| source Caltech (X) |
level |
department Applied and Computational Mathematics (X) |
Analytical methods for the formulation and solution of initial and boundary value problems for ordinary and partial differential equations. Techniques include the use of complex variables, generalized eigenfunction expansions, transform methods and applied spectral theory, linear operators, nonlinear methods, asymptotic and approximate methods, Weiner-Hopf, and integral equations. Instructors: Hou, Fok.
Score: 12.126015 Details | Listing | Web page
Vector spaces, bases, Gram-Schmidt, linear maps and matrices, linear functionals, the transposed matrix and duality, kernel, image and rank, invertibility, triangularization, determinants and multilinear forms, powers of matrices and difference equations, the exponential of a matrix and ODEs, eigenvalues, Gershgorin’s disc theorem, eigenspaces, SVD, polar decomposition. Nilpotent-semisimple decomposition and the Jordan normal form. Symmetric hermitian and positive definite matrices, diagonalizability, unitary matrices, bilinear forms. Hilbert spaces, projections, Riesz theorem, Fourier series, spectrum, self-adjoint operators. Instructor: Hansen.
Score: 12.126015 Details | Listing | Web page
The Lebesgue integral on the line, general measure and integration theory, convergence theorems, Fubini, Tonelli, the Lebesgue integral in n dimensions and the transformation theorem, L
Score: 12.126015 Details | Listing | Web page
Unconstrained optimization: optimality conditions, line search and trust region methods, properties of steepest descent, conjugate gradient, Newton and quasi-Newton methods. Linear programming: optimality conditions, the simplex method, primal-dual interior-point methods. Nonlinear programming: Lagrange multipliers, optimality conditions, logarithmic barrier methods, quadratic penalty methods, augmented Lagrangian methods. Integer programming: cutting plane methods, branch and bound methods, complexity theory, NP complete problems. Not offered 2008–09.
Score: 12.126015 Details | Listing | Web page
Introduction to parallel program design for numerically intensive scientific applications. First term: parallel programming methods; distributed-memory model with message passing using the message passing interface; shared-memory model with threads using open MP; object-based models using a problem-solving environment with parallel objects. Parallel numerical algorithms: numerical methods for linear algebraic systems, such as LU decomposition, QR method, Lanczos and Arnoldi methods, pseudospectra, CG solvers. Second term: parallel implementations of numerical methods for PDEs, including finite-difference, finite-element, and shock-capturing schemes; particle-based simulations of complex systems. Implementation of adaptive mesh refinement. Grid-based computing, load balancing strategies. Not offered 2008–09.
Score: 12.126015 Details | Listing | Web page
Introduction to fundamental ideas and techniques of stochastic analysis and modeling. Random variables, expectation and conditional expectation, joint distributions, covariance, moment generating function, central limit theorem, weak and strong laws of large numbers, discrete time stochastic processes, stationarity, power spectral densities and the Wiener-Khinchine theorem, Gaussian processes, Poisson processes, Brownian motion. The course develops applications in selected areas such as signal processing (Wiener filter), information theory, genetics, queuing and waiting line theory, and finance. Instructor: Owhadi.
Score: 12.126015 Details | Listing | Web page
The aim is to cover the interactions existing between applied mathematics, namely applied and computational harmonic analysis, approximation theory, etc., and statistics and signal processing. The Fourier transform: the continuous Fourier transform, the discrete Fourier transform, FFT, time-frequency analysis, short-time Fourier transform. The wavelet transform: the continuous wavelet transform, discrete wavelet transforms, and orthogonal bases of wavelets. Statistical estimation. Denoising by linear filtering. Inverse problems. Approximation theory: linear/nonlinear approximation and applications to data compression. Wavelets and algorithms: fast wavelet transforms, wavelet packets, cosine packets, best orthogonal bases matching pursuit, basis pursuit. Data compression. Nonlinear estimation. Topics in stochastic processes. Topics in numerical analysis, e.g., multigrids and fast solvers. Instructor: Tropp.
Score: 12.126015 Details | Listing | Web page
For course description, see Mathematics.
Score: 12.126015 Details | Listing | Web page
. For course description, see Mathematics.
Score: 12.126015 Details | Listing | Web page
Graded pass/fail only.
Score: 12.126015 Details | Listing | Web page
Fully nonlinear first-order PDEs, shocks, eikonal equations. Classification of second-order linear equations: elliptic, parabolic, hyperbolic. Well-posed problems. Laplace and Poisson equations; Gauss’s theorem, Green’s function. Existence and uniqueness theorems (Sobolev spaces methods, Perron’s method). Applications to irrotational flow, elasticity, electrostatics, etc. Heat equation, existence and uniqueness theorems, Green’s function, special solutions. Wave equation and vibrations. Huygens’ principle. Spherical means. Retarded potentials. Water waves and various approximations, dispersion relations. Symmetric hyperbolic systems and waves. Maxwell equations, Helmholtz equation, Schrödinger equation. Radiation conditions. Gas dynamics. Riemann invariants. Shocks, Riemann problem. Local existence theory for general symmetric hyperbolic systems. Global existence and uniqueness for the inviscid Burgers’ equation. Integral equations, single- and double-layer potentials. Fredholm theory. Navier-Stokes equations. Stokes flow, Reynolds number. Potential flow; connection with complex variables. Blasius formulae. Boundary layers. Subsonic, supersonic, and transonic flow. Instructor: Bruno.
Score: 12.126015 Details | Listing | Web page
Finite difference and finite volume methods for hyperbolic problems. 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. Interface problems, level set methods for multiphase flows, boundary integral methods, fast summation algorithms, stability issues. Spectral methods: Fourier spectral methods on infinite and periodic domains. Chebyshev spectral methods on finite domains. Spectral element methods and h-p refinement. Multiscale finite element methods for elliptic problems with multiscale coefficients. Not offered 2008–09.
Score: 12.126015 Details | Listing | Web page
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. Instructor: Owhadi.
Score: 12.126015 Details | Listing | Web page
For course description, see Aeronautics.
Score: 12.126015 Details | Listing | Web page
A seminar course in applied and computational mathematics. Weekly lectures on current developments are presented by staff members, graduate students, and visiting scientists and engineers. Graded pass/fail only.
Score: 12.126015 Details | Listing | Web page
First term: complex analysis: analyticity, Laurent series, singularities, branch cuts, contour integration, residue calculus. Second term: ordinary differential equations. Linear initial value problems: Laplace transforms, series solutions. Linear boundary value problems: eigenvalue problems, Fourier series, Sturm-Liouville theory, eigenfunction expansions, the Fredholm alternative, Green’s functions, nonlinear equations, stability theory, Lyapunov functions, numerical methods. Third term: linear partial differential equations: heat equation separation of variables, Fourier transforms, special functions, Green’s functions, wave equation, Laplace equation, method of characteristics, numerical methods. Instructors: Pierce, Meiron.
Score: 12.126015 Details | Listing | Web page
Analytical methods for the formulation and solution of initial and boundary value problems for ordinary and partial differential equations. Techniques include the use of complex variables, generalized eigenfunction expansions, transform methods and applied spectral theory, linear operators, nonlinear methods, asymptotic and approximate methods, Weiner-Hopf, and integral equations. Instructors: Hou, Fok.
Score: 12.126015 Details | Listing | Web page
Vector spaces, bases, Gram-Schmidt, linear maps and matrices, linear functionals, the transposed matrix and duality, kernel, image and rank, invertibility, triangularization, determinants and multilinear forms, powers of matrices and difference equations, the exponential of a matrix and ODEs, eigenvalues, Gershgorin’s disc theorem, eigenspaces, SVD, polar decomposition. Nilpotent-semisimple decomposition and the Jordan normal form. Symmetric hermitian and positive definite matrices, diagonalizability, unitary matrices, bilinear forms. Hilbert spaces, projections, Riesz theorem, Fourier series, spectrum, self-adjoint operators. Instructor: Hansen.
Score: 12.126015 Details | Listing | Web page
The Lebesgue integral on the line, general measure and integration theory, convergence theorems, Fubini, Tonelli, the Lebesgue integral in n dimensions and the transformation theorem, L
Score: 12.126015 Details | Listing | Web page
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 course covers methods such as direct and iterative solution of large linear systems; eigenvalue and vector computations; function minimization; nonlinear algebraic solvers; preconditioning; time-frequency transforms (Fourier, wavelet, etc.); root finding; data fitting; interpolation and approximation of functions; numerical quadrature; numerical integration of systems of ODEs (initial and boundary value problems); finite difference, element, and volume methods for PDEs; level set methods. Programming is a significant part of the course. Instructor: Yan.
Score: 12.126015 Details | Listing | Web page
Unconstrained optimization: optimality conditions, line search and trust region methods, properties of steepest descent, conjugate gradient, Newton and quasi-Newton methods. Linear programming: optimality conditions, the simplex method, primal-dual interior-point methods. Nonlinear programming: Lagrange multipliers, optimality conditions, logarithmic barrier methods, quadratic penalty methods, augmented Lagrangian methods. Integer programming: cutting plane methods, branch and bound methods, complexity theory, NP complete problems. Not offered 2008–09.
Score: 12.126015 Details | Listing | Web page
Introduction to parallel program design for numerically intensive scientific applications. First term: parallel programming methods; distributed-memory model with message passing using the message passing interface; shared-memory model with threads using open MP; object-based models using a problem-solving environment with parallel objects. Parallel numerical algorithms: numerical methods for linear algebraic systems, such as LU decomposition, QR method, Lanczos and Arnoldi methods, pseudospectra, CG solvers. Second term: parallel implementations of numerical methods for PDEs, including finite-difference, finite-element, and shock-capturing schemes; particle-based simulations of complex systems. Implementation of adaptive mesh refinement. Grid-based computing, load balancing strategies. Not offered 2008–09.
Score: 12.126015 Details | Listing | Web page
Introduction to fundamental ideas and techniques of stochastic analysis and modeling. Random variables, expectation and conditional expectation, joint distributions, covariance, moment generating function, central limit theorem, weak and strong laws of large numbers, discrete time stochastic processes, stationarity, power spectral densities and the Wiener-Khinchine theorem, Gaussian processes, Poisson processes, Brownian motion. The course develops applications in selected areas such as signal processing (Wiener filter), information theory, genetics, queuing and waiting line theory, and finance. Instructor: Owhadi.
Score: 12.126015 Details | Listing | Web page
Introduction to fundamental ideas and techniques of statistical modeling, with an emphasis on conceptual understanding and on the analysis of real data sets. Multiple regression: estimation, inference, model selection, model checking. Regularization of ill-posed and rank-deficient regression problems. Cross-validation. Principal component analysis. Discriminant analysis. Resampling methods and the bootstrap. Instructor: Candes.
Score: 12.126015 Details | Listing | Web page
The aim is to cover the interactions existing between applied mathematics, namely applied and computational harmonic analysis, approximation theory, etc., and statistics and signal processing. The Fourier transform: the continuous Fourier transform, the discrete Fourier transform, FFT, time-frequency analysis, short-time Fourier transform. The wavelet transform: the continuous wavelet transform, discrete wavelet transforms, and orthogonal bases of wavelets. Statistical estimation. Denoising by linear filtering. Inverse problems. Approximation theory: linear/nonlinear approximation and applications to data compression. Wavelets and algorithms: fast wavelet transforms, wavelet packets, cosine packets, best orthogonal bases matching pursuit, basis pursuit. Data compression. Nonlinear estimation. Topics in stochastic processes. Topics in numerical analysis, e.g., multigrids and fast solvers. Instructor: Tropp.
Score: 12.126015 Details | Listing | Web page