COURSE DESCRIPTIONS. 1 of 5 8/21/2008 3:15 PM. (S) = Spring and (F) = Fall. All courses are 3 semester hours, unless otherwise noted.

Transcription

1 1 of 5 8/21/2008 3:15 PM COURSE DESCRIPTIONS (S) = Spring and (F) = Fall All courses are 3 semester hours, unless otherwise noted. INTRODUCTORY COURSES: CAAM 210 (BOTH) INTRODUCTION TO ENGINEERING COMPUTATION Modeling, Simulation, and Visualization via MATLAB. Numerical methods: Newton s method in one and several dimensions. Gaussian elimination and optimization. Applications to gene nets, fiber nets, and neural nets. Prerequisite: Math 101. CAAM 335 (BOTH) MATRIX ANALYSIS 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 experiements with vibrating beaded strings. Prerequisites: MATH 212 and CAAM 210. CAAM 336 (BOTH) DIFFERENTIAL EQUATIONS IN SCIENCE AND ENGINEERING 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. Prerequisites: MATH 212 and CAAM 210. CAAM 353 (S) COMPUTATIONAL NUMERICAL ANALYSIS An introductory course in numerical analysis with computer applications. Prerequisite: MATH 212 and CAAM 210. CAAM 378 (F) 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. Prerequisites: MATH 212, and any one of the following: CAAM 335, MATH 211, or MATH 355. ADVANCED COURSES: CAAM 401 (F) 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, operative norms. Prerequisite: MATH 211/212 or permission of instructor. CAAM 402 (S) 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: CAAM 401. CAAM 415 (S) 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 rates in visual neurons. Finally, methods to analyze phase-locking and activity in populations of cells as well as learning algorithms will be considered.

2 2 of 5 8/21/2008 3:15 PM Prerequisite: MATH 211 or CAAM 335. CAAM 420 (F) 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 computing. Prerequisites: CAAM 210; CAAM 335 or 353, or permission of instructor. CAAM 435 (F) 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. Prerequisites: CAAM 210, MATH 212, (CAAM 335 or MATH 355), (CAAM 401 or MATH 321) or permission of instructor. CAAM 436 (F) PARTIAL DIFFERENTIAL EQUATIONS 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. Prerequisites: CAAM 336 or permission of instructor. CAAM 437 (S) METHODS OF MATHEMATICAL PHYSICS Analysis of the solutions of the partial differential equations of continuum physics. First order linear and nonlinear PDE's and systems of PDE's, characteristics, and shocks. Sturm-Liouville problems and Fourier series. Integral transforms: Fourier and Laplace. Integral relations and Green's functions. Asymptotic methods: regular pertubation methods, singular pertubations, and geometric optics. Prerequisites: CAAM 402 and CAAM 436, or permission of instructor. CAAM 402 may be taken concurrently. CAAM 452 (S) NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS (4) Structure and properties of the finite element method for statistics problems in mechanics, electromagnetism, and other field theories. Finite difference methods for initial/boundary value problems of fluid flow, heat transfer, and wave motion. Prerequisite: CAAM 336 or permission of instructor. CAAM 436 recommended. Computer programming in Matlab is required. CAAM 453 (F) 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. Prerequisites: CAAM 335 or permission of the instructor. Computer programming in Matlab is required. CAAM 454 (S) 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 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. Prerequisites: CAAM 453 or permission of the instructor. Computer programming in Matlab is required. CAAM 469 (BOTH) DYNAMICAL SYSTEMS LAB Modeling, simulation, and visualization of dynamical systems in MATLAB. CAAM 470 (S) INTRODUCTION TO GRAPH THEORY 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. CAAM 471 (S) INTRODUCTION TO LINEAR AND INTEGER PROGRAMMING Linear and integer programming involve formulating and solving fundamental optimization models widely used in practice. This course introduces the basic theory, algorithms, and software of linear and integer programming. Topics studied in the linear programming part include polyhedron concepts, simplex methods, duality, sensitivity analysis and decomposition techniques. Building on linear programming, the second part of this course introduces modeling with integer variables and solution methodologies in integer programming including branch-and-bound and cutting-plane techniques. This course will provide a basis for further studies in convex and combinatorial optimization. Prerequisites: Linear Algebra/Matrix Analysis (CAAM 335 or equivalent). CAAM 474 (F) COMBINATORIAL OPTIMIZATION 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. Prerequisites: CAAM 378 or 471 or permission of the instructor.

3 3 of 5 8/21/2008 3:15 PM CAAM 490 (F) INDEPENDENT STUDY (Hours Variable) CAAM 491 (S) INDEPENDENT STUDY (Hours Variable) CAAM 495 (F) SENIOR DESIGN PROJECT I (Hours Variable) Students engage in team-oriented year-long design projects that utilize modeling, analysis, and scientific computing skills to solve a problem motivated by an application in engineering or the physical, biological, or social sciences. Participants attend biweekly seminars addressing research techniques and the effective written and verbal presentation of mathematics. CAAM 498 (F) RESEARCH THEMES IN THE MATHEMATICAL SCIENCES (Hours Variable) A seminar course that will cover selected theme of general research in the mathematical sciences from the perspective of mathematics, computational and applied mathematics, and statistics. The course may be repeated multiple times for credit. Also offered as MATH 498 and STAT 498 CAAM 499 (BOTH) MATH SCIENCES VIGRE SEMINAR (Hours Variable) 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. Also offered as MATH 499 and STAT 499. CAAM 500 (BOTH) GRADUATE RESEARCH SEMINAR (1) Presentations of ongoing projects by CAAM students and faculty. Required of all graduates. CAAM 508 (S) ORDINARY DIFFERENTIAL EQUATIONS Review of the fundamental properties of nonlinear systems, includes nonlinear ordinary differential equations (e.g.: the existence and uniqueness of solutions), Lyapunov stability (e.g.: stability definitions, Lyapunov's direct method, invariance theory, stability of linear systems, Lyapunov's linearization methods, and converse theorems), and input-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. Also offered as MECH 508 and ELEC 508. Not offered every year. CAAM 520 (S) 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: CAAM 420. CAAM 540 (S) APPLIED FUNCTIONAL ANALYSIS Hilbert spaces, Banach spaces, spectral theory, and weak topologies with applications to signal processing, control, and partial differential equations. Prerequisite: CAAM 402 or permission of instructor. CAAM 551 (F) NUMERICAL LINEAR ALGEBRA 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, 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 implicitly shifted QR algorithm, computation of the SVD, generalized eigenvalue problems. Introduction to large scale eigenvalue algorithms and multigrid. Prerequisites: CAAM 453 or permission of the instructor. Computer programming in Matlab and one or more of C, F77, C++, F90 is required. CAAM 552 (F) PARTIAL DIFFERENTIAL EQUATIONS I 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. Prerequisites: CAAM 402, CAAM 436. CAAM 553 (S) PARTIAL DIFFERENTIAL EQUATIONS II 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. Prerequisites: CAAM 402, CAAM 452. CAAM 554 (F) CONVEX OPTIMIZATION 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

4 4 of 5 8/21/2008 3:15 PM 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. Prerequisites: Linear algebra and real analysis. Matlab programming experience will help. No previous background in linear or nonlinear optimization is required. CAAM 560 (F) OPTIMIZATION THEORY Derivation and application of necessity conditions and sufficiency conditions for constrained optimization problems. CAAM 563 (F) ENGINEERING APPROACH TO MATHEMATICAL PROGRAMMING 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. CAAM 564 (S) 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: CAAM 454 or permission of instructor. CAAM 560 recommended (may be taken concurrently). CAAM 583 (F) INTRODUCTION TO RANDOM PROCESSES AND 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: STAT 381 (STAT 581 recommended). Also offered as ELEC 533 and STAT 583. CAAM 590 (F) INDEPENDENT STUDY (Hours Variable) CAAM 591 (S) INDEPENDENT STUDY (Hours Variable) CAAM 600 (S) THESIS WRITING 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). CAAM 640 (BOTH) OPTIMIZATION WITH SIMULATION CONSTRAINTS Course may be repeated for credit. Prerequisites: CAAM 564 or permission of the instructor. CAAM 641 (S) 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. CAAM 651 (S) 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 control systems; linear stability of dynamic applications involving nonnormal matrices. May be repeated for credit. Prerequisite: CAAM 551 or permission of instructor. CAAM 652 (BOTH) TOPICS IN NUMERICAL DIFFERENTIAL EQUATIONS CAAM 654 (BOTH) TOPICS IN OPTIMIZATION CAAM 664 (F) TOPICS IN NONLINEAR PROGRAMMING CAAM 698 (F) RESEARCH THEMES IN THE MATHEMATICAL SCIENCES (Hours Variable) A seminar course that will cover selected theme of general research in the mathematical sciences from the perspective of mathematics, computational and applied mathematics, and statistics. The course may be repeated multiple times for credit. Also offered as MATH 698 and STAT 698

5 5 of 5 8/21/2008 3:15 PM CAAM 699 (BOTH) MATH SCIENCES VIGRE SEMINAR (Hours Variable) 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. Also offered as MATH 699 and STAT 699. CAAM 777 (SUMMER) VISITING RESEARCH TRAINEE (0) CAAM 800 THESIS (Hours Variable) Return to the Top