Prerequisites: ACM 95/100 ab, ACM/IDS 101 ab, ACM 11 or equivalent. This is an introductory course on statistical inference. 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. Mathematical tools will include ordinary, partial and stochastic differential equations, as well as Markov chains and other stochastic processes. Ordinary differential equations: linear initial value problems, linear boundary value problems, Sturm-Liouville theory, eigenfunction expansions, transform methods, Green's functions. Finite difference and finite volume methods for hyperbolic problems. 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. Method of multiple scales for oscillatory systems. 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. 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. Prerequisites: ACM 95/100 or instructor's permission. Introductory Methods of Applied Mathematics for the Physical Sciences. The main goal is to equip science and engineering students with necessary probabilistic tools they can use in future studies and research. May be repeated for credit. The ODE parts include initial and boundary value problems. Address: Mathematics 253-37 | Caltech | Pasadena, CA 91125 Telephone: (626) 395-4335 | Fax: (626) 585-1728 Prerequisites: instructor's permission, which should be obtained sufficiently early to allow time for planning the research. Methods rely heavily on linear algebra, convex optimization, and stochastic modeling. Random walks and the Strong law of large numbers via the theory of martingales and Markov chains. Prerequisites: Ma 1 abc, some familiarity with MATLAB, e.g. Markov Chains, Discrete Stochastic Processes and Applications. This course develops some of the techniques of stochastic calculus and applies them to the theory of financial asset modeling. This course offers an introduction to the theory of Partial Differential Equations (PDEs) commonly encountered across mathematics, engineering and science. Prerequisites: ACM/IDS 104, CMS/ACM/IDS 113. Chebyshev spectral methods on finite domains. This class offers an introduction to the emerging field of randomized algorithms for solving linear algebra problems. Emphasis will be placed on the principles used to develop these models, and on the unity and cross-cutting nature of the mathematical and computational tools used to study them. 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. To enroll in the program, the student should meet and discuss his/her plans with the option representative. Welcome. 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. 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. Stochastic processes: Branching processes, Poisson point processes, Determinantal point processes, Dirichlet processes and Gaussian processes (including the Brownian motion). Prerequisites: Basic differential equations, linear algebra, probability and statistics: ACM/IDS 104, ACM/EE 106 ab, ACM/EE/IDS 116, IDS/ACM/CS 157 or equivalent. In particular, the class will cover techniques based on least-squares and on sparse modeling. Anderson, Thomas Geoffrey (2020) Hybrid Frequency-Time Analysis and Numerical Methods for Time-Dependent Wave Propagation. Matrix factorizations play a central role. The mathematical theory of ordinary and partial differential equations, including a discussion of elliptic regularity, maximal principles, solubility of equations. The mathematical concepts/tools developed will include introductions to random walks, Brownian motion, quadratic variation, and Ito-calculus. Introductory Methods of Computational Mathematics. Problems may include denoising, deconvolution, spectral estimation, direction-of-arrival estimation, array processing, independent component analysis, system identification, filter design, and transform coding. 2019-20: Randomized algorithms for linear algebra. 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. 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. Example topics include discrete optimization, convex and computational algebraic geometry, numerical methods for large-scale optimization, and convex geometry. Introduction to finite element methods. ACM 11 is desired. This course gives an overview of different mathematical models used to describe a variety of phenomena arising in the biological, engineering, physical and social sciences. The course is oriented for upper level undergraduate students in IDS, ACM, and CS and graduate students from other disciplines who have sufficient background in probability and statistics. The course will also emphasize good programming habits and choosing the appropriate language/software for a given scientific task. Not offered 2020-21. The topic must be approved by the project supervisor, and a formal final report must be presented on completion of research. Concepts of risk-neutral pricing and martingale representation are introduced in the pricing of options. Prerequisites: Ma 2, Ma 108a, ACM/IDS 104 or equivalent. 2020. The PDE parts include finite difference and finite element for elliptic/parabolic/hyperbolic equation. Review of numerical stability theory for time evolution. 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 training essential for future careers in applied mathematics in academia, national laboratories, or in industry is provided, especially when combined with graduate work, by successful completion of the requirements for an undergraduate degree in applied and computational mathematics. Topic varies by year. Prerequisites: Ma 108; Ma 109 is desirable. The training essential for future careers in applied mathematics in academia, national laboratories, or in industry is provided, especially when combined with graduate work, by successful completion of the requirements for an undergraduate degree in applied and computational mathematics. 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. 9 units (3-1-5): first term. 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.