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. Topic varies by year. Open only to upper class students. The approximation theory includes data fitting; interpolation using Fourier transform, orthogonal polynomials and splines; least square method, and numerical quadrature. Prerequisites: Math 2/102 and ACM 95 ab or equivalent. 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. 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. 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. This course offers a rigorous introduction to probability and stochastic processes. This course develops some of the techniques of stochastic calculus and applies them to the theory of financial asset modeling. 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 topic lies at the intersection of fields including inverse problems, differential equations, machine learning and uncertainty quantification. Introductory Methods of Applied Mathematics for the Physical Sciences. 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. 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. Not offered 2020-21. Regression: Gaussian vectors, spaces, conditioning, processes, fields and measures will be presented with an emphasis on linear regression. 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. Second term: Applied spectral theory, special functions, generalized eigenfunction expansions, convergence theory. Not offered 2020-21. Spectral methods: Fourier spectral methods on infinite and periodic domains. This course offers an introduction to the theory of Partial Differential Equations (PDEs) commonly encountered across mathematics, engineering and science. 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. Markov Chains, Discrete Stochastic Processes and Applications. 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. Fast spectrally-accurate PDE solvers for linear and nonlinear Partial Differential Equations in general domains. ACM 11 is desired. ACM 11 is desired. The course will also emphasize good programming habits and choosing the appropriate language/software for a given scientific task. The course can be viewed as a statistical analog of CMS/CS/CNS/EE/IDS 155. Address: Mathematics 253-37 | Caltech | Pasadena, CA 91125 Telephone: (626) 395-4335 | Fax: (626) 585-1728 Prerequisites: Ma 1 abc, Ma 2, Ma 3, ACM 11, ACM 95/100 ab or equivalent. Finite difference and finite volume methods for hyperbolic problems. This course provides an introduction to Bayesian Statistics and its applications to data analysis in various fields. Not offered 2020-21. Prerequisites: Ma 3, some familiarity with MATLAB, e.g. Matrix factorizations play a central role.