# Applied and Computational Mathematics

**ACM 11. Introduction to Matlab and Mathematica.** *6 units (2-2-2); 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, object-oriented 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 (rule-based, 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: Lam.

**ACM 95/100 ab. Introductory Methods of Applied Mathematics for the Physical Sciences.** *12 units (4-0-8); 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, Sturm-Liouville 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: Zuev, Meiron.

**ACM 101 ab. Methods of Applied Mathematics.** *12 units (4-0-8); first, second, terms. Prerequisites: Math 2/102 and ACM 95ab.* First term: brief review of the elements of complex analysis and complex-variable methods. Asymptotic expansions, asymptotic evaluation of integrals (Laplace method, stationary phase, steepest descents), perturbation methods, WKB theory, boundary-layer theory, matched asymptotic expansions with first-order and high-order 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 self-adjoint second order operators with variable coefficients in n-dimensional domains. Integral equations, Fredholm theorem, application to Laplace and Maxwell’s equations, harmonicity at infinity, Kelvin transform, conditions of radiation at infinity. Instructor: Bruno.

**ACM 104. Applied Linear Algebra.** *9 units (3-1-5); first term. Prerequisites: Ma 1 abc, Ma 2/102.* This is an intermediate linear algebra course aimed at a diverse group of students, including junior and senior majors in applied mathematics, sciences and engineering. The focus is on applications. Matrix factorizations play a central role. Topics covered include linear systems, vector spaces and bases, inner products, norms, minimization, the Cholesky factorization, least squares approximation, data fitting, interpolation, orthogonality, the QR factorization, ill-conditioned systems, discrete Fourier series and the fast Fourier transform, eigenvalues and eigenvectors, the spectral theorem, optimization principles for eigenvalues, singular value decomposition, condition number, principal component analysis, the Schur decomposition, methods for computing eigenvalues, non-negative matrices, graphs, networks, random walks, the Perron-Frobenius theorem, PageRank algorithm. Instructor: Zuev

**ACM 105. Applied Real and Functional Analysis.** *9 units (3-0-6); second term. Prerequisite: ACM 100 ab or instructor’s permission. *Lebesgue integral on the line, general measure and integration theory; Lebesgue integral in n-dimensions, convergence theorems, Fubini, Tonelli, and the transformation theorem; normed vector spaces, completeness, Banach spaces, Hilbert spaces; dual spaces, Hahn-Banach theorem, Riesz-Frechet theorem, weak convergence and weak solvability theory of boundary value problems; linear operators, existence of the adjoint. Self-adjoint operators, polar decomposition, positive operators, unitary operators; dense subspaces and approximation, the Baire, Banach-Steinhaus, 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: Hoffman.

**ACM/EE 106 ab. Introductory Methods of Computational Mathematics. ***12 units (3-0-9); first, second 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, Gauss-Seidel 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: Lam.

**CMS/ACM 107. Linear Analysis with Applications.*** 12 units (3-3-6); first term. *See course description in Computing and Mathematical Sciences.

**Ec/ACM/CS 112. Bayesian Statistics.** *9 units (3-0-6); third term. *See course description in Economics.

**CMS/ACM 113. Mathematical Optimization.** *9 units (3-0-6); first term. *See course description in Computing and Mathematical Sciences.

**ACM/CS 114 ab. Parallel Algorithms for Scientific Applications.** *9 units (3-0-6); second, third term. Prerequisites: ACM 11, 106 or equivalent. *Introduction to parallel program design for numerically intensive scientific applications. Parallel programming methods; distributed-memory model with message passing using the message passing interface; shared-memory model with threads using open MP, CUDA; 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, CG solvers; parallel implementations of numerical methods for PDEs, including finite-difference, finite-element; particle-based simulations. Performance measurement, scaling and parallel efficiency, load balancing strategies. Not offered 2017–18.

**ACM/EE 116. Introduction to Probability Models.** *9 units (3-1-5); first term. Prerequisites: Ma 2, Ma 3. *This course introduces students to the fundamental concepts, methods, and models of applied probability and stochastic processes. The course is application oriented and focuses on the development of probabilistic thinking and intuitive feel of the subject rather than on a more traditional formal approach based on measure theory. The main goal is to equip science and engineering students with necessary probabilistic tools they can use in future studies and research. Topics covered include sample spaces, events, probabilities of events, discrete and continuous random variables, expectation, variance, correlation, joint and marginal distributions, independence, moment generating functions, law of large numbers, central limit theorem, random vectors and matrices, random graphs, Gaussian vectors, branching, Poisson, and counting processes, general discrete- and continuous-timed processes, auto- and cross-correlation functions, stationary processes, power spectral densities. Instructor: Zuev.

**CMS/ACM/EE 117. Probability and Random Processes. ***12 units (3-0-9); first term. *For course description, see Computation and Mathematical Sciences.

**AM/ACM 127. Calculus of Variations.** *9 units (3-0-6).* For course description, see Applied Mechanics.

**Ma/ACM 142. 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); second, third terms.* For course description, see Mathematics.

**ACM/CS 157. Statistical Inference.** *9 units (3-2-4); third term. Prerequisites: ACM/EE 116, Ma 3.* Statistical Inference is a branch of mathematical engineering that studies ways of extracting reliable information from limited data for learning, prediction, and decision making in the presence of uncertainty. This is an introductory course on statistical inference. The main goals are: develop statistical thinking and intuitive feel for the subject; introduce the most fundamental ideas, concepts, and methods of statistical inference; and explain how and why they work, and when they don’t. Topics covered include summarizing data, fundamentals of survey sampling, statistical functionals, jackknife, bootstrap, methods of moments and maximum likelihood, hypothesis testing, p-values, the Wald, t-, permutation, likelihood ratio tests, multiple testing, scatterplots, simple linear regression, ordinary least squares, interval estimation, prediction, graphical residual analysis. Instructor: Zuev.

**ACM/CS/EE 158. Mathematical Statistics. ***9 units (3-0-6); third term. Prerequisites: CMS/ACM 113, ACM/EE 116 and ACM/CS 157. *Fundamentals of estimation theory and hypothesis testing; minimax analysis, Cramer-Rao bounds, Rao-Blackwell theory, shrinkage in high dimensions; Neyman-Pearson theory, multiple testing, false discovery rate; exponential families; maximum entropy modeling; other advanced topics may include graphical models, statistical model selection, etc. Throughout the course, a computational viewpoint will be emphasized. Not offered 2017–18.

**ACM 159. Inverse Problems and Data Assimilation.** *9 units (3-0-6); first term. Prerequisites: ACM 104, ACM/EE 106 ab, ACM/EE 116, ACM/CS 157 or equivalent.* Models in applied mathematics often have input parameters that are uncertain; observed data can be used to learn about these parameters and thereby to improve predictive capability. The purpose of the course is to describe the mathematical and algorithmic principles of this area. The topic lies at the intersection of fields including inverse problems, differential equations, machine learning and uncertainty quantification. Applications will be drawn from the physical, biological and data sciences. Instructor: Stuart.

**ACM/EE 170. Mathematics of Signal Processing.** *12 units (3-0-9); third term. Prerequisites: ACM 104, CMS/ACM 113, and *ACM/EE 116*; or instructor’s permission.* This course covers classical and modern approaches to problems in signal processing. Problems may include denoising, deconvolution, spectral estimation, direction-of-arrival 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 least-squares and on sparse modeling. Throughout the course, a computational viewpoint will be emphasized. Not offered 2017–18.

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

**ACM 201 ab. Partial Differential Equations.** *12 units (4-0-8); second, third terms. Prerequisite: ACM 11, 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. Not offered 2017–18.

**ACM 204. Topics in Convexity.** *9 units (3-0-6); second term. Prerequisites: 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. Instructor: Tropp.

**ACM 210 ab. Numerical Methods for PDEs.** *9 units (3-0-6); second, third terms. Prerequisite: 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, 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. Not offered 2017–18.

**ACM 213. Topics in Optimization.** *9 units (3-0-6); third term. Prerequisites: ACM 104, CMS/ACM 113.* Material varies year-to-year. Example topics include discrete optimization, convex and computational algebraic geometry, numerical methods for large-scale optimization, and convex geometry. Instructor: Chandrasekaran.

**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: Owhadi.

**ACM 217 ab. Advanced Topics in Stochastic Analysis.** *9 units (3-0-6); second 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. Instructor: Tropp.

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

**ACM 256 ab. Special Topics in Applied Mathematics.** *9 units (3-0-6); first term. Prerequisite: ACM 101 or equivalent.* Introduction to finite element methods. Development of the most commonly used method—continuous, piecewise-linear 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, piecewise-linear 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 H-convergence, Gamma-convergence, G-closure problems, bounds on effective properties, and optimal composites. Not offered 2017–18.

**ACM 257. Special Topics in Financial Mathematics.** *9 units (3-0-6); third term. Prerequisite: 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 Ito-calculus. Connections to PDEs will be made by Feynman-Kac theorems. Concepts of risk-neutral pricing and martingale representation are introduced in the pricing of options. Topics covered will be selected from standard options, exotic options, American derivative securities, term-structure models, and jump processes. Not offered 2017–18.

**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: Hou.

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