__Graduate Advisor__:

Matthias Heinkenschloss

__Undergraduate Advisor__:

Maarten de Hoop

**(S) = Spring and (F) = Fall**

All courses are 3 semester hours, unless otherwise noted.

__INTRODUCTORY COURSES__:

**CAAM 007 (SUMMER) VISITING RESEARCH (0)**

Research conducted by visiting student scholars. Instructor Permission Required. Repeatable for Credit.

**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. Application to problems in science and engineering. Lectures are held Monday and Wednesdays. In a laboratory component held on Fridays, students work in small groups on computational projects led by a Rice Learning Assistant.

*Recommended Prerequisite(s)*: Math 101.

**CAAM 335 (BOTH) MATRIX ANALYSIS**

Equilibria and the solution of linear systems and linear least squares problems. Eigenvalue problem and its application to solve dynamical systems. Singular value decomposition and its applications.

*Recommended Prerequisite(s)*: (MATH 212 or MATH 222) 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.

*Recommended Prerequisite(s)*: (MATH 212 or MATH 222) 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.

*Recommended Prerequisite(s)*: MATH 212 and any one of the following: CAAM 335, MATH 211, or MATH 355.

__ADVANCED COURSES__:

**CAAM 415 (F) THEORETICAL NEUROSCIENCE: FROM CELLS TO LEARNING SYSTEMS**

We present the theoretical foundations of cellular and systems neuroscience from distinctly quantitative point of view. We develop the mathematical and computational tools as they are needed to model, analyze, visualize and interpret a broad range of experimental data.

*Recommended Prerequisite(s)*: CAAM 210 or MATH 211 or CAAM 335.

*Cross-listed with*: ELEC 488 and NEUR 415.

**CAAM 416 (S) NEURAL COMPUTATION**

How does the brain work? Understanding the brain requires sophisticated theories to make sense of the collective actions of billions of neurons and trillions of synapses. Word theories are not enough; we need mathematical theories. The goal of this course is to provide an introduction to the mathematical theories of learning and computation by neural systems. These theories use concepts from dynamical systems (attractors, oscillations, chaos) and concepts from statistics (information, uncertainty, inference) to relate the dynamics and functions of neural networks. We will apply these theories to sensory computation, learning and memory, and motor control. Students will learn to formalize and mathematically answer questions about neural computations, including "what does a network compute?", "how does it compute?", and "why does it compute that way?" Prerequisites: knowledge of calculus, linear algebra, and probability and statistics.

*Cross-listed with*: ELEC 416 and NEUR 416.

**CAAM 423 (F) PARTIAL DIFFERENTIAL EQUATIONS I**

First order systems of partial differential equations. The method of characteristics. Analysis of the solutions of the wave equation, heat equation and Laplace's equation. Integral relations and Green's functions. Potential theory, Dirichlet and Neumann problems. Asymptotic methods: the method of stationary phase, geometrical optics, regular and singular perturbation methods.

*Prerequisite(s)*: MATH 321 and MATH 322.

**CAAM 435 (S) 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.

*Recommended Prerequisite(s)*: CAAM 210 and MATH 212, and (CAAM 335 or MATH 355), and (MATH 302 or MATH 321).

*Cross-listed with*: MATH 435.

**CAAM 436 MODELING 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.

*Recommended Prerequisite(s)*: CAAM 336.

**CAAM 440 (S) APPLIED MATRIX ANALYSIS**

A second course in matrix analysis that presents advanced theoretical results alongside motivating applications. Topics include: properties of Hermitian, positive definite, nonnegative, and stochastic matrices; Perron-Frobenius Theorem; spectral perturbation theory; singular value inequalities; generalized eigenvalue problems; functions of matrices; Lyapunov, Sylvester, and Riccati matrix equations. Applications include dynamical systems, control theory, and Markov chains.

*Recommended Prerequisite(s)*: CAAM 335 or MATH 354 or MATH 355.

__Biennial; Offered in Even Years__

**CAAM 453 (F) NUMERICAL ANALYSIS I**

Construction and application 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, linear least squares problems, and eigenvalue problems via matrix factorizations; Newton and Newton-like methods for nonlinear systems of equations. Computer programming in MATLAB is required.

*Prerequisite(s)*: CAAM 335 and CAAM 336.

**CAAM 454 (S) NUMERICAL ANALYSIS II**

Iterative methods for linear systems of equations including Krylov subspace methods; Newton and Newton-like methods for nonlinear systems of equations; Gradient and Newton-like methods for unconstrained optimization and nonlinear least squares problems; techniques for improving the global convergence of these algorithms; linear programming duality and primal-dual interior-point methods. Credit may not be received for both CAAM 454 and CAAM 554.

*Recommended Prerequisite(s)*: CAAM 453.

**CAAM 469 (BOTH) DYNAMICAL SYSTEMS LAB (1)**

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

**CAAM 471 (S) 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.

*Recommended Prerequisite(s)*: CAAM 335 and CAAM 378.

**CAAM 480 (BOTH) PEDAGOGY FOR RLAS**

A study of content and pedagogy, designed for Rice learning Assistants (RLAs) as they direct their own labs sections of CAAM 210. Topics include analysis of computational science and engineering concepts, issues of problem-based learning (PBL), theories of learning, and fundamental teaching skills. Required for CAAM 210 RLA's.

**CAAM 490 (F) UNDERGRADUATE RESEARCH PROJECTS (1-6)**

Semester-long undergraduate-level research on a topic in Computational and Applied Mathematics.

**CAAM 491 (S) UNDERGRADUATE RESEARCH PROJECTS (1-6)**

Semester-long undergraduate-level research on a topic in Computational and Applied Mathematics.

**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 regular seminars addressing research techniques and the effective written and verbal presentation of mathematics.

**CAAM 496 (S) SENIOR DESIGN PROJECT II (Hours Variable)**

Continuation of CAAM 495. Seminars focus on the presentation of results from design groups and provide guidance on the composition of a substantial project report.
*Recommended Prerequisite(s)*: CAAM 495.

**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.

*Cross-listed with*: MATH 498 and STAT 498.

**CAAM 499 (BOTH) COMPUTATIONAL AND APPLIED MATHEMATICS 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.

**CAAM 501 (F) ANALYSIS I**

Real numbers, completeness, sequences and convergence, compactness, continuity, the derivative, the Riemann integral, fundamental theorem of calculus. Vector spaces, dimension, linear maps, inner products and norms, derivatives in R^d, inverse function theorem, implicit function theorem, multiple integration, change of variable theorem.

**CAAM 502 (S) ANALYSIS II**

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, fundamental theorems on differential equations, Stoke's theorem and relatives.

*Recommended Prerequisite(s)*: CAAM 501.

**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. **Not offered every year.**

*Cross-listed with*: MECH 508 and ELEC 508.

**CAAM 519 (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.

*Recommended Prerequisite(s)*: (CAAM 210 and CAAM 335) or CAAM 353.

**CAAM 520 (S) COMPUTATIONAL SCIENCE II**

Theory and application of the message passing interface for programming scientific computing applications. Introduction to the architecture and programming of multicore and massively parallel processors, including general purpose graphics processing units. Insight for designing efficient numerical algorithms to improve parallelization of memory access and utilization of non-uniform memory architectures. Application interfaces include OpenMP, MPI, CUDA, OpenCL, and parallel numerical algorithm libraries.

*Recommended Prerequisite(s)*: CAAM 519.

**CAAM 523 (S) PARTIAL DIFFERENTIAL EQUATIONS II**

First order of partial differential equations. The method of characteristics. Analysis of the solutions of the wave equation, heat equation and Laplace's equation. Integral relations and Green's functions. Potential theory, Dirichlet and Neumann problems. Asymptotic methods: the method of stationary phase, geometrical optics, regular and singular perturbation methods. Additional course work is required beyond the undergraduate course requirements.

Credit cannot be earned for CAAM 523 and CAAM 423
*Recommended Prerequisite*: MATH 321 AND MATH 322.

**CAAM 535 MODELING 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.

*Recommended Prerequisite(s)*: CAAM 336.

**CAAM 536 (S) NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS**

This course covers various numerical methods for solving partial differential equations: aspects of finite difference methods, finite element methods, finite volume methods, mixed methods, discontinuous Galerkin methods, and meshless methods. Both theoretical convergence and practical implementation of the methods are studied for hyperbolic, elliptic, and parabolic problems.

*Recommended Prerequisite(s)*: CAAM 336.

**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.

*Recommended Prerequisite(s)*: MATH 322.

__Biennial; Offered in Odd Years__

**CAAM 550 (F) NUMERICAL ANALYSIS I**

Construction and application 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, linear least squares problems, and eigenvalue problems via matrix factorizations; Newton and Newton-like methods for nonlinear systems of equations. Computer programming in MATLAB is required.

*Prerequisite(s)*: CAAM 335 and CAAM 336.

**CAAM 551 (S) 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.

*Recommended Prerequisite(s)*: CAAM 453 or (CAAM 550 or CAAM 553)

__Biennial; Offered in Even Years__

**CAAM 552 (F) FOUNDATIONS OF FINITE ELEMENT METHODS**

This course addresses the theory and implementation of finite element methods. Topics include weak solutions of partial differential equations, Sobolev spaces, approximation theory, convergence and reliability of the numerical methods. Continuous and discontinuous finite element methods are considered.

**CAAM 553 (F) ADVANCED 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. Computer programming in MATLAB is required. This course covers fewer topics than CAAM 453 with greater theoretical depth.

*Prerequisite(s)*: CAAM 501 (may be taken concurrently).

**CAAM 554 (S) ADVANCED NUMERICAL ANALYSIS II**

This course covers the same lecture material as CAAM 454, but fosters greater theoretical sophistication through more challenging problem sets and exams. Credit may not be received for both CAAM 454 and CAAM 554.

*Recommended Prerequisite(s)*: CAAM 550 or CAAM 553.

**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.

*Cross-listed with*: 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.

*Recommended Prerequisite(s)*: CAAM 560 (may be taken concurrently) and CAAM 454.

__Biennial; Offered in Even Years__

**CAAM 565 (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 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.

*Recommended Prerequisite(s)*: CAAM 335 and MATH 321

__Biennial; Offered in Odd Years__

**CAAM 567 (S) SIGNAL RECOVERY: THEORY AND SIMULATION**

This course introduces the theory and numerical algorithms for several fundamental signal recovery tasks. Topics include L1 minimization, sparse regression, compressed sensing, orthogonal matching pursuit, proximal operators, ADMM algorithms, Iterative Reweighted Least Squares. Nuclear norm minimization, matrix completion, robust Principal Component Analysis.

*Recommended Prerequisite(s)*: CAAM 378 or MATH 302 or STAT 310.

**CAAM 570 (S) 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 571 (S) LINEAR AND INTEGER PROGRAMMING**

This course covers the same lecture material as CAAM 471, but fosters greater theoretical sophistication through more challenging problem sets and exams. Credit may not be received for both CAAM 471 and CAAM 571.

**CAAM 574 (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.

*Recommended Prerequisite(s)*: CAAM 378 or 471.

__Biennial; Offered in Even Years__

**CAAM 581 (F) MATHEMATICAL PROBABILITY I**

Measure-theoretic foundations of probability for students who need access to advanced mathematical literature in probability and random processes.

*Cross-listed with*: STAT 581.

**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.

*Recommended Prerequisite(s)*: STAT 381 and STAT 581.

*Cross-listed with*: ELEC 533 and STAT 583.

**CAAM 585 (S) STOCHASTIC OPTIMIZAITON**

Stochastic optimization models arise in many contexts. This course focuses on stochastic programs, including stochastic integer programs and multi-stage stochastic programs. It will emphasize the interplay between theory and computational approaches.

*Prerequisite(s)*: CAAM 571

**CAAM 590 (F) GRADUATE RESEARCH PROJECTS (1-15)**

Semester-long graduate-level research on a topic in Computational and Applied Mathematics.

**CAAM 591 (S) GRADUATE RESEARCH PROJECTS (1-15)**

Semester-long graduate-level research on a topic in Computational and Applied Mathematics.

**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(s)*: Advisor approval of topic and consent of the instructor(s).

**CAAM 615 (F) THEORETICAL NEUROSCIENCE I:**

A computational companion lab to NEUR 385. Introduction to Matlab programming. Newton's Method and Rest Potential. Numerical methods for differential equations. Solution and visualization of the Hodgkin Huxley equations. Investigation of threshold, firing rate, channel variety, temperature effects. Calcium handing and synaptic plasticity. Modeling/Analyzing sensory input with application to the auditory system.

*Cross-listed with*: NEUR 615.

**CAAM 620 (BOTH) TOPICS IN COMPUTATIONAL SCIENCE**

Content varies from year to year. Course may be repeated for credit.

**CAAM 640 (BOTH) OPTIMIZATION WITH SIMULATION CONSTRAINTS**

Content varies from year to year. Course may be repeated for credit.

*Recommended Prerequisite(s)*: CAAM 564.

**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 642 (BOTH) TOPICS IN SEISMIC IMAGING**

Content varies from year to year.

**CAAM 643 (BOTH) TOPICS IN GEOMATHEMATICS**

Content varies from year to year.

*Cross-listed with*: ESCI 643.

*Recommended Prerequisite(s)*: CAAM 335 and CAAM 336.

**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.

*Recommended Prerequisite(s)*: CAAM 551.

**CAAM 652 (BOTH) TOPICS IN NUMERICAL DIFFERENTIAL EQUATIONS**

Content varies from year to year.

**CAAM 654 (BOTH) TOPICS IN OPTIMIZATION**

Content varies from year to year.

**CAAM 664 (F) TOPICS IN NONLINEAR PROGRAMMING**

Content varies from year to year.

**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.

*Cross-listed with*: MATH 698 and MATH 498.

**CAAM 699 (BOTH) COMPUTATIONAL AND APPLIED MATHEMATICS 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.

**CAAM 800 RESEARCH AND THESIS (1-15)**
This course is for CAAM MA or PhD students working on their thesis research.