COMPUTATIONAL AND APPLIED MATHEMATICS --Master of Science in Computational and Applied Mathematics --Doctor of Philosophy Facilities for Graduate Work Support facilities for work in computational and applied mathematics include the Kuehne Physics-Mathematics-Astronomy Library, the McKinney Engineering Library, and the Mallet Chemistry Library. Computing facilities available for graduate student and faculty research are also extensive. The Texas Institute for Computational Mechanics maintains an INTEL iPSC/860 parallel computer with sixteen processors. A number of shared and distributed parallel computers maintained by the Department of Computer Sciences are also available, as are workstations in several academic departments in the College of Engineering and the College of Natural Sciences. Faculty members and graduate students also have access to the resources of the Computation Center described in chapter 1. Areas of Study Graduate study in computational and applied mathematics comprises three areas: applicable mathematics, numerical analysis and scientific computations, and mathematical modeling and applications. Within these broad areas, the student may take courses and conduct research in numerical analysis and scientific computing, applicable mathematics, computational mechanics and physics, parallel computing and computer architecture, and mathematical modeling, and in supporting areas in engineering and science that involve mathematical modeling of physical phenomena and engineering systems. Graduate Studies Committee The following faculty members served on the Graduate Studies Committee in the spring semester of 1994-1995. Aristotle Arapostathis Kenneth S. Ball Eric Baker Becker William Beckner Roger T. Bonnecaze Jerry L. Bona Alan C. Bovik James C. Browne Michael D. Bryant Graham F. Carey E. Ward Cheney Alan K. Cline James W. Daniel Rafael de la Llave Charles N. Friedman Donald S. Fussell Vijay K. Garg John E. Gilbert Linda J. Hayes John Kallinderis Hans Koch Takis Konstantopoulos Michael P. Marder Richard A. Matzner Christine M. Maziar Daene C. McKinney Philip J. Morrison J. Tinsley Oden James B. Rawlings Peter J. Rossky William Schelter Kamy Sepehrnoori Jeff S. Shamma Lawrence C. Shepley Ralph E. Showalter Jack B. Swift Harry L. Swinney Toshiki Tajima John L. Tassoulas Jack S. Turner Karen K. Uhlenbeck Robert van de Geijn Mikhail M. Vishik Robert F. Williams Gary L. Wise Robert E. Wyatt David M. Young, Jr. Admission Requirements Students entering the program are expected to have undergraduate degrees in engineering, computer sciences, mathematics, or a natural science such as physics or chemistry. Degree Requirements Each student develops a program of study that includes a substantial component in each of the three areas of concentration listed below. The program must be reviewed and approved by the Graduate Studies Committee. Master of Science in Computational and Applied Mathematics. This program requires completion of thirty-three semester hours of coursework including a research report, or of thirty-six hours of work without a report. At least twenty-four hours must be chosen from the courses listed under the heading "Areas of Concentration" below, with at least six hours in each area. These twenty-four hours of coursework must be taken on the letter-grade basis. Doctor of Philosophy. Before admission to candidacy for the degree, each student develops a program of study that draws courses from each of the three areas of concentration below, to be approved by the Graduate Studies Subcommittee. The student must also pass an examination in each area. With approval of the Graduate Studies Subcommittee, a student may substitute participating departmental qualifying requirements in one or more areas. In addition to meeting the area requirements, the student must prepare a written dissertation proposal. Oral presentation of the proposal and an oral examination are required. A dissertation is required of every candidate, followed by a final oral examination covering the dissertation and the general field of the dissertation. Areas of Concentration AREA A: APPLICABLE MATHEMATICS Computational and Applied Mathematics 381D, Complex Analysis Computational and Applied Mathematics 381M, Methods of Mathematical Physics Computational and Applied Mathematics 381N, Methods of Mathematical Physics Computational and Applied Mathematics 381R, Real and Abstract Analysis Computational and Applied Mathematics 381S, Real and Abstract Analysis Computational and Applied Mathematics 384K, Theory of Probability Computational and Applied Mathematics 384L, Theory of Probability Computational and Applied Mathematics 384R, Mathematical Statistics Computational and Applied Mathematics 384S, Mathematical Statistics Computational and Applied Mathematics 386M, Functional Analysis in Theoretical Mechanics Computational and Applied Mathematics 386N, Qualitative Methods in Nonlinear Mechanics Computational and Applied Mathematics 391, Introductory Dynamical Systems Computational and Applied Mathematics 391C, Topics in Analysis Topics include Fourier Analysis Wavelets Advanced Dynamical Systems Computational and Applied Mathematics 393C, Topics in Applied Mathematics Topics include Methods of Applied Mathematics Approximation Theory Introduction to Partial Differential Equations Monotone Operators and Nonlinear Partial Differential Equations Abstract Cauchy Problems and Nonlinear Partial Differential Equations Nonlinear Wave Equations Euler and Navier-Stokes Equations Pseudodifferential and Fourier Integral Operators Statistical Mechanics Ergodic Theory Quantum Mechanics Bifurcation Theory Computational and Applied Mathematics 394C, Topics in Probability and Statistics Topics include Nonparametric Statistics Advanced Probability AREA B: NUMERICAL ANALYSIS AND SCIENTIFIC COMPUTATION Computational and Applied Mathematics 380N, Algorithms for Parallel and Distributed Computation Computational and Applied Mathematics 381C, Computational Physics Computational and Applied Mathematics 382L, Numerical Methods in Petroleum and Geosystems Engineering Computational and Applied Mathematics 383, Special Topics in Petroleum and Geosystems Engineering Topic 1: Numerical Solution of Time-Dependent Problems Topic 16: Topics in Computational Methods Computational and Applied Mathematics 383C, Numerical Analysis: Linear Algebra Computational and Applied Mathematics 383D, Numerical Analysis: Interpolation, Approximation, Quadrature, and Differential Equations Computational and Applied Mathematics 384G, Computer Graphics Computational and Applied Mathematics 386K, Numerical Treatment of Differential Equations Computational and Applied Mathematics 393D, Topics in Numerical Analysis Computational and Applied Mathematics 393M, Numerical Solution of Elliptic Partial Differential Equations Computational and Applied Mathematics 393N, Numerical Methods for Flow and Transport Problems Computational and Applied Mathematics 394F, Finite Element Methods Computational and Applied Mathematics 394G, Computational Techniques in Finite Elements Computational and Applied Mathematics 394H, Advanced Theory of Finite Element Methods Computational and Applied Mathematics 195T, 295T, 395T, Topics in Computer Sciences: Parallel Computations Topics include Parallel Methods for Numerical Algorithms Parallel Algorithms and Architecture Parallel Programming Computational and Applied Mathematics 397, Topics in Computational and Applied Mathematics Topics include Supercomputing in Computational Mechanics Grid Generation, Adaptive Grids, and Multigrids Advanced Computational Fluid Mechanics AREA C: MATHEMATICAL MODELING AND APPLICATIONS Courses in mathematical modeling and applications are offered in several areas of science and engineering. Four groups of courses from which an Area C concentration might be developed are given as examples below. Other examples are fluid dynamics, control theory, semiconductors, celestial mechanics, chemistry, and the theory of relativity. Concentrations in mathematical modeling and applications are also available in various departments; these may be included in computational and applied mathematics degree programs with the approval of the Graduate Studies Committee. Solid and Continuum Mechanics Engineering Mechanics 380, Theory of Plasticity Engineering Mechanics 388, Solid Mechanics I Engineering Mechanics 388F, Fracture Mechanics Engineering Mechanics 388L, Solid Mechanics II Engineering Mechanics 394V, Wave Propagation I Fluid Mechanics Aerospace Engineering 382Q, Fluid Mechanics Topic 4: Turbulent Fluid Mechanics Topic 6: Finite Difference Methods in Computational Fluid Dynamics Topic 7: Advanced Problems in Compressible Flow Aerospace Engineering 382R, Aerodynamics Topic 1: Low-Speed Aerodynamics Topic 2: High-Speed Aerodynamics Topic 3: Hypersonic Aerodynamics Topic 4: Theoretical Gas Dynamics Topic 5: Advanced Computational Methods Chemical Engineering 381N, Fluid Flow and Heat Transfer Engineering Mechanics 387, Foundations of Fluid Mechanics Mechanical Engineering 380Q, Mathematical Methods in Engineering Topic 3: Perturbation Methods Topic 5: Spectral Methods in Fluid Dynamics Mechanical Engineering 381P, Dynamics of Fluids Topic 1: Incompressible Flow I: Theory Topic 3: Dynamics of Turbulent Flow Topic 5: Incompressible Flow II: Applications Topic 6: Modeling of Turbulent Flows Nonlinear Dynamics Aerospace Engineering 388P, Celestial Mechanics Topic 1: Hamiltonian Mechanics Topic 2: Celestial Mechanics I Topic 3: Celestial Mechanics II Topic 4: Satellite Theory Topic 5: Theory of Orbits I Topic 6: Theory of Orbits II Topic 10: Regularization Engineering Mechanics 381, Advanced Dynamics System and Control Theory Aerospace Engineering 381P, System Theory Topic 1: Linear Systems Analysis Topic 2: Multivariable Control Systems Topic 3: Optimal Control Theory I Topic 5: Optimal Control Theory II Topic 6: Statistical Estimation Theory Topic 7: Advanced Topics in Estimation Theory Topic 8: Stochastic Estimation and Control Electrical Engineering 380K, Introduction to System Theory Electrical Engineering 380N, Topics in System Theory Topic 1: Nonlinear Systems: Input-Output Properties Topic 2: Nonlinear Systems: Geometric Theory Topic 5: Stochastic Control Theory Topic 6: Stochastic Dynamical Systems Mechanical Engineering 390R, Engineering Statistics and Probability Topic 2: Time-Series Analysis Topic 4: Applied Stochastic Processes Topic 5: Regression and Analysis of Variance Mechanical Engineering 391Q, Operations Research Topic 1: Nonlinear Programming Topic 2: Dynamic Programming For More Information Campus address: T. U. Taylor Hall (TAY) 3.104A, Phone (512) 471- 9717, Fax (512) 471-8694 Mailing address: Graduate Program in Computational and Applied Mathematics, Department of Computer Sciences, The University of Texas at Austin, Austin, Texas 78712 WWW: http://www.ticam.utexas.edu/ Graduate Courses The faculty expects to offer the following courses in the academic years 1995-1996 and 1996-1997; however, all courses are not taught each semester or summer session. Students should consult the Course Schedule, published before registration, and the supplement to the Course Schedule, published before classes begin, to determine which courses and topics will be offered during a particular semester or summer session. These publications also may reflect changes that have been made to the courses listed here since this catalog was printed. Unless otherwise stated below, each course meets for three lecture hours a week for one semester. COMPUTATIONAL AND APPLIED MATHEMATICS: CAM CAM 380N. Algorithms for Parallel and Distributed Computation. Same as Electrical Engineering 380N (Topic 8: Algorithms for Parallel and Distributed Computation). Prerequisite: Graduate standing, and Electrical Engineering 380K or consent of the graduate adviser. CAM 381C. Computational Physics. Same as Physics 381C. Dynamical and statical descriptions and solutions of many-body, nonlinear physical systems by computation. Theory of computation and applications to various branches of physics. Prerequisite: Graduate standing; and Physics 385K and 387K, or consent of instructor. CAM 381D. Complex Analysis. Same as Mathematics 381D. Prerequisite: Graduate standing and consent of instructor or the graduate adviser. CAM 381M. Methods of Mathematical Physics. Same as Physics 381M. Theory of analytic functions; linear algebra and vector spaces; orthogonal functions; ordinary differential equations; partial differential equations; Green's functions; complex variables. Prerequisite: Graduate standing. CAM 381N. Methods of Mathematical Physics. Same as Physics 381N. Continuation of Computational and Applied Mathematics 381M. Topology, functional analysis, approximation methods, group theory, differential manifolds. Prerequisite: Graduate standing, and Computational and Applied Mathematics 381M or Physics 381M. CAM 381R. Real and Abstract Analysis. Same as Mathematics 381C. Only one of the following may be counted: Computational and Applied Mathematics 681EA, 381R, Mathematics 681CA. Naive set theory, cardinal and ordinal numbers, metric spaces, topological spaces, measure and integration over abstract spaces, Lebesgue's theory of integration and differentiation on the real line, elements of functional analysis. Prerequisite: Graduate standing and consent of instructor or the graduate adviser. CAM 381S. Real and Abstract Analysis. Same as Mathematics 381E. Only one of the following may be counted: Computational and Applied Mathematics 681EB, 381S, Mathematics 681CB. Continuation of Computational and Applied Mathematics 381R. Prerequisite: Graduate standing, consent of instructor, and Computational and Applied Mathematics 381R (or 681EA) or Mathematics 381C (or 681CA). CAM 382L. Numerical Methods in Petroleum and Geosystems Engineering. Same as Petroleum and Geosystems Engineering 382L. The use of numerical methods and computers in the solution of petroleum and geosystems engineering problems. Prerequisite: Graduate standing. CAM 383. Special Topics in Petroleum and Geosystems Engineering. May be repeated for credit when the topics vary. Recent literature on petroleum production practice and petroleum and geosystems engineering problems. Prerequisite: Graduate standing in computational and applied mathematics, engineering, or geology. Students seeking to enroll in any seminar must present technical prerequisites satisfactory to the instructor. Topic 1: Numerical Solution of Time-Dependent Problems. Same as Petroleum and Geosystems Engineering 383 (Topic 10: Numerical Solution of Time-Dependent Problems). Topic 2: Topics in Computational Methods. Same as Petroleum and Geosystems Engineering 383 (Topic 16: Topics in Computational Methods). CAM 383C. Numerical Analysis: Linear Algebra. Same as Computer Sciences 383C and Mathematics 383E. Computational and Applied Mathematics 383C and Mathematics 383C (Numerical Analysis: Linear Algebra) may not both be counted. Survey of numerical methods in linear algebra: floating-point computation, solution of linear equations, least squares problems, algebraic eigenvalue problems. Prerequisite: Graduate standing; either consent of instructor or Mathematics 311 or 340L; and one of the following: Mathematics 318M, 368K, Computer Sciences 367. CAM 383D. Numerical Analysis: Interpolation, Approximation, Quadrature, and Differential Equations. Same as Computer Sciences 383D and Mathematics 383F. Computational and Applied Mathematics 383D and Mathematics 383D (Numerical Analysis: Interpolation, Approximation, Quadrature, and Differential Equations) may not both be counted. Survey of numerical methods for interpolation, functional approximation, integration, and solution of differential equations. Prerequisite: Graduate standing; either consent of instructor or Mathematics 427K and 365C (or 655A); and Computational and Applied Mathematics 383C, Computer Sciences 383C, or Mathematics 383E (or 383C [Numerical Analysis: Linear Algebra]). CAM 384G. Computer Graphics. Same as Computer Sciences 384G. Advanced material in computer graphics, including in-depth treatments of techniques for realistic image synthesis, advanced geometric modeling methods, animation and dynamic simulation, scientific visualization, and high-performance graphics architectures. Prerequisite: Graduate standing; and Computer Sciences 354 or another introductory course in computer graphics, or equivalent background and consent of instructor. CAM 384K. Theory of Probability. Same as Mathematics 385C. Only one of the following may be counted: Computational and Applied Mathematics 684CA, 384K, Mathematics 684CA. Prerequisite: Graduate standing and consent of instructor. CAM 384L. Theory of Probability. Same as Mathematics 385D. Only one of the following may be counted: Computational and Applied Mathematics 684CB, 384L, Mathematics 684CB. Prerequisite: Graduate standing, consent of instructor, and Computational and Applied Mathematics 384K (or 684CA) or Mathematics 385C (or 684CA). CAM 384R. Mathematical Statistics. Same as Mathematics 384C. Only one of the following may be counted: Computational and Applied Mathematics 684DA, 384R, Mathematics 684DA. Fourier analysis, wavelets, and advanced dynamical systems; numerical analysis and scientific computation. Prerequisite: Graduate standing and consent of instructor. CAM 384S. Mathematical Statistics. Same as Mathematics 384D. Only one of the following may be counted: Computational and Applied Mathematics 684DB, 384S, Mathematics 684DB. Continuation of Computational and Applied Mathematics 384R. Prerequisite: Graduate standing, consent of instructor, and Computational and Applied Mathematics 384R (or 684DA) or Mathematics 384C (or 684DA). CAM 386K. Numerical Treatment of Differential Equations. Same as Computer Sciences 386K and Mathematics 383G. Computational and Applied Mathematics 386K and Mathematics 386K may not both be counted. The analysis of numerical methods for solving ordinary and partial differential equations. Prerequisite: Graduate standing; and Computational and Applied Mathematics 383D, Computer Sciences 383D, Mathematics 368K, 383F (or 383D [Numerical Analysis: Interpolation, Approximation, Quadrature, and Differential Equations]), or consent of instructor. CAM 386M. Functional Analysis in Theoretical Mechanics. An introduction to modern concepts in functional analysis and linear operator theory, with emphasis on their application to problems in theoretical mechanics; topological and metric spaces, norm linear spaces, theory of linear operators on Hilbert spaces, applications to boundary value problems in elasticity and dynamical systems. Prerequisite: Graduate standing, Engineering Mechanics 386L, and Mathematics 365C (or 665A). CAM 386N. Qualitative Methods in Nonlinear Mechanics. A study of methods for assessing the qualitative behavior of solutions to equations governing nonlinear continuum mechanics. Prerequisite: Graduate standing, and Computational and Applied Mathematics 386M or Engineering Mechanics 386M. CAM 391. Introductory Dynamical Systems. Prerequisite: Graduate standing. CAM 391C. Topics in Analysis. Same as Mathematics 391C. May be repeated for credit when the topics vary. Some sections are offered on the credit/no credit basis only; these are identified in the Course Schedule. Recent topics have included measure and integration, real variables; complex analysis, functional analysis, ordinary differential equations, partial differential equations, integral transforms, operator theory, approximation theory, abstract harmonic analysis. Prerequisite: Graduate standing and consent of instructor. CAM 393C. Topics in Applied Mathematics. Same as Mathematics 393C. May be repeated for credit when the topics vary. Some sections are offered on the credit/no credit basis only; these are identified in the Course Schedule. Recent topics have included potential theory, calculus of variations, applications of tensor analysis, integral equations; special functions of mathematical physics, game theory and linear programming, mathematical programming, relativity theory, mathematical physics, survey of applied mathematics. Prerequisite: Graduate standing and consent of instructor. CAM 393D. Topics in Numerical Analysis. Same as Computer Sciences 393D and Mathematics 393D. May be repeated for credit when the topics vary. Recent topics have included numerical methods in ordinary differential equations, numerical methods in partial differential equations, computational problems in linear algebra, numerical solution of systems of equations, numerical methods in functional approximation, numerical integration. Prerequisite: Graduate standing and consent of instructor. CAM 393M. Numerical Solution of Elliptic Partial Differential Equations. Same as Computer Sciences 393N and Mathematics 393N. The numerical solution of large systems of linear algebraic equations arising in the solution of elliptic partial differential equations by discretization methods. Prerequisite: Graduate standing; and Computational and Applied Mathematics 386K, Computer Sciences 386K, Mathematics 383G (or 386K), or consent of instructor. CAM 393N. Numerical Methods for Flow and Transport Problems. Approximate solution methods for flow and transport problems in engineering and applied science. Finite element, finite difference, and residual methods for linear and nonlinear problems. Prerequisite: Graduate standing. CAM 394C. Topics in Probability and Statistics. Same as Mathematics 394C. May be repeated for credit when the topics vary. Some topics are offered on the credit/no credit basis only; these are identified in the Course Schedule. Recent topics have included nonparametric statistics and advanced probability. Prerequisite: Graduate standing and consent of instructor. CAM 394F. Finite Element Methods. Same as Aerospace Engineering 384P (Topic 4: Finite Element Methods) and Engineering Mechanics 394F. Derivation and implementation of the finite element method; basic coding techniques; application to problems of stress and diffusion. Prerequisite: Graduate standing and consent of instructor. CAM 394G. Computational Techniques in Finite Elements. Organization and data management in finite element codes; element models and calculations; equation solving; preprocessing and postprocessing. Prerequisite: Graduate standing, and Aerospace Engineering 384P (Topic 4: Finite Element Methods), Computational and Applied Mathematics 394F, or Engineering Mechanics 394F. CAM 394H. Advanced Theory of Finite Element Methods. Contemporary topics in the theory and application of finite element methods. Prerequisite: Graduate standing, Computational and Applied Mathematics 394F or Engineering Mechanics 394F, and Engineering Mechanics 386L or the equivalent. CAM 195T, 295T, 395T. Topics in Computer Sciences: Parallel Computations. Same as Computer Sciences 195T, 295T, 395T (Topic 1: Parallel Computations). Computational and Applied Mathematics 195T is offered on the credit/no credit basis only. Prerequisite: Graduate standing. One, two, or three lecture hours a week for one semester. CAM 397. Topics in Computational and Applied Mathematics. May be repeated for credit. Offered on the credit/no credit basis only. Prerequisite: Graduate standing. Conference course. CAM 398R. Master's Report. Offered on the letter-grade basis only. Preparation of report to fulfill the requirement for the master's degree under the report option. Prerequisite: Graduate standing in computational and applied mathematics and consent of the graduate adviser. Independent study. CAM 399R, 699R, 999R. Dissertation. Offered on the letter-grade basis only. Prerequisite: Admission to candidacy for the doctoral degree. Independent study. CAM 399W, 699W, 999W. Dissertation. Offered on the letter-grade basis only. Prerequisite: Computational and Applied Mathematics 399R, 699R, or 999R. Independent study.