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. ACM 11 is desired. Students will learn what methods for statistical learning exist, how and why they work (not just what tasks they solve and in what built-in functions they are implemented), and when they are expected to perform poorly. Not offered 2020-21. 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. This class studies mathematical optimization from the viewpoint of convexity. Executive Officer of Applied and Computational Math, Caltech, July 2000 - June, 2006. Advanced topics in applied and computational mathematics that will vary according to student and instructor interest. Stability analysis will be covered with numerical PDE. 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. Caltech’s Computing & Mathematical Sciences department offers an interdisciplinary program of graduate study in applied and computational mathematics leading to the Ph.D. degree. Prerequisites: Ma 1 abc, Ma 2 or equivalents. CS 1 or prior programming experience recommended. 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. Overview of measure theory. 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. This course develops some of the techniques of stochastic calculus and applies them to the theory of financial asset modeling. 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. This program is designed to give students a thorough training in fundamental computational and applied mathematics and to develop their research ability in a specific application field. Multiscale finite element methods for elliptic problems with multiscale coefficients. Written report required. 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. Welcome. The mathematical concepts/tools developed will include introductions to random walks, Brownian motion, quadratic variation, and Ito-calculus.