# 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/IDS 101 ab. Methods of Applied Mathematics. ***12 units (4-0-8); first, second terms. Prerequisites: Math 2/102 and ACM 95 ab or equivalent.* 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, generalized eigenfunction expansions, convergence theory. Gibbs and Runge phenomena and their resolution. Chebyshev expansion and Fourier Continuation methods. Review of numerical stability theory for time evolution. Fast spectrally-accurate PDE solvers for linear and nonlinear Partial Differential Equations in general domains. Integral-equations methods for linear partial differential equation in general domains (Laplace, Helmholtz, Schrödinger, Maxwell, Stokes). Homework problems in both 101 a and 101 b include theoretical questions as well as programming implementations of the mathematical and numerical methods studied in class. Instructor: Bruno.

**ACM/IDS 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. Prerequisites: Ma 2, Ma 108a, ACM/IDS 104 or equivalent. *This course is about the fundamental concepts in real and functional analysis that are vital for many topics and applications in mathematics, physics, computing and engineering. The aim of this course is to provide a working knowledge of functional analysis with an eye especially for aspects that lend themselves to applications. The course gives an overview of the interplay between different functional spaces and focuses on the following three key concepts: Hahn-Banach theorem, open mapping and closed graph theorem, uniform boundedness principle. Other core concepts include: normed linear spaces and behavior of linear maps, completeness, Banach spaces, Hilbert spaces, Lp spaces, duality of normed spaces and dual operators, dense subspaces and approximations, hyperplanes, compactness, weak and weak* convergence. More advanced topics include: spectral theory, compact operators, theory of distributions (generalized functions), Fourier analysis, calculus of variations, Sobolev spaces with applications to PDEs, weak solvability theory of boundary value problems. Instructor: Hoffmann.

**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. Instructors: Lam, Hou.

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

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

**CMS/ACM/IDS 113. Mathematical Optimization. **12 *units (3-3-6). *See course description in Computing and Mathematical Sciences.

**ACM/CS/IDS 114. Parallel Algorithms for Scientific Applications. ***9 units (3-0-6). 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 2018–19.

**ACM/EE/IDS 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/IDS 117. Probability and Random Processes. ***12 units (3-0-9). *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/IDS 144 ab. Probability.** *9 units (3-0-6); second, third terms.* For course description, see Mathematics.

A**CM/IDS 154. Inverse Problems and Data Assimilation. ***9 units (3-0-6); first term. Prerequisites: Basic differential equations, linear algebra, probability and statistics: ACM/IDS 104, ACM/EE 106 ab, ACM/EE/IDS 116, ACM/CS/IDS 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. Not offered 2018–19.

**ACM/CS/IDS 157. Statistical Inference. ***9 units (3-2-4); third term. Prerequisites: ACM/EE/IDS 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, Student’s t-, permutation, and likelihood ratio tests, multiple testing, scatterplots, simple linear regression, ordinary least squares, interval estimation, prediction, graphical residual analysis. Instructor: Zuev.

**ACM/CS/EE/IDS 158. Mathematical Statistics. ***9 units (3-0-6); third term. Prerequisites: CMS/ACM/IDS 113, ACM/EE/IDS 116 and ACM/CS/IDS 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 2018–19.

**ACM/EE/IDS 170. Mathematics of Signal Processing. ***12 units (3-0-9); third term. Prerequisites: ACM/IDS 104, CMS/ACM/IDS 113, and ACM/EE/IDS 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. Instructor: Hassibi.

**CS/ACM 177 ab. Discrete Differential Geometry: Theory and Applications. ***9 units (3-3-3). *For course description, see Computer Science.

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

**ACM 201. Partial Differential Equations. **1*2 units (4-0-8); first term. Prerequisites: ACM 95/100 ab, ACM/IDS 101 ab, ACM 11 or equivalent.* This course offers an introduction to the theory of Partial Differential Equations (PDEs) commonly encountered across mathematics, engineering and science. The goal of the course is to study properties of different classes of linear and nonlinear PDEs (elliptic, parabolic and hyperbolic) and the behavior of their solutions using tools from functional analysis with an emphasis on applications. We will discuss representative models from different areas such as: heat equation, wave equation, advection-reaction-diffusion equation, conservation laws, shocks, predator prey models, Burger’s equation, kinetic equations, gradient flows, transport equations, integral equations, Helmholtz and Schrödinger equations and Stoke’s flow. In this course you will use analytical tools such as Gauss’s theorem, Green’s functions, weak solutions, existence and uniqueness theory, Sobolev spaces, well-posedness theory, asymptotic analysis, Fredholm theory, Fourier transforms and spectral theory. More advanced topics include: Perron’s method, applications to irrotational flow, elasticity, electrostatics, special solutions, vibrations, Huygens’ principle, Eikonal equations, spherical means, retarded potentials, water waves, various approximations, dispersion relations, Maxwell equations, gas dynamics, Riemann problems, single- and double-layer potentials, Navier-Stokes equations, Reynolds number, potential flow, boundary layer theory, subsonic, supersonic and transonic flow. Instructors: Hoffmann, Hosseini.

**ACM/IDS 204. Topics in Linear Algebra and Convexity. ***12 units (3-0-9); first term. Prerequisites: ACM/IDS 104 and CMS/ACM/IDS 113; or instructor’s permission.* Topic varies by year. 2018-2019: Convexity. This class offers an overview of discrete and continuous aspects of convex geometry with some computational applications. Material may include geometry of convex sets and functions, facial geometry of convex sets, convexity in infinite dimensions, polarity and duality theory, ellipsoids, polytopes, lattices and lattice points, geometric probability. Instructor: Tropp.

**ACM 210. Numerical Methods for PDEs. ***9 units (3-0-6); 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. Instructor: Hou.

**ACM/IDS 213. Topics in Optimization. ***9 units (3-0-6); third term. Prerequisites: ACM/IDS 104, CMS/ACM/IDS 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. Not offered 2018–19.

**ACM/IDS 216. Markov Chains, Discrete Stochastic Processes and Applications. ***9 units (3-0-6); second term. Prerequisites: ACM/EE/IDS 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/EE/IDS 217. Advanced Topics in Stochastic Analysis. ***9 units (3-0-6); third term. Prerequisites: ACM/CMS/EE/IDS 117.* 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: Stuart.

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

**ACM 256. Special Topics in Applied Mathematics.** *9 units (3-0-6); first term. Prerequisite: ACM/IDS 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. Instructor: Hoffmann.

**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 2018–19.

**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. Not offered 2018–19.

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