This course covers classical and modern approaches to problems in signal processing. A basic knowledge of probability and statistics as well as transform methods for solving PDEs is assumed. The course focuses on the mathematics and statistics of methods developed for learning from data. Mathematical tools will include ordinary, partial and stochastic differential equations, as well as Markov chains and other stochastic processes. The sequence covers the introductory methods in both theory and implementation of numerical linear algebra, approximation theory, ordinary differential equations, and partial differential equations. Second term: Applied spectral theory, special functions, generalized eigenfunction expansions, convergence theory. Material varies year-to-year. 2020. Probability Theory and Stochastic Processes. CS 1 or prior programming experience recommended. Random walks and the Strong law of large numbers via the theory of martingales and Markov chains. Not offered 2020-21. Assignments will require mathematical proofs, programming, and computer simulation. Development and analysis of algorithms used in the solution of fluid mechanics problems. The ODE parts include initial and boundary value problems. Prerequisites: For 140 a, Ma 108 b is strongly recommended. Executive Officer of Applied and Computational Math, Caltech, July 2000 - June, 2006. Emphasis is placed both on rigorous mathematical development and on applications to control theory, data analysis and partial differential equations. 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. Prerequisites: ACM 95/100 or instructor's permission. Stochastic processes: Branching processes, Poisson point processes, Determinantal point processes, Dirichlet processes and Gaussian processes (including the Brownian motion). Review of numerical stability theory for time evolution. Graded pass/fail. Concepts of risk-neutral pricing and martingale representation are introduced in the pricing of options. Introductory Methods of Computational Mathematics. 2019-20: Randomized algorithms for linear algebra. 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. Kernel and variational methods in numerical approximation, signal processing and learning will also be covered through their connections with Gaussian process regression. Complete programs will be worked out with faculty advisers. Prerequisites: ACM 95/100 ab or equivalent. 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. Not offered 2020-21. Ordinary differential equations: linear initial value problems, linear boundary value problems, Sturm-Liouville theory, eigenfunction expansions, transform methods, Green's functions. This is an introductory course on statistical inference. In general, approval is contingent on good academic performance by the student and demonstrated ability for handling the heavier course load. Topics covered include duality and representation of convex sets; linear and semidefinite programming; connections to discrete, network, and robust optimization; relaxation methods for intractable problems; as well as applications to problems arising in graphs and networks, information theory, control, signal processing, and other engineering disciplines. The purpose of the course is to describe the mathematical and algorithmic principles of this area. To enroll in the program, the student should meet and discuss his/her plans with the option representative. 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. Address: Mathematics 253-37 | Caltech | Pasadena, CA 91125 Telephone: (626) 395-4335 | Fax: (626) 585-1728 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. Method of multiple scales for oscillatory systems. Prerequisites: ACM/IDS 104, CMS/ACM/IDS 113, and ACM/EE/IDS 116; or instructor's permission. 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. First term: Brief review of the elements of complex analysis and complex-variable methods. and other important topics. 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. Prerequisites: Ma 3, some familiarity with MATLAB, e.g. ACM 11 is desired. The method of characteristics. Not offered 2020-21. Topics include probability measures, random variables and expectation, independence, concentration inequalities, distances between probability measures, modes of convergence, laws of large numbers and central limit theorem, Gaussian and Poisson approximation, conditional expectation and conditional distributions, filtrations, and discrete-time martingales. Problems may include denoising, deconvolution, spectral estimation, direction-of-arrival estimation, array processing, independent component analysis, system identification, filter design, and transform coding. Prerequisites: Ma 3, ACM/EE/IDS 116 or equivalent. The focus is on applications. Prerequisites: Ae/APh/CE/ME 101 abc or equivalent; ACM 100 abc or equivalent. Overview of measure theory. Programming is a significant part of the course. 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 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. Prerequisites: Ma 1 abc, some familiarity with MATLAB, e.g. The mathematical concepts/tools developed will include introductions to random walks, Brownian motion, quadratic variation, and Ito-calculus. Written report required. 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. 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. This course offers an introduction to the theory of Partial Differential Equations (PDEs) commonly encountered across mathematics, engineering and science. The fields of application include a wide range of areas such as fluid mechanics, materials science, and mathematical biology, engineering applications, image processing, and mathematical finance. 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. welcome to math Caltech's mathematics program brings together faculty, researchers, and students who have a breadth of interests and expertise in the use and analysis of numbers, and who are interested in collaborating with colleagues across fields to solve some of the most complicated problems of our time. Topics covered will be selected from standard options, exotic options, American derivative securities, term-structure models, and jump processes.

caltech applied math

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caltech applied math 2020