Option Info | Course Websites | Current Class Schedule

ACM 11. Introduction to Matlab and Mathematica. 3 units (1-1-1); first term. Prerequisites: Ma 1 abc, Ma 2 ab. Matlab: basic syntax; linear algebra computation; visualization; control flow; numerical analysis, including curve fitting, interpolation, differentiation, integration, optimization and solving nonlinear equations; script and function with m-file; strings; file input/output; arrays and structures; optimizing performance by vectorization; fast fourier transform and ode solvers.  Mathematica: basic syntax; numerical calculations; algebraic computations, including transforming and simplifying algebraic expressions; symbolic mathematics, including calculus, inequalities, power series, differential equations, limits and integral transforms; graphics and sound; special functions; linear algebra; numerical analysis; functions and programs; lists; input and output in notebooks.  Instructor: Chung

ACM 95/100 abc. Introductory Methods of Applied Mathematics. 12 units (4-0-8); first, second, third terms. Prerequisites: Ma 1 abc, Ma 2 ab, or equivalents. Introduction to functions of a complex variable; linear ordinary differential equations; special functions; eigenfunction expansions; integral transforms; linear partial differential equations and boundary value problems. Instructors: Pierce, Meiron.
This course is listed on Caltech's Moodle site. To access the Moodle course information, you will need your IMSS login and an enrollment key from a course TA.

Schematic of a synthetic DNA walker for molecular transport.

ACM 101 abc. Methods of Applied Mathematics I. 9 units (3-0-6); first, second, third terms. Prerequisite: ACM 95/100 abc. 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. Instructor: Bruno, Fok.

ACM 104. Linear Algebra. 9 units (3-0-6); second term. Prerequisite: ACM 100 abc or instructor's permission. 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: Fok.

ACM 105. Applied Real and Functional Analysis. 9 units (3-0-6); third term. Prerequisite: ACM 100 abc or instructor's permission. 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, LP spaces, convolution, Fourier transform and Sobolev spaces with application to PDEs, the convolution theorem, Friedrich’s mollifiers, dense subspaces and approximation, normed vector spaces, completeness, Banach spaces, linear operators, the Baire, Banach-Steinhaus, open mapping and closed graph theorems with applications to differential and integral equations, dual spaces, weak convergence and weak solvability theory of boundary value problems, spectral theory of compact operators. Instructor: Chung.

ACM 106 abc. Introductory Methods of Computational Mathematics. 9 units (3-0-6); first, second, third terms. Prerequisites: Ma 1 abc, Ma 2ab, ACM 95/100 abc or equivalent. 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. Instructors: Schröder, Chung.

ACM 113. Introduction to Optimization. 9 units (3-0-6); first term. Prerequisite: ACM 100 abc or instructor's permission. 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. Instructor: Candes.

ACM/CS 114 ab. Parallel Algorithms for Scientific Applications. 9 units (3-0-6); second, third terms. Prerequisites: ACM 106 or equivalent. 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 2007-08.

ACM/EE 116. Introduction to Stochastic Processes and Modeling. 9 units (3-0-6); first term. Prerequisite: Ma2ab or instructor's permission. 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, finance. Instructor: Owhadi.

ACM/ESE 118. Methods in Applied Statistics and Data Analysis. 9 units (3-0-6); second term. Prerequisite: Ma 2 or another introductory course in probability and statistics. 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: Tropp.

ACM 126 ab. Wavelets and Modern Signal Processing. 9 units (3-0-6); second, third terms. Prerequisites: ACM 104, ACM 105 or undergraduate equivalent, or instructor’s permission. 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. Not offered 2007-08.

Ma/ACM 142 abc. Ordinary and Partial Differential Equations. 9 units (3-0-6); first, second, third terms. Prerequisite: Ma 108. Ma 109 is desirable. The mathematical theory of ordinary and partial differential equations, including a discussion of elliptic regularity, maximal principles, solubility of equations. The method of characteristics. Not offered 2007-08.

Ma/ACM 144 ab. Probability. 9 units (3-0-6); first, second terms. Overview of measure theory. Random walks and the Strong law of large numbers via the theory of martingales and Markov chains. Characteristic functions and the central limit theorem. Poisson process and Brownian motion. Not offered 2007-08.

ACM 151 ab. Asymptotic and Perturbation Methods. 9 units (3-0-6); first, second terms. Prerequisite: ACM 101 abc or equivalent, may be taken concurrently with instructor's permission. Approximation methods for formulating and solving applied problems, with examples taken from various fields of science. Applications to various linear and nonlinear ordinary and partial differential equations. Singular and multiscale perturbation techniques, boundary-layer theory, coordinate straining, a method of averaging. Bifurcation theory, amplitude equations, and nonlinear stability. Not offered 2007-08.

ACM 190. Reading and Independent Study. Units by arrangement. Graded pass/fail only. Instructor: Staff.

ACM 201 ab. Partial Differential Equations. 12 units (4-0-8); first, second terms. Prerequisite: ACM 101 abc or instructor's permission. 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 hyper-bolic 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. Not offered 2007-08.

ACM 210 ab. Numerical Methods for PDEs. 9 units (3-0-6); first, second terms. Prerequisites: ACM 106, or instructor's permission. 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. Instructor: Hou.

ACM 216. Markov Chains, Discrete Stochastic Processes and Applications. 9 units (3-0-6); second term. Prerequisite: ACM/EE 116 or equivalent. 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 their 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.

ACM 217/EE 164. Advanced Topics in Stochastic Analysis: Stochastic Control. 9 units (3-0-6); third term. Prerequisite: ACM 216 or equivalent. This course introduces the fundamentals of stochastic differential equations and stochastic control in continuous time. The aim of the course is to develop the basic mathematical tools, and to demonstrate these through concrete applications in engineering, finance and physics. Topics include Brownian motion, Ito calculus and stochastic differential equations, Girsanov and martingale representation theorems, stochastic stability, linear and nonlinear filtering, optimal control, optimal stopping, impulse control, and changepoint detection. Instructor: Hassibi.

Ae/ACM 232 abc. Computational Fluid Dynamics. 9 units (3-0-6); first, second, third terms. Prerequisites: Ae/APh/CE/ME 101 abc or equivalent; ACM 100 abc or AM 125 abc, or equivalent; ACM 104, ACM 105, or equivalent. Introduction to the use of numerical methods in the solution of fluid mechanics problems. First term: review of basic numerical techniques: interpolation, integration, application for systems of ordinary differential equations, stability and accuracy. Treatment of partial differential equations in one space variable. Nonlinear convective-diffusive and convective-dispersive phenomena. Treatment of discontinuous solutions. Second term: survey of finite difference, finite element, and spectral approximations for the solution of the incompressible Navier-Stokes equations in two and three dimensions. Numerical study of problems of hydrodynamic stability, transition, and turbulence. Third term: methods for the numerical solution of the compressible Euler and Navier-Stokes equations in one, two, and three dimensions. Finite-difference and finite-volume methods. Methods based on solution of the Riemann problem. Flux-splitting. Shock-capturing methods and related stability problems. Implicit artificial viscosity for the Euler equations. Total variation diminishing approximations. Not offered 2007-08.

ACM 256 ab. Special Topics in Applied Mathematics. 9 units (3-0-6); second, third terms. Prerequisites: ACM 101 or equivalent. (a) Topics on Finite Element Methods. We provide an introduction to finite element methods in two parts. In the first part of the quarter, we give a careful development of the most commonly used method -- continuous, piecewise-linear finite elements on triangles for scalar elliptic partial differential equations. More attention will be given to practical (a posteriori) error estimation techniques and adaptive improvement than one generally finds in such a course -- many of the results, or at least the proofs, will be quite recent. To compensate for this emphasis on error estimation, not much attention will be given to the practical aspects of discretizing PDEs in a finite element code (data structures, quadrature rules, etc.) or on solution techniques for the resulting linear systems (multigrid methods, conjugate gradient methods, etc.). The remainder of the course will be devoted a more abstract formulation of finite element methods, with a few concrete examples of important equations which are not adequately treated by continuous, piecewise-linear finite elements, together with choices of finite elements which are appropriate for those problems. Instructor: Ovall   (b) Homogenization and Optimal Design. The course provides an introduction to the modern theory of homogenization, with emphasis on applications to the optimal design of composite materials in the settings of conductivity and linear elasticity. Topics covered include: periodic homogenization, G- and H-convergence, Gamma-convergence, G-closure problems, bounds on effective properties, and optimal composites. Instructor: Albin.

ACM 270. Advanced Topics in Applied and Computational Mathematics. Hours and units by arrangement. Advanced topics in applied and computational mathematics that will vary according to student and instructor interest. May be repeated for credit.

ACM 290 abc. Applied and Computational Mathematics Colloquium. 1 unit (1-0-0); first, second, third terms. 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. Instructor: Staff.

ACM 300. Research in Applied and Computational Mathematics. Units by arrangement. Instructor: Staff.

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