Applied and Computational Mathematics

ACM 95/100 abc. Introductory Methods of Applied Mathematics. 12 units (4-0-8); first, second, third terms. Prerequisites: Ma 1 abc, Ma 2 ab, or equivalents. Introduction to functions of a complex variable; linear ordinary differential equations; special functions; eigenfunction expansions; integral transforms; linear partial differential equations and boundary value problems. Instructors: Meiron, Pierce.

ACM 101 abc. Methods of Applied Mathematics I. 9 units (3-0-6); first, second, third terms. Prerequisite: ACM 95/100 abc. Analytical methods for the formulation and solution of initial and boundary value problems for ordinary and partial differential equations. Techniques include the use of complex variables, generalized eigenfunction expansions, transform methods and applied spectral theory, linear operators, nonlinear methods, asymptotic and approximate methods, Weiner-Hopf, and integral equations. Instructor: Bruno.

ACM 104. Linear Algebra. 9 units (3-0-6); second term. Prerequisite: ACM 100 abc or instructor’s permission. Vector spaces, bases, Gram-Schmidt, linear maps and matrices, linear functionals, the transposed matrix and duality, kernel, image and rank, invertibility, triangularization, determinants and multilinear forms, powers of matrices and difference equations, the exponential of a matrix and ODEs, eigenvalues, Gershgorin’s disc theorem, eigenspaces, SVD, polar decomposition. Nilpotent-semisimple decomposition and the Jordan normal form. Symmetric hermitian and positive definite matrices, diagonalizability, unitary matrices, bilinear forms. Hilbert spaces, projections, Riesz theorem, Fourier series, spectrum, self-adjoint operators. Instructor: Chung.

ACM 105. Applied Real and Functional Analysis. 9 units (3-0-6); first term. Prerequisite: ACM 100 abc or instructor’s permission. The Lebesgue integral on the line, general measure and integration theory, convergence theorems, Fubini, Tonelli, the Lebesgue integral in n dimensions and the transformation theorem, L_P spaces, convolution, Fourier transform and Sobolev spaces with application to PDEs, the convolution theorem, Friedrich’s mollifiers, dense subspaces and approximation, normed vector spaces, completeness, Banach spaces, linear operators, the Baire, Banach-Steinhaus, open mapping and closed graph theorems with applications to differential and integral equations, dual spaces, weak convergence and weak solvability theory of boundary value problems, spectral theory of compact operators. Instructor: Chung.

ACM 106 abc. Introductory Methods of Computational Mathematics. 9 units (3-0-6); first, second, third terms. Prerequisites: Ma 1 abc, Ma 2 ab, ACM 95/100 abc or equivalent. The sequence covers the introductory methods in both theory and implementation of numerical linear algebra, approximation theory, ordinary differential equations, and partial differential equations. The course covers methods such as direct and iterative solution of large linear systems; eigenvalue and vector computations; function minimization; nonlinear algebraic solvers; preconditioning; time-frequency transforms (Fourier, wavelet, etc.); root finding; data fitting; interpolation and approximation of functions; numerical quadrature; numerical integration of systems of ODEs (initial and boundary value problems); finite difference, element, and volume methods for PDEs; level set methods. Programming is a significant part of the course. Instructors: Candes, Fok.

ACM 113. Introduction to Optimization. 9 units (3-0-6); first term. Prerequisites: ACM 95/100 abc, ACM 104 or equivalent, or instructor’s permission. Unconstrained optimization: optimality conditions, line search and trust region methods, properties of steepest descent, conjugate gradient, Newton and quasi-Newton methods. Linear programming: optimality conditions, the simplex method, primal-dual interior-point methods. Nonlinear programming: Lagrange multipliers, optimality conditions, logarithmic barrier methods, quadratic penalty methods, augmented Lagrangian methods. Integer programming: cutting plane methods, branch and bound methods, complexity theory, NP complete problems. Not offered 2006–07.

ACM/CS 114 ab. Parallel Algorithms for Scientific Applications. 9 units (3-0-6); second, third terms. Prerequisites: ACM 106 or equivalent. Introduction to parallel program design for numerically intensive scientific applications. First term: parallel programming methods; distributed-memory model with message passing using the message passing interface; shared-memory model with threads using open MP; object-based models using a problem-solving environment with parallel objects. Parallel numerical algorithms: numerical methods for linear algebraic systems, such as LU decomposition, QR method, Lanczos and Arnoldi methods, pseudospectra, CG solvers. Second term: parallel implementations of numerical methods for PDEs, including finite-difference, finite-element, and shock-capturing schemes; particle-based simulations of complex systems. Implementation of adaptive mesh refinement. Grid-based computing, load balancing strategies. Not offered 2006–07.

ACM/EE 116. Introduction to Stochastic Processes and Modeling. 9 units (3-0-6); first term. Prerequisite: Ma 2 ab or instructor’s permission.Introduction to fundamental ideas and techniques of stochastic analysis and modeling. Random variables, expectation and conditional expectation, joint distributions, covariance, moment generating function, central limit theorem, weak and strong laws of large numbers, discrete time stochastic processes, stationarity, power spectral densities and the Wiener-Khinchine theorem, Gaussian processes, Poisson processes, Brownian motion. The course develops applications in selected areas such as signal processing (Wiener filter), information theory, genetics, queuing and waiting line theory, and finance. Instructor: Owhadi.

ACM/ESE 118. Methods in Applied Statistics and Data Analysis. 9 units (3-0-6); second term. Prerequisite: Ma 2 or another introductory course in probability and statistics. Introduction to fundamental ideas and techniques of statistical modeling, with an emphasis on conceptual understanding and on the analysis of real data sets. Multiple regression: estimation, inference, model selection, model checking. Regularization of ill-posed and rank-deficient regression problems. Cross-validation. Principal component analysis. Discriminant analysis. Resampling methods and the bootstrap. Instructor: Schneider.

ACM 126 ab. Wavelets and Modern Signal Processing. 9 units (3-0-6); second, third terms. Prerequisites: ACM 104, ACM 105 or undergraduate equivalent, or instructor’s permission. The aim is to cover the interactions existing between applied mathematics, namely applied and computational harmonic analysis, approximation theory, etc., and statistics and signal processing. The Fourier transform: the continuous Fourier transform, the discrete Fourier transform, FFT, time-frequency analysis, short-time Fourier transform. The wavelet transform: the continuous wavelet transform, discrete wavelet transforms, and orthogonal bases of wavelets. Statistical estimation. Denoising by linear filtering. Inverse problems. Approximation theory: linear/nonlinear approximation and applications to data compression. Wavelets and algorithms: fast wavelet transforms, wavelet packets, cosine packets, best orthogonal bases matching pursuit, basis pursuit. Data compression. Nonlinear estimation. Topics in stochastic processes. Topics in numerical analysis, e.g., multigrids and fast solvers. Not offered 2006–07.

Ma/ACM 142 abc. Ordinary and Partial Differential Equations. 9 units (3-0-6). For course description, see Mathematics.

Ma/ACM 144 ab. Probability. 9 units (3-0-6). For course description, see Mathematics.

ACM 151 ab. Asymptotic and Perturbation Methods. 9 units (3-0-6); first, second terms. Prerequisite: ACM 101 abc or equivalent, may be taken concurrently with instructor’s permission. Approximation methods for formulating and solving applied problems, with examples taken from various fields of science. Applications to various linear and nonlinear ordinary and partial differential equations. Singular and multiscale perturbation techniques, boundary-layer theory, coordinate straining, a method of averaging. Bifurcation theory, amplitude equations, and nonlinear stability. Not offered 2006–07.

ACM 190. Reading and Independent Study. Units by arrangement. Graded pass/fail only.

ACM 201 ab. Partial Differential Equations. 12 units (4-0-8); first, second terms. Prerequisite: ACM 101 abc or instructor’s permission. Fully nonlinear first-order PDEs, shocks, eikonal equations. Classification of second-order linear equations: elliptic, parabolic, hyperbolic. Well-posed problems. Laplace and Poisson equations; Gauss’s theorem, Green’s function. Existence and uniqueness theorems (Sobolev spaces methods, Perron’s method). Applications to irrotational flow, elasticity, electrostatics, etc. Heat equation, existence and uniqueness theorems, Green’s function, special solutions. Wave equation and vibrations. Huygens’ principle. Spherical means. Retarded potentials. Water waves and various approximations, dispersion relations. Symmetric hyperbolic systems and waves. Maxwell equations, Helmholtz equation, Schrödinger equation. Radiation conditions. Gas dynamics. Riemann invariants. Shocks, Riemann problem. Local existence theory for general symmetric hyperbolic systems. Global existence and uniqueness for the inviscid Burgers’ equation. Integral equations, single- and double-layer potentials. Fredholm theory. Navier-Stokes equations. Stokes flow, Reynolds number. Potential flow; connection with complex variables. Blasius formulae. Boundary layers. Subsonic, supersonic, and transonic flow. Instructor: Meiron.

ACM 210 ab. Numerical Methods for PDEs. 9 units (3-0-6); second, third terms. Prerequisite: ACM 106 or instructor’s permission. Finite difference and finite volume methods for hyperbolic problems. Stability and error analysis of nonoscillatory numerical schemes: i) linear convection: Lax equivalence theorem, consistency, stability, convergence, truncation error, CFL condition, Fourier stability analysis, von Neumann condition, maximum principle, amplitude and phase errors, group velocity, modified equation analysis, Fourier and eigenvalue stability of systems, spectra and pseudospectra of nonnormal matrices, Kreiss matrix theorem, boundary condition analysis, group velocity and GKS normal mode analysis; ii) conservation laws: weak solutions, entropy conditions, Riemann problems, shocks, contacts, rarefactions, discrete conservation, Lax-Wendroff theorem, Godunov’s method, Roe’s linearization, TVD schemes, high-resolution schemes, flux and slope limiters, systems and multiple dimensions, characteristic boundary conditions; iii) adjoint equations: sensitivity analysis, boundary conditions, optimal shape design, error analysis. Interface problems, level set methods for multiphase flows, boundary integral methods, fast summation algorithms, stability issues. Spectral methods: Fourier spectral methods on infinite and periodic domains. Chebyshev spectral methods on finite domains. Spectral element methods and h-p refinement. Multiscale finite element methods for elliptic problems with multiscale coefficients. Instructor: Hou.

ACM 216. Markov Chains, Discrete Stochastic Processes and Applications. 9 units (3-0-6); second term. Prerequisite: ACM/EE 116 or equivalent. Stable laws, Markov chains, classification of states, ergodicity, von Neumann ergodic theorem, mixing rate, stationary/equilibrium distributions and convergence of Markov chains, Markov chain Monte Carlo and its applications to scientific computing, Metropolis Hastings algorithm, coupling from the past, martingale theory and discrete time martingales, rare events, law of large deviations, Chernoff bounds. Instructor: Candes.

ACM 217. Advanced Topics in Stochastic Analysis. 9 units (3-0-6); third term. Prerequisite: ACM 216 or equivalent. The topic of this course changes from year to year and is expected to cover areas such as stochastic differential equations, stochastic control, statistical estimation and adaptive filtering, empirical processes and large deviation techniques, concentration inequalities and their applications. Examples of selected topics for stochastic differential equations include continuous time Brownian motion, Ito’s calculus, Girsanov theorem, stopping times, and applications of these ideas to mathematical finance and stochastic control. Instructors: van Handel, Bouten.

Ae/ACM 232 abc. Computational Fluid Dynamics. 9 units (3-0-6). For course description, see Aeronautics.

ACM 256. Special Topics in Applied Mathematics: Large Deviations Techniques and Applications. 9 units (3-0-6); third term. Prerequisites: ACM 101 or equivalent. Large deviations is a very active field of probability theory dealing with rare events, more precisely the asymptotic computation of small probabilities on an exponential scale. Its applications range from statistics, computing, statistical physics, and mathematical finance to DNA analysis, communication, and control. Instructor: Owhadi.

ACM 290 abc. Applied and Computational Mathematics Colloquium. 1 unit (1-0-0); first, second, third terms. A seminar course in applied and computational mathematics. Weekly lectures on current developments are presented by staff members, graduate students, and visiting scientists and engineers. Graded pass/fail only.

ACM 300. Research in Applied and Computational Mathematics. Units by arrangement.