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

third term
Prerequisites: Ma 1 abc. 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 ab
Introductory Methods of Applied Mathematics for the Physical Sciences
12 units (408)

second, third terms
Prerequisites: Ma 1 abc, Ma 2 or equivalents.
Complex analysis: analyticity, Laurent series, contour integration, residue calculus. Ordinary differential equations: linear initial value problems, linear boundary value problems, SturmLiouville theory, eigenfunction expansions, transform methods, Green's functions. Linear partial differential equations: heat equation, separation of variables, Laplace equation, transform methods, wave equation, method of characteristics, Green's functions.
Instructors:
Meiron, Pierce
ACM 101 ab
Methods of Applied Mathematics
12 units (408)

first, second, terms
Prerequisites: Math 2/102 and ACM 95ab.
First term: 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 with firstorder and highorder matching. Method of multiple scales for oscillatory systems. Second term: applied spectral theory, special functions, Hilbert spaces and linear operators, generalized eigenfunction expansions, convergence theory. Transform methods, distributions, Fourier Transform and Sobolev Spaces. Eigensystems and spectral theory for selfadjoint second order operators with variable coefficients in ndimensional domains. Integral equations, Fredholm theorem, application to Laplace and Maxwell's equations, harmonicity at infinity, Kelvin transform, conditions of radiation at infinity.
Instructor:
Bruno
CMS/ACM 104
Linear Algebra and Applied Operator Theory
12 units (309)

first term
Prerequisites: Undergraduate prerequistes: Ma 1 abc (analytic track), Ma 2, and ACM 95 ab; or instructor's permission.
This course introduces the theory and applications of linear algebra and linear analysis. Lectures and homework will require the ability to understand and produce mathematical proofs. Theoretical topics may include topology of metric spaces, structure of Banach and Hilbert spaces, examples of normed spaces, duality, structure of linear operators, spectral theory, functional calculus for linear operators, and calculus in Banach spaces. Applications will be drawn from signal processing, numerical analysis, optimization, approximation, differential equations, control, and other areas. Emphasis will be placed on geometry and convexity.
Instructor:
Tropp
ACM 105
Applied Real and Functional Analysis
9 units (306)

second term
Prerequisites: ACM 100 ab 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.
Instructor:
Tropp
ACM 106 ab
Introductory Methods of Computational Mathematics
12 units (309)

second, third terms
Prerequisites: Ma 1 abc, Ma 2, Ma 3, ACM 11, ACM 95/100 ab 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 linear algebra parts covers basic methods such as direct and iterative solution of large linear systems, including LU decomposition, splitting method (Jacobi iteration, GaussSeidel iteration); eigenvalue and vector computations including the power method, QR iteration and Lanczos iteration; nonlinear algebraic solvers. The approximation theory includes data fitting; interpolation using Fourier transform, orthogonal polynomials and splines; least square method, and numerical quadrature. The ODE parts include initial and boundary value problems. The PDE parts include finite difference and finite element for elliptic/parabolic/hyperbolic equation. Stability analysis will be covered with numerical PDE. Programming is a significant part of the course.
Instructor:
Hou
CMS/ACM 113
Mathematical Optimization
9 units (306)

first term
Prerequisites: ACM 95/100 ab, ACM 11, or instructor's permission.
Corequisite: It is suggested that students take CMS/ACM 104 concurrently. This class studies mathematical optimization from the viewpoint of convexity. Topics covered include duality and representation of convex sets; linear and semidefinite programming; connections to discrete, network, and robust optimization; relaxation methods for intractable problems; as well as applications to problems arising in graphs and networks, information theory, control, signal processing, and other engineering disciplines.
Instructor:
Chandrasekaran
ACM/CS 114 ab
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. Not offered 201516.
CMS/ACM 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
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 201516.
Ma/ACM 142
Ordinary and Partial Differential Equations
9 units (306)

second 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 ab
Probability
9 units (306)

first, second terms
Prerequisites: For 144a, Ma 108b is strongly recommended; for 144b, 108b and 144a are prerequisite.
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:
Makarov, Fathizadeh
ACM/EE 170
Mathematics of Signal Processing
12 units (309)

third term
Prerequisites: CMS/ACM 104, CMS/ACM 113, and CMS/ACM 116; or instructor's permission.
This course covers classical and modern approaches to problems in signal processing. Problems may include denoising, deconvolution, spectral estimation, directionofarrival estimation, array processing, independent component analysis, system identification, filter design, and transform coding. Methods rely heavily on linear algebra, convex optimization, and stochastic modeling. In particular, the class will cover techniques based on leastsquares and on sparse modeling. Throughout the course, a computational viewpoint will be emphasized. Not offered 201516.
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 201516.
ACM 204
Topics in Convexity
9 units (306)

second term
Prerequisites: CMS/ACM 104 and CMS/ACM 113; or instructor's permission.
The content of this course varies from year to year among advanced subjects in linear algebra, convex analysis, and related fields. Specific topics for the class include matrix analysis, operator theory, convex geometry, or convex algebraic geometry. Lectures and homework will require the ability to understand and produce mathematical proofs. Not overed 201516.
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. Not offered 201516.
ACM 213
Topics in Optimization
9 units (306)

third term
Prerequisites: CMS/ACM 104, CMS/ACM 113.
Material varies yeartoyear. Example topics include discrete optimization, convex and computational algebraic geometry, numerical methods for largescale optimization, and convex geometry.
Instructor:
Chandrasekaran
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)

third term
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. Not offered 201516.
ACM/CS/EE 218
Statistical Inference
9 units (306)

third term
Prerequisites: CMS/ACM 104 and CMS/ACM/EE 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. Not offered 201516.
Ae/ACM/ME 232 ab
Computational Fluid Dynamics
9 units (306)

first, second 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 20152016.
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 201516.
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 201516.
ACM 270
Advanced Topics in Applied and Computational Mathematics
Hours and units by arrangement

second, third terms
Advanced topics in applied and computational mathematics that will vary according to student and instructor interest. May be repeated for credit.
Instructor:
Staff
Published Date:
July 28, 2022