): course 11B or Economics 11B or Mathematics 11B or 19B or 20B. (General Education Code(s): SR.) *The Staff,* *R. Prado,* *J. Lee,* *B. Sanso,* *D. Draper,* *A. Kottas*

**132.** **Classical and Bayesian Inference.** **W**

Introduction to statistical inference at a calculus-based level: maximum likelihood estimation, sufficient statistics, distributions of estimators, confidence intervals, hypothesis testing, and Bayesian inference. Students cannot receive credit for this course and course 206. (Formerly Statistical Inference.) Prerequisite(s): course 131 or Computer Engineering 107. (General Education Code(s): SR.) *The Staff,* *R. Prado,* *J. Lee,* *D. Draper,* *A. Rodriguez,* *A. Kottas*

**147.** **Computational Methods and Applications.** **W**

Applications of computational methods to solving mathematical problems using Matlab. Topics include solution of nonlinear equations, linear systems, differential equations, sparse matrix solver, and eigenvalue problems. Prerequisite(s): course 10 or 10A, or Mathematics 21. Knowledge of differential equations is recommended (course 20 or 20A, or Mathematics 24). (General Education Code(s): MF.) *D. Venturi,* *H. Wang*

**156.** **Linear Regression.** *****

Covers simple linear regression, multiple regression, and analysis of variance models. Students learn to use the software package R to perform the analysis, and to construct a clear technical report on their analysis, readable by either scientists or nontechnical audiences. (Formerly *Linear Statistical Models*.) Prerequisite(s): course 132 and satisfaction of the Entry Level Writing and Composition requirements. Enrollment limited to 30. *H. Lee*

**198.** **Independent Study or Research.** **F,W,S**

Students submit petition to sponsoring agency. May be repeated for credit. *The Staff*

**198F.** **Independent Study or Research** **(2 credits).** **F,W,S**

Students submit petition to sponsoring agency. May be repeated for credit. *The Staff*

## Graduate Courses

**200.** **Research and Teaching in AMS** **(3 credits).** *****

Basic teaching techniques for teaching assistants, including responsibilities and rights; resource materials; computer skills; leading discussions or lab sessions; presentation techniques; maintaining class records; and grading. Examines research and professional training, including use of library; technical writing; giving talks in seminars and conferences; and ethical issues in science and engineering. Enrollment restricted to graduate students. *A. Kottas,* *The Staff*

**202.** **Linear Models in SAS.** *****

Case study-based course teaches statistical linear modeling using the SAS software package. Teaches generalized linear models; linear regression; analysis of variance/covariance; analysis of data with random effects and repeated measures. Prerequisite(s): course 156 or 256, or permission of instructor. Enrollment restricted to graduate students. *B. Mendes,* *The Staff*

**203.** **Introduction to Probability Theory.** **F**

Introduces probability theory and its applications. Requires a multivariate calculus background, but has no measure theoretic content. Topics include: combinatorial analysis; axioms of probability; random variables (discrete and continuous); joint probability distributions; expectation and higher moments; central limit theorem; law of large numbers; and Markov chains. Students cannot receive credit for this course and course 131 or Computer Engineering 107. Enrollment restricted to graduate students, or by permission of the instructor. *The Staff,* *B. Sanso,* *A. Kottas,* *R. Prado*

**204.** **Introduction to Statistical Data Analysis.** **F**

Presents tools for exploratory data analysis (EDA) and statistical modeling in R. Topics include numerical and graphical tools for EDA, linear and logistic regression, ANOVA, PCA, and tools for acquiring and storing large data. No R knowledge is required. Enrollment is restricted to graduate students. Enrollment limited to 30. *A. Rodriguez,* *R. Prado*

**205B.** **Intermediate Classical Inference.** *****

Statistical inference from a frequentist point of view. Properties of random samples; convergence concepts applied to point estimators; principles of statistical inference; obtaining and evaluating point estimators with particular attention to maximum likelihood estimates and their properties; obtaining and evaluating interval estimators; and hypothesis testing methods and their properties. (Formerly Statistical Inference.) Prerequisite(s): course 203 or equivalent. Enrollment restricted to graduate students. *The Staff,* *R. Guhaniyogi,* *D. Draper,* *B. Sanso*

**206.** **Applied Bayesian Statistics.** **W**

Introduces Bayesian statistical modeling from a practitioner's perspective. Covers basic concepts (e.g., prior-posterior updating, Bayes factors, conjugacy, hierarchical modeling, shrinkage, etc.), computational tools (Markov chain Monte Carlo, Laplace approximations), and Bayesian inference for some specific models widely used in the literature (linear and generalized linear mixed models). (Formerly Classical and Bayesian Inference.) Prerequisite(s): course 131 or 203, or by permission of the instructor. Enrollment is restricted to graduate students. *R. Prado,* *A. Kottas,* *D. Draper,* *A. Rodriguez,* *(F)* *The Staff*

**206B.** **Intermediate Bayesian Inference.** **W**

Bayesian statistical methods for inference and prediction including: estimation; model selection and prediction; exchangeability; prior, likelihood, posterior, and predictive distributions; coherence and calibration; conjugate analysis; Markov Chain Monte Carlo methods for simulation-based computation; hierarchical modeling; Bayesian model diagnostics, model selection, and sensitivity analysis. Prerequisite(s): course 203. Enrollment restricted to graduate students; undergraduates may enroll by permission of the instructor. *The Staff,* *J. Lee,* *A. Rodriguez,* *R. Prado*

**207.** **Intermediate Bayesian Statistical Modeling.** **S**

Hierarchical modeling, linear models (regression and analysis of variance) from the Bayesian point of view, intermediate Markov chain Monte Carlo methods, generalized linear models, multivariate models, mixture models, hidden Markov models. Prerequisite(s): courses 206 or 206B; enrollment restricted to graduate students or by permission of instructor. *The Staff,* *D. Draper,* *B. Sanso,* *R. Prado*

**209.** **Foundations of Scientific Computing.** **F**

Covers the fundamental aspects of scientific computing for research. Introduces algorithmic development; programming (including the use of compilers, libraries, debugging, optimization, and code publication); computational infrastructure; and data-analysis tools. Students gain hands-on experience through practical assignments. Enrollment is restricted to graduate students; undergraduates may enroll by permission of the instructor. *The Staff,* *D. Lee,* *N. Brummell*

**211.** **Foundations of Applied Mathematics.** **F**

Accelerated class reviewing fundamental applied mathematical methods for all sciences. Topics include: multivariate calculus, linear algebra, Fourier series and integral transform methods, complex analysis, and ordinary differential equations. Enrollment restricted to graduate students. *The Staff,* *N. Brummell,* *J. Katznelson*

**212A.** **Applied Mathematical Methods I.** **W**

Focuses on analytical methods for partial differential equations (PDEs), including: the method of characteristics for first-order PDEs; canonical forms of linear second-order PDEs; separation of variables; Sturm-Liouville theory; Green's functions. Illustrates each method using applications taken from examples in physics. Course 211 or equivalent is strongly recommended as preparation. Enrollment restricted to graduate students; undergraduates are encouraged to take this class with permission of instructor. *The Staff,* *N. Brummell,* *P. Garaud,* *H. Wang*

**212B.** **Applied Mathematical Methods II.** **S**

Covers perturbation methods: asymptotic series, stationary phase and expansion of integrals, matched asymptotic expansions, multiple scales and the WKB method, Padé approximants and improvements of series. Prerequisite(s): course 212A. Offered in alternate academic years. *The Staff,* *N. Brummell,* *P. Garaud,* *H. Wang*

**213A.** **Numerical Linear Algebra.** **W**

Focuses on numerical solutions to classic problems of linear algebra. Topics include: LU, Cholesky, and QR factorizations; iterative methods for linear equations; least square, power methods, and QR algorithms for eigenvalue problems; and conditioning and stability of numerical algorithms. Provides hands-on experience in implementing numerical algorithms for solving engineering and scientific problems. Basic knowledge of mathematical linear algebra is assumed. *The Staff,* *Q. Gong,* *P. Garaud*

**213B.** **Numerical Methods for the Solution of Differential Equations.** **S**

Introduces the numerical solutions of ordinary and partial differential equations (ODEs and PDEs). Focuses on the derivation of discrete solution methods for a variety of differential equations, and their stability and convergence. Also provides hands-on experience in implementing such numerical algorithms for the solution of engineering and scientific problems using MATLAB software. The class consists of lectures and hands-on programming sections. Basic mathematical knowledge of ODEs and PDEs is assumed, and a basic working knowledge of programming in MATLAB is expected. Enrollment restricted to graduate students. *The Staff,* *D. Lee,* *H. Wang*

**214.** **Applied Dynamical Systems.** **F**

Introduces continuous and discrete dynamical systems. Topics include: fixed points; stability; limit cycles; bifurcations; transition to and characterization of chaos; and fractals. Examples drawn from sciences and engineering; founding papers of the subject are studied. Students cannot receive credit for this course and course 114. Enrollment restricted to graduate students; undergraduates may enroll by permission of instructor. Enrollment limited to 15. *The Staff,* *P. Garaud,* *D. Venturi,* *D. Milutinovic,* *Q. Gong*

**215.** **Stochastic Modeling in Biology.** *****

Application of differential equations and probability and stochastic processes to problems in cell, organismal, and population biology. Topics include: life-history theory, behavioral ecology, and population biology. Students may not receive credit for this course and course 115. Enrollment restricted to graduate students; undergraduates may enroll by permission of the instructor. *The Staff*

**216.** **Stochastic Differential Equations.** **W**

Introduction to stochastic differential equations and diffusion processes with applications to biology, biomolecular engineering, and chemical kinetics. Topics include Brownian motion and white noise, gambler's ruin, backward and forward equations, and the theory of boundary conditions. Enrollment restricted to graduate students; undergraduates may enroll by permission of the instructor. *H. Wang,* *The Staff*

**217.** **Introduction to Fluid Dynamics.** **F**

Covers fundamental topics in fluid dynamics at the graduate level: Euler and Lagrange descriptions of continuum dynamics; conservation laws for inviscid and viscous flows; potential flows; exact solutions of the Navier-Stokes equation; boundary layer theory; gravity waves. Students cannot receive credit for this course and course 107. Enrollment restricted to graduate students; undergraduates may enroll by permission of the instructor. *N. Brummell,* *The Staff*

**221.** **Bayesian Decision Theory.** *****

Explores conceptual and theoretical bases of statistical decision making under uncertainty. Focuses on axiomatic foundations of expected utility, elicitation of subjective probabilities and utilities, and the value of information and modern computational methods for decision problems. Prerequisite(s): course 206. Enrollment restricted to graduate students. *D. Draper,* *B. Sanso*

**223.** **Time Series Analysis.** **W**

Graduate level introductory course on time series data and models in the time and frequency domains: descriptive time series methods; the periodogram; basic theory of stationary processes; linear filters; spectral analysis; time series analysis for repeated measurements; ARIMA models; introduction to Bayesian spectral analysis; Bayesian learning, forecasting, and smoothing; introduction to Bayesian Dynamic Linear Models (DLMs); DLM mathematical structure; DLMs for trends and seasonal patterns; and autoregression and time series regression models. Prerequisite(s): course 206B, or by permission of instructor. Enrollment restricted to graduate students. *R. Prado*

**225.** **Multivariate Statistical Methods.** **S**

Introduction to statistical methods for analyzing data sets in which two or more variables play the role of outcome or response. Descriptive methods for multivariate data. Matrix algebra and random vectors. The multivariate normal distribution. Likelihood and Bayesian inferences about multivariate mean vectors. Analysis of covariance structure: principle components, factor analysis. Discriminant, classification and cluster analysis. Prerequisite(s): course 206 or 206B, or by permission of instructor. Enrollment restricted to graduate students. *The Staff,* *J. Lee,* *D. Draper*

**227.** **Waves and Instabilities in Fluids.** **W**

Advanced fluid dynamics course introducing various types of small-amplitude waves and instabilities that commonly arise in geophysical and astrophysical systems. Topics covered include, but are not limited to: pressure waves, gravity waves, Rossby waves, interfacial instabilities, double-diffusive instabilities, and centrifugal instabilities. Advanced mathematical methods are used to study each topic. Undergraduates are encouraged to take this course with permission of the instructor. Prerequisite(s): courses 212A and 217. Offered in alternate academic years. *P. Garaud*

**230.** **Numerical Optimization.** **S**

Introduces numerical optimization tools widely used in engineering, science, and economics. Topics include: line-search and trust-region methods for unconstrained optimization, fundamental theory of constrained optimization, simplex and interior-point methods for linear programming, and computational algorithms for nonlinear programming. Basic knowledge of linear algebra is assumed. Enrollment is restricted to graduate students. *Q. Gong*

**231.** **Nonlinear Control Theory.** **S**

Covers analysis and design of nonlinear control systems using Lyapunov theory and geometric methods. Includes properties of solutions of nonlinear systems, Lyapunov stability analysis, effects of perturbations, controllability, observability, feedback linearization, and nonlinear control design tools for stabilization. Prerequisite(s): basic knowledge of mathematical analysis and ordinary differential equations is assumed. Enrollment restricted to graduate students; undergraduates may enroll by permission of the instructor. Offered in alternate academic years. *Q. Gong*

**232.** **Applied Optimal Control.** *****

Introduces optimal control theory and computational optimal control algorithms. Topics include: calculus of variations, minimum principle, dynamic programming, HJB equation, linear-quadratic regulator, direct and indirect computational methods, and engineering application of optimal control. Prerequisite(s): course 114 or 214, or Computer Engineering 240 or 241, or Mathematics 145. Enrollment restricted to graduate students. Offered in alternate academic years. *Q. Gong*

**236.** **Motion Coordination of Robotic Networks.** *****

Comprehensive introduction to motion coordination algorithms for robotic networks. Emphasis on mathematical tools to model, analyze, and design cooperative strategies for control, robotics, and sensing tasks. Topics include: continuous and discrete-time evolution models, proximity graphs, performance measures, invariance principles, and coordination algorithms for rendezvous, deployment, flocking, and consensus. Techniques and methodologies are introduced through application setups from multi-agent robotic systems, cooperative control, and mobile sensor networks. Enrollment restricted to graduate students. Enrollment limited to 15. *The Staff*

**238.** **Fundamentals of Uncertainty Quantification in Computational Science and Engineering.** *****

Computing the statistical properties of nonlinear random system is of fundamental importance in many areas of science and engineering. Introduces students to state-of-the-art methods for uncertainty propagation and quantification in model-based computations, focusing on the computational and algorithmic features of these methods most useful in dealing with systems specified in terms of stochastic ordinary and partial differential equations. Topics include: polynomial chaos methods (gPC and ME-gPC), probabilistic collocation methods (PCM and ME-PCM), Monte-Carlo methods (MC, quasi-MC, multi-level MC), sparse grids (SG), probability density function methods, and techniques for dimensional reduction. Basic knowledge of probability theory and elementary numerical methods for ODEs and PDEs is recommended. Prerequisite(s): course 203 or equivalent, and course 213B or equivalent. Enrollment restricted to graduate students. *D. Venturi*

**241.** **Bayesian Nonparametric Methods.** **S**

Theory, methods, and applications of Bayesian nonparametric modeling. Prior probability models for spaces of functions. Dirichlet processes. Polya trees. Nonparametric mixtures. Models for regression, survival analysis, categorical data analysis, and spatial statistics. Examples drawn from social, engineering, and life sciences. Prerequisite(s): course 207. Enrollment restricted to graduate students. *A. Rodriguez,* *A. Kottas*

**245.** **Spatial Statistics.** *****

Introduction to the analysis of spatial data: theory of correlation structures and variograms; kriging and Gaussian processes; Markov random fields; fitting models to data; computational techniques; frequentist and Bayesian approaches. Prerequisite(s): course 207. Enrollment restricted to graduate students. *B. Sanso,* *H. Lee*

**250.** **An Introduction to High Performance Computing.** **S**

Designed for STEM students and others. Through hands-on practice, this course introduces high-performance parallel computing, including the concepts of multiprocessor machines and parallel computation, and the hardware and software tools associated with them. Students become familiar with parallel concepts and the use of MPI and OpenMP together with some insight into the use of heterogeneous architectures (CPU, CUDA, OpenCL), and some case-study problems. Enrollment restricted to graduate students; undergraduates may enroll by permission of the instructor. *The Staff,* *D. Lee,* *N. Brummell,* *S. Dong*

**256.** **Linear Statistical Models.** **S**

Theory, methods, and applications of linear statistical models. Review of simple correlation and simple linear regression. Multiple and partial correlation and multiple linear regression. Analysis of variance and covariance. Linear model diagnostics and model selection. Case studies drawn from natural, social, and medical sciences. Course 205 strongly recommended as a prerequisite. Undergraduates are encouraged to take this class with permission of instructor. Prerequisite(s): course 205A or 205B or permission of instructor. Enrollment restricted to graduate students. *The Staff,* *R. Prado,* *R. Guhaniyogi,* *A. Rodriguez,* *J. Lee,* *B. Sanso*

**260.** **Computational Fluid Dynamics.** *****

Introduces modern computational approaches to solving the differential equations that arise in fluid dynamics, particularly for problems involving discontinuities and shock waves. Examines the fundamentals of the mathematical foundations and computation methods to obtain solutions. Focuses on writing practical numerical codes and analyzing their results for a full understanding of fluid phenomena. Prerequisite(s): Basic knowledge of computer programming languages is assumed. Enrollment is restricted to graduate students. Offered in alternate academic years. *The Staff,* *D. Lee,* *N. Brummell*

**261.** **Probability Theory with Markov Chains.** **F**

Introduction to probability theory: probability spaces, expectation as Lebesgue integral, characteristic functions, modes of convergence, conditional probability and expectation, discrete-state Markov chains, stationary distributions, limit theorems, ergodic theorem, continuous-state Markov chains, applications to Markov chain Monte Carlo methods. Prerequisite(s): course 205B or by permission of instructor. Enrollment restricted to graduate students. *A. Kottas*

**263.** **Stochastic Processes.** *****

Includes probabilistic and statistical analysis of random processes, continuous-time Markov chains, hidden Markov models, point processes, Markov random fields, spatial and spatio-temporal processes, and statistical modeling and inference in stochastic processes. Applications to a variety of fields. Prerequisite(s): course 205A, 205B, or 261, or by permission of instructor. *The Staff,* *A. Rodriguez,* *A. Kottas*

**266A.** **Data Visualization and Statistical Programming in R** **(3 credits).** **F**

Introduces students to data visualization and statistical programming techniques using the R language. Covers the basics of the language, descriptive statistics, visual analytics, and applied linear regression. Enrollment is by permission of the instructor. (Also offered as Computer Science 266A. Students cannot receive credit for both courses.) Enrollment limited to 30. *A. Rodriguez,* *S. Lodha*

**266B.** **Advanced Statistical Programming in R** **(3 credits).** **W,S**

Teaches students already familiar with the R language advanced tools such as interactive graphics, interfacing with low-level languages, package construction, debugging, profiling, and parallel computation. (Also offered as Computer Science 266B. Students cannot receive credit for both courses.) Prerequisite(s): Applied Mathematics and Statistics 266A or Computer Science 266A. Enrollment limited to 30. *A. Rodriguez,* *S. Lodha*

**266C.** **Introduction to Data Wrangling** **(3 credits).** **W,S**

Introduces students to concepts and tools associated with data collection, curation, manipulation, and cleaning including an introduction to relational databases and SQL, regular expressions, API usage, and web scraping using Python. (Also offered as Computer Science 266C. Students cannot receive credit for both courses.) Prerequisite(s): Applied Mathematics and Statistics 266A or Computer Science 266A. Enrollment limited to 30. *A. Rodriguez,* *S. Lodha*

**268.** **Advanced Bayesian Computation.** **W**

Teaches some advanced techniques in Bayesian Computation. Topics include Hamiltonian Monte Carlo; slice sampling; sequential Monte Carlo; assumed density filtering; expectation propagation; stochastic gradient descent; approximate Markov chain Monte Carlo; variational inference; and stochastic variational inference. Prerequisite(s): course 207, or by permission of the instructor. Enrollment restricted to graduate students; undergraduates may enroll by permission of the instructor. *R. Guhaniyogi*

**274.** **Generalized Linear Models.** **F**

Theory, methods, and applications of generalized linear statistical models; review of linear models; binomial models for binary responses (including logistical regression and probit models); log-linear models for categorical data analysis; and Poisson models for count data. Case studies drawn from social, engineering, and life sciences. Prerequisite(s): course 205A, 205B, or 256. Enrollment restricted to graduate students. *A. Kottas,* *The Staff*

**275.** **Magnetohydrodynamics.** **S**

Studies the interaction of fluid motion and magnetic fields in electrically conducting fluids, with applications in many natural and man-made flows ranging from, for example, planetary physics and astrophysics to industrial metallurgic engineering. (Also offered as Earth Sciences 275. Students cannot receive credit for both courses.) Prerequisite(s): course 107 or 217. Course 227 suggested. Enrollment restricted to graduate students. Offered in alternate academic years. *N. Brummell,* *P. Garaud*

**276.** **Bayesian Survival Analysis and Clinical Design.** *****

Introduction to Bayesian statistical methods for survival analysis and clinical trial design: parametric and semiparametric models for survival data, frailty models, cure rate models, the design of clinical studies in phase I/II/III. Prerequisite(s): course 207 or by permission of instructor. Enrollment restricted to graduate students. *J. Lee*

**280A.** **Seminar in Mathematical and Computational Biology** **(2 credits).** *****

Weekly seminar on mathematical and computational biology. Participants present research findings in organized and critical fashion, framed in context of current literature. Students present own research on a regular basis. Enrollment restricted to graduate students. Enrollment limited to 20. May be repeated for credit. *The Staff*

**280B.** **Seminars in Statistical and Applied Mathematical Modeling** **(2 credits).** **F,W,S**

Weekly seminar series covering topics of current research in applied mathematics and statistics. Permission of instructor required. Enrollment restricted to graduate students. (Formerly *Seminar in Applied Mathematics and Statistics*.) May be repeated for credit. *The Staff*

**280C.** **Seminar in Geophysical and Astrophysical Fluid Dynamics** **(2 credits).** **F,W,S**

Weekly seminar/discussion group on geophysical and astrophysical fluid dynamics covering both analytical and computational approaches. Participants present research progress and findings in semiformal discussions. Students must present their own research on a regular basis. Enrollment restricted to graduate students; undergraduates may enroll by permission of the instructor. May be repeated for credit. *P. Garaud,* *D. Lee,* *N. Brummell*

**280D.** **Seminar in Bayesian Statistical Methodology** **(2 credits).** *****

Weekly seminar/discussion group on Bayesian statistical methods, covering both analytical and computational approaches. Participants present research progress and finding in semiformal discussions. Students must present their own research on a regular basis. Enrollment restricted to graduate students. May be repeated for credit. *The Staff*

**285.** **Seminar in Career Skills** **(2 credits).** **F**

Seminar in career skills for applied mathematicians and statisticians. Learn about professional activities such as the publication process, grant proposals, and the job market. Enrollment restricted to graduate students, typically within two years of their expected Ph.D. completion date. *H. Lee,* *(F)* *The Staff*

**290A.** **Topics in Mathematical and Computational Biology** **(2 credits).** *****

Focuses on applications of mathematical and computational methods with particular emphasis on advanced methods applying to organismal biology or resource management. Students read current literature, prepare critiques, and conduct projects. Enrollment restricted to graduate students. Enrollment limited to 20. May be repeated for credit. *The Staff*

**290B.** **Advanced Topics in the Numerical Solution of PDEs.** *****

Modern practical methods for the numerical solution of partial differential equations. Methods considered depend on the expertise of the instructor, but are covered in-depth and up to the cutting-edge of practical contemporary implementation. Content could be method-based (e.g., spectral methods, finite-element methods) or topic-based (e.g., simulations of turbulence). Some programming and numerical analysis (e.g., course 213) highly recommended. Enrollment restricted to graduate students; undergraduates may enroll by permission of the instructor. *The Staff,* *N. Brummell,* *P. Garaud,* *H. Wang*

**291.** **Advanced Topics in Bayesian Statistics** **(3 credits).** *****

Advanced study of research topics in the theory, methods, or applications of Bayesian statistics. The specific subject depends on the instructor. Enrollment restricted to graduate students and by permission of instructor. May be repeated for credit. *The Staff*

**296.** **Masters Project** **(2 credits).** **F,W,S**

Independent completion of a masters project under faculty supervision. Students submit petition to sponsoring agency. Enrollment restricted to graduate students. May be repeated for credit. *The Staff*

**297.** **Independent Study or Research.** **F,W,S**

Independent study or research under faculty supervision. Students submit petition to sponsoring agency. Enrollment restricted to graduate students. *The Staff*

**297F.** **Independent Study** **(2 credits).** **F,W,S**

Independent study or research under faculty supervision. Students submit petition to sponsoring agency. Enrollment restricted to graduate students. May be repeated for credit. *The Staff*

**299.** **Thesis Research.** **F,W,S**

Thesis research under faculty supervision. Students submit petition to sponsoring agency. Enrollment restricted to graduate students. *The Staff*

Revised: 09/01/17