Course Schedule and Catalog

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Distribution Group: Distribution Group III

Credit Hours: 3

Prerequisite(s): MATH 101

Description: INTRODUCTION TO ENGINEERING COMPUTATION ***** 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 neutral nets.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 212 AND CAAM 210

Description: MATRIX ANALYSIS ***** Equilibria and the solution of linear and linear least squares problems. Dynamical systems and the eigenvalue problem with the Jordan form and Laplace transform via complex integration. ***** Prerequisite(s): MATH 212 and CAAM 210.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 212 AND CAAM 210

Description: DIFFERENTIAL EQUATIONS IN SCIENCE AND ENGINEERING ***** Classical solution techniques for ordinary and partial differential equations. Green's functions, fourier series, finite element method for initial and boundary value problems arising in diffusion and wave propagation phenomena. ***** Prerequisite(s): MATH 212 and CAAM 210.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture/Laboratory

Credit Hours: 3

Prerequisite(s): MATH 210

Description: COMPUTATIONAL NUMERICAL ANALYSIS ***** An introductory course in numerical analysis with computer applications. ***** Prerequite(s): MATH 212 and CAAM 210.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 212

Description: INTRODUCTION TO OPERATIONS RESEARCH AND OPTIMIZATION ***** 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. ***** Prerequisite(s): MATH 212, and any one of the following: CAAM 335, MATH 211 or MATH 355.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 211 AND MATH 212

Description: ANALYSIS I ***** Real numbers completeness, sequences and convergence, compactness, continuity, the derivative, the Riemann integral, fundamental theorem of calculus. Vectors spaces, dimension, linear maps, inner products and norms. ***** Prerequisite(s): MATH 211 and 212 or permission of instructor.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 401

Description: ANALYSIS II ***** 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. ***** Prerequisite(s): CAAM 401.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): MATH 210 OR CAAM 335

Description: THEORETICAL NEUROSCIENCE ***** 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 rate in visual neurons. Finally, methods to analyze phase-locking and activity in populations of cells as well as learning algorithms will be considered. ***** Prerequisite(s): MATH 211 or CAAM 335.

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

Description: COMPUTATIONAL SCIENCE I ***** 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 computation. ***** Prerequisite(s): CAAM 210; CAAM 335 or 353, or permission of instructor.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

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

Description: DYNAMICAL SYSTEMS- THEORY AND COMPUTATION ***** 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. ***** Prerequisite(s): CAAM 210, MATH 212, (CAAM 335 or MATH 355), (CAAM 401 or MATH 321) *****

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 336

Description: PDES OF MATHEMATICAL PHYSICS ***** 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. ***** Prerequisite(s): CAAM 336 or permission of instructor.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Seminar

Credit Hours: 3

Description: METHODS OF MATHEMATICAL PHYSICS ***** 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. ***** Prerequisite(s): CAAM 402 and CAAM 436, or permission of instructor. CAAM 402 may be taken concurrently.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture/Laboratory

Credit Hours: 4

Prerequisite(s): CAAM 336

Description: NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS ***** 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. CAAM 436 recommended. ***** Prerequisite(s): CAAM 336 or permission of instructor. Computer programming in Matlab is required.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 353

Description: NUMERICAL ANALYSIS I ***** 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. ***** Prerequisite(s): CAAM 335 or permission of the instructor. Computer programming in MATLAB is required.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 453

Description: NUMERICAL ANALYSIS II ***** Iterative methods for linear systems of equations including Krylov subspace methods; gradient method for unconstrained optimization; Newton and Newton-like methods for nonlinear system of equations, unconstrained optimization and nonlinear least squares problems; techniques for improving the global convergence of these algorithms. Theoretical and practical considerations for these algoritms will be discussed. ***** Prerequisite(s): CAAM 453 or permission of the instructor. Computer programming in MATLAB required.

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: OPTIMIZATION THEORY ***** Derivation and application of necessity conditions and sufficiency conditions for constrained optimization problems. ***** Prerequisite(s): MATH 212 and CAAM 335 or MATH 355.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 454

Description: NUMERICAL OPTIMIZATION ***** 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). ***** Prerequisite(s): CAAM 454 or permission of instructor. CAAM 460 recommended (may be taken concurrently).

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 1

Description: DYNAMICAL SYSTEMS LAB ***** Modeling, simulation and visualization of dynamical systems in MATLAB.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Distribution Group: Distribution Group II

Credit Hours: 3

Prerequisite(s): CAAM 378 OR CAAM 464

Description: INTEGER AND COMBINATORIAL OPTIMIZATION ***** 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. ***** Prerequisite(s): CAAM 378 or 464 or permission of the instructor. ***** Also offered as ECON 475.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Independent Study

Credit Hours: 1 TO 6

Description: INDEPENDENT STUDY ***** No description.

Department: *Computational & Applied Math*

Grade Mode: Satisfactory/Unsatisfactory

Course Type: Independent Study

Credit Hours: 1 TO 6

Description: INDEPENDENT STUDY ***** No description. Repeatable for Credit.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Seminar

Credit Hours: 1 TO 6

Description: MATHEMATICAL SCIENCES VIGRE SEMINAR ***** 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, and data simulation. The topics change each semester. ***** Also offered as STAT 499 and MATH 499. Repeatable for Credit.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Seminar

Credit Hours: 1

Description: GRADUATE RESEARCH SEMINAR ***** Presentations of ongoing projects by CAAM students and faculty. Repeatable for Credit.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Description: ORDINARY DIFFERENTIAL EQUATIONS ***** Review of the fundamental properties of nonlinear systems, includes nonlinear ordinary differential equations (e.g., the existence and uniquences of solution), Lyapunov stability (e.g., stability of definitions, Lyapunov's direct method, invariance theory, stability of linear systems, Lyapunov's linerizatin methods, and converse theorems), and inpur-output stability (e.g., the small gain theorem and passivity theorem), as well as case studies showing applications to nonlinear and adaptive control and robotics. Course not offered every year. ***** Also offered as MECH 508 and ELEC 508.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 420

Description: COMPUTATIONAL SCIENCE II ***** 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. ***** Prerequisite(s): CAAM 420.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Description: ADVANCED STATISTICAL INFERENCE ***** See STAT 533.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 402

Description: APPLIED FUNCTIONAL ANALYSIS ***** Hilbert spaces, Banach spaces, spectral theory, and weak topologies with applications to signal processing, control, and partial differential equations. ***** Prerequisite(s): CAAM 402 or permission of instructor.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 454

Description: NUMERICAL LINEAR ALGEBRA ***** Direct methods for large, sparse linear system; regularization of ill-conditioned lest squares problems; backward error analysis of basic algorithms for linear equations and least squares, condition estimation. Preconditioned iterative methods for linear systems (CG, GMRES, BiCGstab, QMR); matrix theory including spectral decompositions, Schur form, eigenvalue perturbations, and the geometry of subspaces. Eigenvalue algorithms, Sylvester's equation, the implicity shifted QR algorithm, computation of the SVD, generalized eigenvalue problems. Introduction to large scale eigen value algoritms and multigrid. ***** Prerequisite(s): CAAM 454 or permission of the instructor. Computer programming in Matlab and one or more of C, F77, C++, F90 is required.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 402 AND CAAM 436

Description: PARTIAL DIFFERENTIAL EQUATIONS ***** Analysis of boundary and initial value problems. Dirichlet problem for Laplace's equation, variational formulation, Rayleigh-Ritz principle, Sobolev spaces, weak solutions, convergence of the finite element method, interior and boundary regularity, heat equation and the Gaussian kernel, energy estimates, maximum principle, stability, consistency, and convergence of numerical methods, the Fourier transform, Fourier synthesis of Green's functions for the wave equation, von Neumann analysis of finite difference methods for waves. ***** Prerequisite(s): CAAM 402, CAAM 436.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Description: ENGINEERING APPROACH TO MATH PROGRAM ***** 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. ***** Also offered as MECH 563. ***** Instructor(s): Miele

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Description: MATHEMATICAL PROBABILITY I ***** Measure-theoretic foundations of probability for students who need access to advanced mathematical literature in probability and random processes.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): STAT 381

Description: INTRODUCTION TO RANDOM PROCESSES & APPLICATIONS ***** Review of basic probability and the formulation, analysis, representation, and application of some random 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. ***** Prerequisite(s): STAT 381 (STAT 581 recommended). ***** Also offered as ELEC 533 and STAT 583.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Independent Study

Credit Hours: 1 TO 15

Description: INDEPENDENT STUDY ***** No description. Repeatable for Credit.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Independent Study

Credit Hours: 1 TO 15

Description: INDEPENDENT STUDY ***** No description. Repeatable for Credit.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 460 AND CAAM 454

Description: OPTIMIZATION WITH SIMULATION CONSTRAINTS ***** Content varies from year to year. Course may be repeated for credit. ***** Prerequisites): CAAM 464 or permission of the instructor. Repeatable for Credit.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Seminar

Credit Hours: 3

Description: TOPICS IN INVERSE PROBLEMS ***** Theoretical, computational and practical issues for inverse problems in science and engineering. Selected topics will vary depending on instructor and student interests. May be repeated for credit. ***** Prerequisite(s): Permission of instructor. Repeatable for Credit.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Prerequisite(s): CAAM 551

Description: TOPICS IN NUMERICAL LINEAR ALGEBRA ***** 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; dimensions reduction for large dynamical contol systems; applications involving nonnormal matrices. May be repeated for credit. ***** Prerequisite(s): CAAM 551 or permission of instructor. Repeatable for Credit.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Description: TOPICS IN NUMERICAL DIFFERENTIAL EQUATIONS ***** Content varies from year to year. Repeatable for Credit.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Lecture

Credit Hours: 3

Description: TOPICS IN OPTIMIZATION ***** Content varies from year to year. Course may be repeated for credit. Repeatable for Credit.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Seminar

Credit Hours: 3

Description: TOPICS IN NONLINEAR PROGRAMMING ***** Content varies from year to year.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Seminar

Credit Hours: 1 TO 9

Description: MATHEMATICAL SCIENCES VIGRE SEMINAR ***** 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, and data simulation. The topics change each semester. ***** Also offered as STAT 699 and MATH 699. Repeatable for Credit.

Department: *Computational & Applied Math*

Grade Mode: Standard Letter

Course Type: Research

Credit Hours: 1 TO 15

Description: THESIS ***** No description. Repeatable for Credit.