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ACM 11
Introduction to Matlab and Mathematica
6 units (222)

third term
Prerequisites: Ma 1 abc, Ma 2, Ma 3. CS 1 or prior programming experience recommended.
Matlab: basic syntax and development environment; debugging; help interface; basic linear algebra; visualization and graphical output; control flow; vectorization; scripts, and functions; file i/o; arrays, structures, and strings; numerical analysis (topics may include curve fitting, interpolation, differentiation, integration, optimization, solving nonlinear equations, fast Fourier transform, and ODE solvers); and advanced topics (may include writing fast code, parallelization, objectoriented features). Mathematica: basic syntax and the notebook interface, calculus and linear algebra operations, numerical and symbolic solution of algebraic and differential equations, manipulation of lists and expressions, Mathematica programming (rulebased, functional, and procedural) and debugging, plotting, and visualization. The course will also emphasize good programming habits and choosing the appropriate language/software for a given scientific task.
Instructor:
Staff
ACM 95/100 abc
Introductory Methods of Applied Mathematics
12 units (408)

first, second, third terms
Prerequisites: Ma 1 abc, Ma 2, Ma 3 (may be taken concurrently), or equivalents.
First term: complex analysis: analyticity, Laurent series, singularities, branch cuts, contour integration, residue calculus. Second term: ordinary differential equations. Linear initial value problems: Laplace transforms, series solutions. Linear boundary value problems: eigenvalue problems, Fourier series, SturmLiouville theory, eigenfunction expansions, the Fredholm alternative, Green's functions, nonlinear equations, stability theory, Lyapunov functions, numerical methods. Third term: linear partial differential equations: heat equation separation of variables, Fourier transforms, special functions, Green's functions, wave equation, Laplace equation, method of characteristics, numerical methods.
Instructors:
Pierce, Meiron, Staff
ACM 101 abc
Methods of Applied Mathematics I
9 units (306)

first, second, third terms
Prerequisites: See below.
Analytical methods for the formulation and solution of initial and boundary value problems for ordinary and partial differential equations. First term: Prerequisites: ACM 95/100 abc or equivalent. Brief review of the elements of complex analysis and complexvariable methods. Asymptotic expansions, asymptotic evaluation of integrals (Laplace method, stationary phase, steepest descents), perturbation methods, WKB theory, boundarylayer theory,matched asymptotic expansions. The methodologies discussed will be illustrated by means of applications in various areas of science and engineering. Second term: (taught concurrently with CDS 140 a and AM 125 b). Prerequisites: ACM 95/100 ab or equivalent. Basic topics in dynamics in Euclidean space, including equilibria, stability, Lyapunov functions, periodic solutions, PoincarÃ©Bendixon theory, and PoincarÃ© maps. Additional topics may include attractors, structural stability and simple bifurcations, including Hopf bifurcations. Third term: Prerequisites: ACM 95/100 abc or equivalent. Applied spectral theory, linear operators, generalized eigenfunction expansions, convergence theory, transform methods, Fredholm theory and integral equation methods. The methodologies discussed will be illustrated by means of applications in various areas of science and engineering.
Instructors:
Bruno, Murray, Bruno
ACM 104
Linear Algebra and Applied Operator Theory
9 units (306)

first term
Prerequisites: ACM 100 abc or instructor's permission.
Linear spaces, subspaces, spans of sets, linear independence, bases, dimensions; linear transformations and operators, examples, nullspace/kernel, rangespace/image, onetoone and onto, isomorphism and invertibility, ranknullity theorem; products of linear transformations, left and right inverses, generalized inverses. Adjoints of linear transformations, singularvalue decomposition and MoorePenrose inverse; matrix representation of linear transformations between finitedimensional linear spaces, determinants, multilinear forms; metric spaces: examples, limits and convergence of sequences, completeness, continuity, fixedpoint (contraction) theorem, open and closed sets, closure; normed and Banach spaces, inner product and Hilbert spaces: examples, CauchySchwarz inequality, orthogonal sets, GramSchmidt orthogonalization, projections onto subspaces, best approximations in subspaces by projection; bounded linear transformations, principle of superposition for infinite series, wellposed linear problems, norms of operators and matrices, convergence of sequences and series of operators; eigenvalues and eigenvectors of linear operators, including their properties for self adjoint operators, spectral theorem for selfadjoint and normal operators; canonical representations of linear operators (finitedimensional case), including diagonal and Jordan form, direct sums of (generalized) eigenspaces. Schur form; functions of linear operators, including exponential, using diagonal and Jordan forms, CayleyHamilton theorem. Taught concurrently with CDS 201.
Instructor:
Beck
ACM 105
Applied Real and Functional Analysis
9 units (306)

second term
Prerequisites: ACM 100 abc or instructor's permission.
Lebesgue integral on the line, general measure and integration theory; Lebesgue integral in ndimensions, convergence theorems, Fubini, Tonelli, and the transformation theorem; normed vector spaces, completeness, Banach spaces, Hilbert spaces; dual spaces, HahnBanach theorem, RieszFrechet theorem, weak convergence and weak solvability theory of boundary value problems; linear operators, existence of the adjoint. Selfadjoint operators, polar decomposition, positive operators, unitary operators; dense subspaces and approximation, the Baire, BanachSteinhaus, open mapping and closed graph theorems with applications to differential and integral equations; spectral theory of compact operators; LP spaces, convolution; Fourier transform, Fourier series; Sobolev spaces with application to PDEs, the convolution theorem, Friedrich's mollifiers. Not offered 201314.
ACM 106 abc
Introductory Methods of Computational Mathematics
9 units (306)

first, second, third terms
Prerequisites: Ma 1 abc, Ma 2, Ma 3, ACM 11, 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; timefrequency 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:
Ames, Li
ACM 113
Introduction to Optimization
9 units (306)

second term
Prerequisites: ACM 95/100 abc, ACM 11, 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 quasiNewton methods. Linear programming: optimality conditions, the simplex method, primaldual interiorpoint 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.
Instructor:
Chandrasekaran
ACM/CS 114
Parallel Algorithms for Scientific Applications
9 units (306)

second, third term
Prerequisites: ACM 11, 106 or equivalent.
Introduction to parallel program design for numerically intensive scientific applications. Parallel programming methods; distributedmemory model with message passing using the message passing interface; sharedmemory model with threads using open MP, CUDA; objectbased models using a problemsolving environment with parallel objects. Parallel numerical algorithms: numerical methods for linear algebraic systems, such as LU decomposition, QR method, CG solvers; parallel implementations of numerical methods for PDEs, including finitedifference, finiteelement; particlebased simulations. Performance measurement, scaling and parallel efficiency, load balancing strategies.
Instructor:
Aivazis
ACM/EE 116
Introduction to Stochastic Processes and Modeling
9 units (306)

first term
Prerequisites: Ma 2, Ma 3 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 WienerKhinchine 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 (306)

second term
Prerequisites: 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. Simple and multiple regression: estimation, inference, model checking. Analysis of variance, comparison of models, model selection. Principal component analysis. Linear discriminant analysis. Generalized linear models and logistic regression. Resampling methods and the bootstrap.
Instructor:
Staff
ACM 126 ab
Wavelets and Modern Signal Processing
9 units (306)

second, third terms
Prerequisites: ACM 11, 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, timefrequency analysis, shorttime 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 201314.
AM/ACM 127
Calculus of Variations
9 units (306)

third Term
Prerequisites: ACM 95/100.
First and second variations; EulerLagrange equation; Hamiltonian formalism; action principle; HamiltonJacobi theory; stability; local and global minima; direct methods and relaxation; isoperimetric inequality; asymptotic methods and gamma convergence; selected applications to mechanics, materials science, control theory and numerical methods. Not offered 20132014.
Ma/ACM 142
Ordinary and Partial Differential Equations
9 units (306)

first term
Prerequisites: Ma 108; Ma 109 is desirable.
The mathematical theory of ordinary and partial differential equations, including a discussion of elliptic regularity, maximal principles, solubility of equations. The method of characteristics.
Instructor:
Frank
Ma/ACM 144 abc
Probability
9 units (306)

first, second, third terms
Overview of measure theory. Random walks and the Strong law of large numbers via the theory of martingales and Markov chains. Characteristic functions and the central limit theorem. Poisson process and Brownian motion. Topics in statistics.
Instructors:
Alberts, Chipeniuk
ACM 190
Reading and Independent Study
Units by arrangement
Graded pass/fail only.
ACM 201 ab
Partial Differential Equations
12 units (408)

first, second terms
Prerequisites: ACM 11, 101 abc or instructor's permission.
Fully nonlinear firstorder PDEs, shocks, eikonal equations. Classification of secondorder linear equations: elliptic, parabolic, hyperbolic. Wellposed 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, SchrÃ¶dinger 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 doublelayer potentials. Fredholm theory. NavierStokes equations. Stokes flow, Reynolds number. Potential flow; connection with complex variables. Blasius formulae. Boundary layers. Subsonic, supersonic, and transonic flow. Not offered 201314.
ACM/CDS 202
Geometry of Nonlinear Systems
9 units (306)

third term
Prerequisites: CDS 201 or AM 125 a.
Basic differential geometry, oriented toward applications in control and dynamical systems. Topics include smooth manifolds and mappings, tangent and normal bundles. Vector fields and flows. Distributions and Frobenius's theorem. Matrix Lie groups and Lie algebras. Exterior differential forms, Stokes' theorem. Not offered 201314.
ACM 210 ab
Numerical Methods for PDEs
9 units (306)

second, third terms
Prerequisites: ACM 11, 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, LaxWendroff theorem, Godunov's method, Roe's linearization, TVD schemes, highresolution 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 hp refinement. Multiscale finite element methods for elliptic problems with multiscale coefficients.
Instructor:
Hou
ACM 216
Markov Chains, Discrete Stochastic Processes and Applications
9 units (306)

second term
Prerequisites: 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:
Owhadi
ACM 217 ab
Advanced Topics in Stochastic Analysis
9 units (306)

first, third terms
Prerequisites: 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:
Beck, Tropp
ACM/CS/EE 218
Statistical Inference
306

third term
Prerequisites: ACM 104 and ACM 116, or instructor's permission.
Fundamentals of estimation theory and hypothesis testing; Bayesian and nonBayesian approaches; minimax analysis, CramerRao bounds, shrinkage in high dimensions; Kalman filtering, basics of graphical models; statistical model selection. Throughout the course, a computational viewpoint will be emphasized.
Instructor:
Chandrasekaran
Ae/ACM/ME 232 abc
Computational Fluid Dynamics
9 units (306)

first, second, third terms
Prerequisites: Ae/APh/CE/ME 101 abc or equivalent; ACM 100 abc or equivalent.
Development and analysis of algorithms used in the solution of fluid mechanics problems. Numerical analysis of discretization schemes for partial differential equations including interpolation, integration, spatial discretization, systems of ordinary differential equations; stability, accuracy, aliasing, Gibbs and Runge phenomena, numerical dissipation and dispersion; boundary conditions. Survey of finite difference, finite element, finite volume and spectral approximations for the numerical solution of the incompressible and compressible Euler and NavierStokes equations, including shockcapturing methods. Not offered 201314.
ACM 256 ab
Special Topics in Applied Mathematics
9 units (306)

first term
Prerequisites: ACM 101 or equivalent.
Introduction to finite element methods. Development of the most commonly used methodcontinuous, piecewiselinear finite elements on triangles for scalar elliptic partial differential equations; practical (a posteriori) error estimation techniques and adaptive improvement; formulation of finite element methods, with a few concrete examples of important equations that are not adequately treated by continuous, piecewiselinear finite elements, together with choices of finite elements that are appropriate for those problems. Homogenization and optimal design. Topics covered include periodic homogenization, G and Hconvergence, Gammaconvergence, Gclosure problems, bounds on effective properties, and optimal composites. Not offered 201314.
ACM 257
Special Topics in Financial Mathematics
9 units (306)

third term
Prerequisites: ACM 95/100 or instructor's permission. A basic knowledge of probability and statistics as well as transform methods for solving PDEs is assumed.
This course develops some of the techniques of stochastic calculus and applies them to the theory of financial asset modeling. The mathematical concepts/tools developed will include introductions to random walks, Brownian motion, quadratic variation, and Itocalculus. Connections to PDEs will be made by FeynmanKac theorems. Concepts of riskneutral pricing and martingale representation are introduced in the pricing of options. Topics covered will be selected from standard options, exotic options, American derivative securities, termstructure models, and jump processes. Not offered 201314.
ACM 270
Advanced Topics in Applied and Computational Mathematics
Hours and units by arrangement

third term
Advanced topics in applied and computational mathematics that will vary according to student and instructor interest. May be repeated for credit.
Instructor:
Ames
ACM 290 abc
Applied and Computational Mathematics Colloquium
1 unit

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.
Published Date:
July 28, 2022