Course Schedule and Catalog

Long Title: INTRODUCTION TO ENGINEERING COMPUTATION

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Distribution Group: Distribution Group III

Credit Hours: 3

Prerequisite(s): MATH 101

Description: Modeling, Simulation, and Visualization via MATLAB. Numerical methods: Newton's method in one and several dimensions. Gaussian elimination and optimization. Application to gene nets, fiber nets, and neural nets. There is a HelpLab on Thursday nights and TrueLab on Friday afternoons. Plan to be in lab for about one hour per Friday (anytime between 1 and 4pm)

Long Title: MATRIX ANALYSIS

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3 TO 4

Prerequisite(s): MATH 212 AND CAAM 210

Description: Equilibria and the solution of linear systems and linear least squares problems. Dynamical systems and the eigenvalue problem with the Jordan form and Laplace transform via complex integration. Optional 1-credit laboratory motivates concepts from the course via physical experiments involving circuits, spring networks, and vibrating mechanical systems.

Long Title: DIFFERENTIAL EQUATIONS IN SCIENCE AND ENGINEERING

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3 TO 4

Prerequisite(s): MATH 212 AND CAAM 210

Description: Classical and numerical solution techniques for ordinary and partial differential equations. Fourier series and the finite element method for initial and boundary value problems arising in diffusion and wave propagation phenomena. Optional 1-credit laboratory motivates concepts from the course via physical experiments with vibrating musical instrument strings.

Long Title: COMPUTATIONAL NUMERICAL ANALYSIS

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture/Laboratory

Credit Hours: 3

Prerequisite(s): (MATH 212 OR MATH 222)

Description: An introductory course in numerical analysis with computer applications. Topics include floating point arithmetic; algorithms for the solution of linear systems, linear least square problems, and nonlinear equations; interpolation; Fourier transform; numerical integration; numerical solution of ordinary differential equations. Computer programming in Matlab is required.

Long Title: INTRODUCTION TO OPERATIONS RESEARCH AND OPTIMIZATION

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 212 AND (CAAM 335 OR MATH 211 OR MATH 355)

Description: Formulation and solution of mathematical models in management, economics, engineering and science applications in which one seeks to minimize or maximize an objective function subject to constraints, including models in linear, nonlinear and integer programming; basic solution methods for these optimization models; problem solving using a modeling language and optimization software.

Long Title: ANALYSIS I

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 211 AND MATH 212 or permission of instructor

Description: Real numbers, completeness, sequences and convergence, compactness, continuity, the derivative, the Riemann integral, fundamental theorem of calculus. Vector spaces, dimension, linar maps, inner products and norms.

Long Title: ANALYSIS II

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 401

Description: Continuation of Analysis I. Vector spaces of functions, sequences and series, convergence. Continuity and differentiability of functions of several variables, the derivative as a linear map, the contraction mapping principle, inverse and implicit function theorems, fundamental theorems on differential equations, multivariable integration, Stoke's theorem and relatives. Credit may not be received for both CAAM 402 and CAAM 502.

Long Title: THEORETICAL NEUROSCIENCE

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 211 OR CAAM 335

Description: This course introduces current theoretical methods used to model the properties of nerve cells and the processing of information by neuronal networks. Concrete examples that can be implemented using MATLAB will be emphasized. The starting point is the passive cable properties of single neurons and the Hodgkin-Huxley model of action potential generation. Subsequently, models of synaptic transmission and active properties of dendritic trees will be considered. This will be followed by stochastic properties of single neurons and information encoding using mean and instantaneous firing rates in visual neurons. Finally, methods to analyze phase-locking and activity in populations of cells as well as learning algorithms will be considered. Cross-list: NEUR 415.

Long Title: COMPUTATIONAL SCIENCE I

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture/Laboratory

Credit Hours: 3

Prerequisite(s): CAAM 210 AND CAAM 335 OR CAAM 353 or permission of instructor

Description: Scientific programming using high level languages, including C, Fortran, and C++. Emphasis on use of numerical libraries. Basic techniques of project planning, source management, documentation, program construction, i/o, visualization. Object-oriented design for numerical computing.

Long Title: DYNAMICAL SYSTEMS

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 210 AND MATH 212 AND (CAAM 335 OR MATH 335) AND (CAAM 401 OR MATH 321)

Description: Existence and uniqueness for solutions of ordinary differential equations and difference equations, linear systems, nonlinear systems, stability, periodic solutions, bifurcation theory. Theory and theoretical examples are complemented by computational, model driven examples from biological and physical sciences. Cross-list: MATH 435.

Long Title: PARTIAL DDIFFERENTIAL EQUATIONS OF MATHEMATICAL PHYSICS

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 336 or permission of instructor

Description: Derivation and properties of solutions of the partial differential equations of continuum physics. Basic concepts of continuum mechanics, ideal fluids, Navier-Stokes equations, linear elasticity, acoustics, basic principles of thermodynamics, Newtonian heat flow, porous flow, Maxwell's equations, electrical circuits.

Long Title: METHODS OF MATHEMATICAL PHYSICS

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Seminar

Credit Hours: 3

Prerequisite(s): CAAM 402 AND CAAM 436 or permission of instructor

Description: Analysis of the solutions of the partial differential equations of continuum physics. First order linear and non-linear PDE's and systems of PDE's, characteristics, shocks. Sturm-Liouville problems and Fourier series. Integral transforms: Fourier and Laplace. Integral relations and Green's functions. Asymptotic methods: regular perturbation methods, singular perturbations, geometric optics. CAAM 402 may be taken concurrently.

Long Title: SEISMOLOGY I

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 211 AND PHYS 101 AND PHYS 102

Description: Principles of elastic wave propagation, the determination of Earth structure, and the understanding of earthquake systems. Cross-list: ESCI 461.

Long Title: NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture/Laboratory

Credit Hours: 4

Prerequisite(s): CAAM 336 or permission of instructor

Description: Structure and properties of the finite element method for static problems in mechanics, electromagnetism, and other field theories. Finite difference methods for initial/boundary value problems of fluid flow, heat transfer, and wave motion. Computer programming in MATLAB is required. Recommended prerequisite(s): CAAM 436.

Long Title: NUMERICAL ANALYSIS I

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 335 or permission of instructor

Description: Construction and analysis of numerical algorithms for root finding, interpolation and approximation of functions, quadrature, and the solution of differential equations; fundamentals of computer arithmetic; solution of linear systems, least squares problems, and eigenvalue problems via matrix factorizations; the singular value decomposition (SVD) and basic sensitivity analysis. Computer programming in MATLAB is required. Credit may not be received for both CAAM 453 and CAAM 553.

Long Title: NUMERICAL ANALYSIS II

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 453 or permission of instructor

Description: Iterative methods for linear systems of equations including Krylov subspace methods; gradient method for unconstrained optimization; Newton and Newton-like methods for nonlinear systems of equations, unconstrained optimization and nonlinear least squares problems; techniques for improving the global convergence of these algorithms. Theoretical and practical considerations for these algorithms will be discussed. Computer programming in MATLAB is required.

Long Title: OPTIMIZATION THEORY

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 212 AND (CAAM 335 OR MATH 355)

Description: Derivation and application of necessity conditions and sufficiency conditions for constrained optimization problems.

Long Title: NUMERICAL OPTIMIZATION

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 454 or permission of instructor

Description: Numerical algorithms for constrained optimization problems in engineering and sciences, including simplex and interior-point methods for linear programming, penalty, barrier, augmented Lagrangian and SQP methods for nonlinear programming. Recommended prerequisite(s): CAAM 460 (may be taken concurrently).

Long Title: DYNAMICAL SYSTEMS LAB

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 1

Description: Modeling, simulation and visualization of dynamical systems in MATLAB.

Long Title: INTRODUCTION TO GRAPH THEORY

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Description: Study of the structure and properties of graphs, together with a variety of applications. Includes paths, cycles, trees, connectivity, matchings, colorings, planarity, directed graphs, and algorithms. Some knowledge of linear algebra is recommended.

Long Title: COMBINATORIAL OPTIMIZATION

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 378 OR CAAM 464 or permission of instructor

Description: General theory and approaches for solving combinatorial optimization problems are studied. Specific topics include basic polyhedral theory, minimum spanning tress, shortest paths, network flow, matching and matroids. The course also cover the traveling salesman problem.

Long Title: INTEGER PROGRAMMING

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 378 OR CAAM 464 or permission of instructor

Description: Modeling and solving optimization problems with discrete components, graphs and networks; network flow problems; minimum spanning trees; basic polyhedral theory; the knapsack problem; the plant location problem; the set packing problem; computational complexity; branch and bound; cutting planes; Lagrangian relaxation and Bender's decomposition. Cross-list: ECON 475.

Long Title: INDEPENDENT STUDY

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Independent Study

Credit Hours: 1 TO 6

Description:  Repeatable for Credit.

Long Title: INDEPENDENT STUDY

Department: *Computational & Applied Math*

Grade Mode: Satisfactory/Unsatisfactory

Course Type: Independent Study

Credit Hours: 1 TO 6

Description:  Repeatable for Credit.

Long Title: COMPUTATIONAL AND APPLIED MATHEMATICS SEMINAR

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Seminar

Credit Hours: 1 TO 6

Description: This course prepares a student for research in the mathematical sciences on a specific topic. Each section is dedicated to a different topic. Current topics include bioinformatics, biomathematics, computational finance, simulation driven optimization, data simulation, and spectral optimization in rational mechanics. The topics may vary each semester. Cross-list: MATH 499, STAT 499. Repeatable for Credit.

Long Title: GRADUATE RESEARCH SEMINAR

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Seminar

Credit Hours: 1

Description: Presentations of ongoing projects by CAAM students and faculty. Repeatable for Credit.

Long Title: NONLINEAR SYSTEMS

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Description: Mathematical background and fundamental properties of nonlinear systems: Vector norms, matrix norms, matrix measures, existence and uniqueness of solutions of ordinary differential equations. Linearization, second order systems, periodic solutions, approximate methods. Lyapunov stability: Stability definitions, Lyapunov's direct method, invariance theory, stability of linear systems, Lyapunov's linearization method, converse theorems. Selected topics in nonlinear systems analysis and nonlinear control from: Input/Output stability: Small gain theorem, passivity theorem. Perturbation theory, averaging, and singular perturbations. Feeback linearization control. Other methods in the control of nonlinear systems such as backstepping, sliding mode and other Lyapunov-based design methods. Advanced nonlinear and adaptive robot control. Cross-list: ELEC 508, MECH 508.

Long Title: COMPUTATIONAL SCIENCE II

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 420

Description: Vector, shared-memory, and message-passing parallel computer architectures. Numerical linear algebra for these architectures. Memory hierarchy issues, analysis and enhancement of performance, and use of programming tools and environments. Application interfaces including OpenMP and MPI, parallel numerical algorithms and scientific visualization.

Long Title: ADVANCED STATISTICAL INFERENCE

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Long Title: APPLIED FUNCTIONAL ANALYSIS

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 402 or permission of instructor

Description: Hilbert spaces, Banach spaces, spectral theory, and weak topologies with applications to signal processing, control, and partial differential equations.

Long Title: SEISMOLOGY II

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 441 OR ESCI 461

Description: Review elastodynamics. Calculation of synthetic seismograms for acoustic and elastic media using reflectivity, asymptotic and finite difference methods. Migration of reflection data by finite differences, FK and Kirchhoff methods. Travel time inversion.

Long Title: NUMERICAL LINEAR ALGEBRA

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 453 or permission of instructor

Description: Direct methods for large, sparse linear systems; regularization of ill-conditioned least squares problems; backward error analysis of basic algorithms for linear equations and least squares, sensitivity and conditioning of linear systems and least square problems; condition estimation. Preconditioned iterative methods for linear systems (CG, GMRES, BiCGstab, QMR); multigrid methods. Matrix theory including spectral decompositions, Schur form, eigenvalue perturbation theory, and the geometry of subspaces. Eigenvalue algorithms, Sylvester and Lyapunov equations, the implicitly shifted QR algorithm, computation of the SVD, generalized eigenvalue problems. Introduction to large scale eigenvalue algorithms. Proficiency in MATLAB and acquaintance with one or more of C, F77, C++, F90 is required.

Long Title: PARTIAL DIFFERENTIAL EQUATIONS

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 402 AND CAAM 452

Description: Analysis of boundary and initial value problems. Dirichlet problem for Laplace's equation, variational formulation, Rayleigh-Ritz principle, Sobolev spaces, weak solutions, interior and boundary regularity, convergence of the finite element method, heat equation and the Gaussian kernel, maximum principle, stability, consistency, and convergence of numerical methods.

Long Title: PARTIAL DIFFERENTIAL EQUATIONS II

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 402 AND CAAM 436

Description: Mathematical analysis of wave propagation. Hyperbolic systems, energy estimates, existence of weak solutions, domains of dependence and influence, Green's functions, geometric optics, bound any value problems, numerical methods. Recommended courses: CAAM 540 and CAAM 552.

Long Title: CONVEX OPTIMIZATION

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Description: Convex optimization problems arise in communication, system theory, VLSI, CAD, finance, inventory, network optimization, computer vision, learning, statistics, etc, even though oftentimes convexity may be hidden and unrecognized. Recent advances in interior-point methodology have made it much easier to solve these problems and various solvers are now available. This course will introduce the basic theory and algorithms for convex optimization, as well as its many applications to computer science, engineering, management science and statistics. Topics covered: Convex sets and functions, convex conic programming (including linear, quadratic, second-order cone and semidefinite programming), duality theory, interior-point algorithms, and various applications. Recommended Prerequisite(s): Linear algebra and basic analysis. Matlab programming experience will help. No previous background in linear or nonlinear optimization is required.

Long Title: OPTIMIZATION THEORY

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Description: Derivation and application of necessity conditions and sufficiency conditions for constrained optimization problems.

Long Title: ENGINEERING APPROACH TO MATH PROGRAM

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Description: Study of the minimization of functions of variables that are either unconstrained, subject to equality constraints, subject to inequality constraints, or subject to both equality and inequality constraints. Includes analytical and computational methods. Cross-list: MECH 563.

Long Title: NUMERICAL OPTIMIZATION

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 454

Description: Numerical algorithms for constrained optimization problems in engineering and sciences, including simplex and interior-point methods for linear programing, penalty, barrier, augmented Lagrangian and SQP methods for nonlinear programming. Recommended: CAAM 460 (may be taken concurrently)

Long Title: MATHEMATICAL PROBABILITY I

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Description: Measure-theoretic foundations of probability for students who need access to advanced mathematical literature in probability and random processes. Cross-list: STAT 581.

Long Title: INTRODUCTION TO RANDOM PROCESSES AND APPLICATIONS

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): STAT 381

Description: Review of basic probability and the formulation, analysis, representation, and application of some standard random processes. Include sequences of random variables, random vectors and estimation, basic concepts of random processes, random processes in linear systems, expansions of random processes, Wiener filtering, spectral representation of random processes, and white-noise integrals. Cross-list: ELEC 533, STAT 583. Recommended prerequisite(s): STAT 581.

Long Title: INDEPENDENT STUDY

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Independent Study

Credit Hours: 1 TO 15

Description:  Repeatable for Credit.

Long Title: INDEPENDENT STUDY

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Independent Study

Credit Hours: 1 TO 15

Description:  Repeatable for Credit.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Seminar

Credit Hours: 3

Description: Assists the student in preparation of the CAAM MA/PhD thesis and in other writing projects. Stucture of a scientific paper, effective approaches to technical writing, building literature review, results, and discussion sections, how to write a good abstract, oral presentation skills. Pre-requisite: Advisor approval topic and consent of the instructor(s). Instructor Permission Required.

Long Title: THESIS WRITING

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Seminar

Credit Hours: 3

Description: Assists the student in preparation of the CAAM MA/PhD thesis and in other writing projects. Structure of a scientific paper, effective approaches to technical writing, building literature review, results, and discussion sections, how to write a good abstract, oral presentation skills. Prerequisite: Advisor approval of topic and consent of the instructor(s). Instructor Permission Required.

Long Title: OPTIMIZATION WITH SIMULATION CONSTRAINTS

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 464 or permission of instructor

Description: Content varies from year to year. Repeatable for Credit.

Long Title: TOPICS IN INVERSE PROBLEMS

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Seminar

Credit Hours: 3

Description: Theoretical, computational and practical issues for inverse problems in science and engineering. Selected topics will vary depending on instructor and student interests. Instructor Permission Required. Repeatable for Credit.

Long Title: TOPICS IN NUMERICAL LINEAR ALGEBRA

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Seminar

Credit Hours: 1 TO 3

Prerequisite(s): CAAM 551 or permission of instructor

Description: Selected topics will vary depending on instructor and student interests. Derivation and analysis of Krylov and subspace iteration methods for large eigenvalue problems (Lanczos, Arnoldi, Jacobi-Davidson algorithms); preconditioning for linear systems and eigenvalue problems (incomplete LU, domain decomposition, multigrid); convergence analysis including potential theory and pseudospectra. Applications: regularization of discrete inverse problems; dimension reduction for large dynamical control systems; effects on non-normality on behavior of dynamical systems and iterative processes. Repeatable for Credit.

Long Title: TOPICS IN NUMERICAL DIFFERENTIAL EQUATIONS

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Description: Content varies from year to year. Repeatable for Credit.

Long Title: TOPICS IN OPTIMIZATION

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Description: Content varies from year to year. Repeatable for Credit.

Long Title: TOPICS IN NONLINEAR PROGRAMMING

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Seminar

Credit Hours: 3

Description: Content varies from year to year.

Long Title: MATHEMATICAL SCIENCES VIGRE SEMINAR

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Seminar

Credit Hours: 1 TO 9

Description: This course prepares a student for research in the mathematical sciences on a specific topic. Each section is dedicated to a different topic. Current topics include bioinformatics, biomathematics, computational finance, simulation driven optimization, data simulation, and spectral optimization in rational mechanics. The topics may vary each semester. Cross-list: MATH 699, STAT 699. Repeatable for Credit.

Long Title: VISITING RESEARCH TRAINEE

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Research

Credit Hours: 0

Long Title: THESIS

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Research

Credit Hours: 1 TO 15

Description:  Repeatable for Credit.