Courses 2024-25

This course is intended to serve as a practical introduction to the methods of computational science and engineering for students in all majors. The goal is to provide students exposure to and hands-on experience with commonly-used computational methods in science and engineering, with theoretical considerations confined to a level appropriate for first-year undergraduate students. Topics covered include computational simulation by discretization in space and time, numerical solution of linear and nonlinear equations, optimization, uncertainty quantification, and function approximation via interpolation and regression. Emphasis is on understanding trade-offs between computational effort and accuracy, and on developing working knowledge of how these tools can be used to solve a wide range of problems arising in applied math, science, and engineering. Assignments and in-class activities use MATLAB. No prior experience with MATLAB expected.

Individual research project, carried out under the supervision of a member of the ACM faculty (or other faculty as approved by the ACM undergraduate option representative). Projects must include significant design effort. Written report required. Open only to upper class students. Not offered on a pass/fail basis.

Supervised research or development in ACM by undergraduates. The topic must be approved by the project supervisor, and a formal final report must be presented on completion of research. Graded pass/fail.

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.

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, Schroedinger, 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.

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.

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 cover 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 equations. Study of numerical PDE will include stability analysis. Programming is a significant part of the course.

Part a: Covers the basic algebraic, geometric, and topological properties of normed linear spaces, inner-product spaces and linear maps. Emphasis is placed both on rigorous mathematical development and on applications to control theory, data analysis and partial differential equations. Topics: Completeness, Banach spaces (l_p, L_p), Hilbert spaces (weighted l_2, L_2 spaces), introduction to Fourier transform, Fourier series and Sobolev spaces, Banach spaces of linear operators, duality and weak convergence, density, separability, completion, Schauder bases, continuous and compact embedding, compact operators, orthogonality, Lax-Milgram, Spectral Theorem and SVD for compact operators, integral operators, Jordan normal form. Part b: Continuation of ACM 107a, developing new material and providing further details on some topics already covered. Emphasis is placed both on rigorous mathematical development and on applications to control theory, data analysis and partial differential equations.Topics: Review of Banach spaces, Hilbert spaces, Linear Operators, and Duality, Hahn-Banach Theorem, Open Mapping and Closed Graph Theorem, Uniform Boundedness Principle, The Fourier transform (L1, L2, Schwartz space theory), Sobolev spaces (W^s,p, H^s), Sobolev embedding theorem, Trace theorem Spectral Theorem, Compact operators, Ascoli Arzela theorem, Contraction Mapping Principle, with applications to the Implicit Function Theorem and ODEs, Calculus of Variations (differential calculus, existence of extrema, Gamma-convergence, gradient flows) Applications to Inverse Problems (Tikhonov regularization, imaging applications).

This course provides an introduction to Bayesian Statistics and its applications to data analysis in various fields. Topics include: discrete models, regression models, hierarchical models, model comparison, and MCMC methods. The course combines an introduction to basic theory with a hands-on emphasis on learning how to use these methods in practice so that students can apply them in their own work. Previous familiarity with frequentist statistics is useful but not required.

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.

This course begins with an overview of measure theory, followed by topics that include random walks, the strong law of large numbers, the central limit theorem, martingales, Markov chains, characteristic functions, Poisson processes, and Brownian motion. Towards the end, some further topics may be covered, such as stochastic calculus, stochastic differential equations, Gaussian processes, random graphs, Markov chain mixing, random matrix theory, and interacting particle systems.

The mathematical theory of ordinary and partial differential equations, including a discussion of elliptic regularity, maximal principles, solubility of equations. The method of characteristics. Part b not offered 2024-25.

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.

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.

Graded pass/fail only.

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. Offered 2024-25.

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.

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 Navier-Stokes equations, including shock-capturing methods.

Measure transport is a rich mathematical topic at the intersection of analysis, probability and optimization. The core idea behind this theory is to rearrange the mass of a reference measure to match a target measure. In particular, optimal transport seeks a rearrangement that transports mass with minimal cost. The theory of optimal transport dates back to Monge in 1781, with significant advancements by Kantorovich in 1942 and later in the '90s, e.g. by Brenier. In recent years, measure transport has become an indispensable tool for representing probability distributions and for defining measures of similarity between distributions. These methods enjoy applications in image retrieval, signal and image representation, inverse problems, cancer detection, texture and color modelling, shape and image registration, and machine learning, to name a few. This class will introduce the foundations of measure transport, present its connections and applications in various fields, and lastly explore modern computational methods for finding discrete and continuous transport maps, e.g. Sinkhorn's algorithm and normalizing flows.

Advanced topics in applied and computational mathematics that will vary according to student and instructor interest. May be repeated for credit.