# Applied & Computational Math (ACM) Courses (2021-22)

ACM 11.
Introduction to Computational Science and Engineering.
6 units (2-2-2):
third term.
Prerequisites: Ma 1ab, Co-requisite Ma 1c. CS 1 or prior programming experience recommended.
This course is intended to serve as a practical introduction to the methods of computational science and engineering for students in all majors. The goal is to provide students exposure to and hands-on experience with commonly-used computational methods in science and engineering, with theoretical considerations confined to a level appropriate for first-year undergraduate students. Topics covered include computational simulation by discretization in space and time, numerical solution of linear and nonlinear equations, optimization, uncertainty quantification, and function approximation via interpolation and regression. Emphasis is on understanding trade-offs between computational effort and accuracy, and on developing working knowledge of how these tools can be used to solve a wide range of problems arising in applied math, science, and engineering. Assignments and in-class activities use MATLAB. No prior experience with MATLAB expected.
Instructor: Qian.

ACM 80 abc.
Undergraduate Thesis.
9 units:
first, second, third terms.
Prerequisites: instructor's permission, which should be obtained sufficiently early to allow time for planning the research.
Individual research project, carried out under the supervision of a member of the ACM faculty (or other faculty as approved by the ACM undergraduate option representative). Projects must include significant design effort. Written report required. Open only to upper class students. Not offered on a pass/fail basis.
Instructor: Staff.

ACM 81 abc.
Undergraduate Projects in Applied and Computational Mathematics.
Units are assigned in accordance with work accomplished:
first, second, third terms.
Prerequisites: Consent of supervisor is required before registering.
Supervised research or development in ACM by undergraduates. The topic must be approved by the project supervisor, and a formal final report must be presented on completion of research. Graded pass/fail.
Instructor: Staff.

ACM 95/100 ab.
Introductory Methods of Applied Mathematics for the Physical Sciences.
12 units (4-0-8):
second, third terms.
Prerequisites: Ma 1 abc, Ma 2 or equivalents.
Complex analysis: analyticity, Laurent series, contour integration, residue calculus. Ordinary differential equations: linear initial value problems, linear boundary value problems, Sturm-Liouville theory, eigenfunction expansions, transform methods, Green's functions. Linear partial differential equations: heat equation, separation of variables, Laplace equation, transform methods, wave equation, method of characteristics, Green's functions.
Instructors: Zuev, Meiron.

ACM/IDS 101 ab.
Methods of Applied Mathematics.
12 units (4-4-4):
first, second terms.
Prerequisites: Math 2/102 and ACM 95 ab or equivalent.
First term: Brief review of the elements of complex analysis and complex-variable methods. Asymptotic expansions, asymptotic evaluation of integrals (Laplace method, stationary phase, steepest descents), perturbation methods, WKB theory, boundary-layer theory, matched asymptotic expansions with first-order and high-order matching. Method of multiple scales for oscillatory systems. Second term: Applied spectral theory, special functions, generalized eigenfunction expansions, convergence theory. Gibbs and Runge phenomena and their resolution. Chebyshev expansion and Fourier Continuation methods. Review of numerical stability theory for time evolution. Fast spectrally-accurate PDE solvers for linear and nonlinear Partial Differential Equations in general domains. Integral-equations methods for linear partial differential equation in general domains (Laplace, Helmholtz, Schroedinger, Maxwell, Stokes). 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.
Instructor: Bruno.

ACM/IDS 104.
Applied Linear Algebra.
9 units (3-1-5):
first term.
Prerequisites: Ma 1 abc, some familiarity with MATLAB, e.g. ACM 11 is desired.
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. The focus is on applications. Matrix factorizations play a central role. 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.
Instructor: Zuev.

ACM 105.
Applied Real and Functional Analysis.
9 units (3-0-6):
second term.
Prerequisites: Ma 2, Ma 108a, ACM/IDS 104 or equivalent.
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. The aim of this course is to provide a working knowledge of functional analysis with an eye especially for aspects that lend themselves to 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. Other core concepts include: normed linear spaces and behavior of linear maps, completeness, Banach spaces, Hilbert spaces, Lp spaces, duality of normed spaces and dual operators, dense subspaces and approximations, hyperplanes, compactness, weak and weak* convergence. More advanced topics include: spectral theory, compact operators, theory of distributions (generalized functions), Fourier analysis, calculus of variations, Sobolev spaces with applications to PDEs, weak solvability theory of boundary value problems.

ACM/EE 106 ab.
Introductory Methods of Computational Mathematics.
12 units (3-0-9):
first, second terms.
Prerequisites: For ACM/EE 106a, Ma 1 abc, Ma 2, Ma 3, ACM 11; for ACM/EE 106b, ACM 95/100 ab 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 linear algebra parts cover 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. The approximation theory includes data fitting; interpolation using Fourier transform, orthogonal polynomials and splines; least square method, and numerical quadrature. The ODE parts include initial and boundary value problems. The PDE parts include finite difference and finite element for elliptic/parabolic/hyperbolic equations. Study of numerical PDE will include stability analysis. Programming is a significant part of the course.
Instructor: Hou.

CMS/ACM/IDS 107.
Linear Analysis with Applications.
12 units (3-0-9):
first term.
Prerequisites: ACM/IDS 104 or equivalent, Ma 1 b or equivalent.
Covers the basic algebraic, geometric, and topological properties of normed linear spaces, inner-product spaces, and linear maps. Emphasis is placed both on rigorous mathematical development and on applications to control theory, data analysis and partial differential equations.
Instructor: Stuart.

ACM 109.
Mathematical Modelling.
9 units (3-0-6):
third term.
Prerequisites ACM 95/100 ab or equivalent. 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. 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. Applications will include quantum, atomistic and continuum modeling of materials; epidemics, reacting-diffusing systems; crowd modeling and opinion formation. Mathematical tools will include ordinary, partial and stochastic differential equations, as well as Markov chains and other stochastic processes. Not offered 2021-2022.
Instructor: Stuart.

Ec/ACM/CS 112.
Bayesian Statistics.
9 units (3-0-6):
second term.
Prerequisites: Ma 3, ACM/EE/IDS 116 or equivalent.
This course provides an introduction to Bayesian Statistics and its applications to data analysis in various fields. Topics include: discrete models, regression models, hierarchical models, model comparison, and MCMC methods. The course combines an introduction to basic theory with a hands-on emphasis on learning how to use these methods in practice so that students can apply them in their own work. Previous familiarity with frequentist statistics is useful but not required.
Instructor: Rangel.

CMS/ACM/IDS 113.
Mathematical Optimization.
12 units (3-0-9):
first term.
Prerequisites: ACM 11 and ACM 104, or instructor's permission.
This class studies mathematical optimization from the viewpoint of convexity. 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.
Instructor: Chandrasekaran.

ACM/EE/IDS 116.
Introduction to Probability Models.
9 units (3-1-5):
first term.
Prerequisites: Ma 3 or EE 55, some familiarity with MATLAB, e.g. ACM 11, is desired.
This course introduces students to the fundamental concepts, methods, and models of applied probability and stochastic processes. 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. The main goal is to equip science and engineering students with necessary probabilistic tools they can use in future studies and research. Topics covered include sample spaces, events, probabilities of events, discrete and continuous random variables, expectation, variance, correlation, joint and marginal distributions, independence, moment generating functions, law of large numbers, central limit theorem, random vectors and matrices, random graphs, Gaussian vectors, branching, Poisson, and counting processes, general discrete- and continuous-timed processes, auto- and cross-correlation functions, stationary processes, power spectral densities.
Instructor: Zuev.

CMS/ACM 117.
Probability Theory and Stochastic Processes.
12 units (3-0-9):
first term.
Prerequisites: ACM/IDS 104, ACM/EE/IDS 116 or instructor's permission.
This course offers a rigorous introduction to probability and stochastic processes. Emphasis is placed on the interaction between inequalities and limit theorems, as well as contemporary applications in computing and mathematical sciences. 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.
Instructor: Tropp.

ACM 118.
Stochastic Processes and Regression.
12 units (3-0-9):
second term.
Prerequisites: CMS/ACM/IDS 107 or equivalent, CMS 117 or equivalent, or permission of the instructor.
Stochastic processes: Branching processes, Poisson point processes, Determinantal point processes, Dirichlet processes and Gaussian processes (including the Brownian motion). Regression: Gaussian vectors, spaces, conditioning, processes, fields and measures will be presented with an emphasis on linear regression. Kernel and variational methods in numerical approximation, signal processing and learning will also be covered through their connections with Gaussian process regression.
Instructor: Owhadi.

CMS/ACM/EE 122.
Mathematical Optimization.
12 units (3-0-9):
first term.
Prerequisites: ACM 11 and ACM 104, or instructor's permission.
This class studies mathematical optimization from the viewpoint of convexity. 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.
Instructor: Chandrasekaran.

AM/ACM 127.
Calculus of Variations.
9 units (3-0-6):
third term.
Prerequisites: ACM 95/100.
First and second variations; Euler-Lagrange equation; Hamiltonian formalism; action principle; Hamilton-Jacobi theory; stability; local and global minima; direct methods and relaxation; isoperimetric inequality; asymptotic methods and gamma convergence; selected applications to mechanics, materials science, control theory and numerical methods.
Instructor: Bhattacharya.

Ma/ACM/IDS 140 ab.
Probability.
9 units (3-0-6):
second, third terms.
Prerequisites: For 140 a, Ma 108 b is strongly recommended.
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. Topics in statistics.
Instructors: Vigneaux, Tamuz.

Ma/ACM 142 ab.
Ordinary and Partial Differential Equations.
9 units (3-0-6):
second, third terms.
Prerequisites: 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.
Instructors: Angelopoulos, Isett.

ACM/IDS 154.
Inverse Problems and Data Assimilation.
9 units (3-0-6):
first term.
Prerequisites: Basic differential equations, linear algebra, probability and statistics: ACM/IDS 104, ACM/EE 106 ab, ACM/EE/IDS 116, IDS/ACM/CS 157 or equivalent.
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. The purpose of the course is to describe the mathematical and algorithmic principles of this area. The topic lies at the intersection of fields including inverse problems, differential equations, machine learning and uncertainty quantification. Applications will be drawn from the physical, biological and data sciences.
Instructor: Stuart.

IDS/ACM/CS 157.
Statistical Inference.
9 units (3-2-4):
third term.
Prerequisites: ACM/EE/IDS 116, Ma 3.
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. This is an introductory course on statistical inference. 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. Topics covered include summarizing data, fundamentals of survey sampling, statistical functionals, jackknife, bootstrap, methods of moments and maximum likelihood, hypothesis testing, p-values, the Wald, Student's t-, permutation, and likelihood ratio tests, multiple testing, scatterplots, simple linear regression, ordinary least squares, interval estimation, prediction, graphical residual analysis.
Instructor: Zuev.

IDS/ACM/CS 158.
Fundamentals of Statistical Learning.
9 units (3-3-3):
third term.
Prerequisites: Ma 3 or ACM/EE/IDS 116, IDS/ACM/CS 157.
The main goal of the course is to provide an introduction to the central concepts and core methods of statistical learning, an interdisciplinary field at the intersection of statistics, machine learning, information and data sciences. The course focuses on the mathematics and statistics of methods developed for learning from data. 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. 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 course can be viewed as a statistical analog of CMS/CS/CNS/EE/IDS 155. Topics covered include supervised and unsupervised learning, regression and classification problems, linear regression, subset selection, shrinkage methods, logistic regression, linear discriminant analysis, resampling techniques, tree-based methods, support-vector machines, and clustering methods. Not offered 2021-2022.

ACM/EE/IDS 170.
Mathematics of Signal Processing.
12 units (3-0-9):
third term.
Prerequisites: ACM/IDS 104, CMS/ACM/IDS 113, and ACM/EE/IDS 116; or instructor's permission.
This course covers classical and modern approaches to problems in signal processing. Problems may include denoising, deconvolution, spectral estimation, direction-of-arrival estimation, array processing, independent component analysis, system identification, filter design, and transform coding. Methods rely heavily on linear algebra, convex optimization, and stochastic modeling. In particular, the class will cover techniques based on least-squares and on sparse modeling. Throughout the course, a computational viewpoint will be emphasized. Not offered 2021-2022.
Instructor: Hassibi.

CS/ACM 177 a.
Discrete Differential Geometry: Theory and Applications.
9 units (3-3-3):
second term.
Working knowledge of multivariate calculus and linear algebra as well as fluency in some implementation language is expected. Subject matter covered: differential geometry of curves and surfaces, classical exterior calculus, discrete exterior calculus, sampling and reconstruction of differential forms, low dimensional algebraic and computational topology, Morse theory, Noether's theorem, Helmholtz-Hodge decomposition, structure preserving time integration, connections and their curvatures on complex line bundles. Applications include elastica and rods, surface parameterization, conformal surface deformations, computation of geodesics, tangent vector field design, connections, discrete thin shells, fluids, electromagnetism, and elasticity.
Instructors: SchrÃ¶der, Desbrun.

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

ACM 201.
Partial Differential Equations.
12 units (4-0-8):
first term.
Prerequisites: ACM 95/100 ab, ACM/IDS 101 ab, ACM 11 or equivalent.
This course offers an introduction to the theory of Partial Differential Equations (PDEs) commonly encountered across mathematics, engineering and science. 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. 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 SchrÃ¶dinger equations and Stoke's flow. 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. 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. Not offered 2021-2022.

ACM/IDS 204.
Topics in Linear Algebra and Convexity.
9 units (3-0-6):
second term.
Prerequisites: CMS/ACM 107 or instructor's permission.
Topic varies by year. 2021-2022: Matrix Analysis. This class develops some foundational results about matrices that have wide application in computational mathematics, statistics, engineering, and related fields. Tools from analysis (real, complex, functional, convex) play a key role. Topics may include multilinear algebra, majorization and doubly stochastic matrices, variational principles for eigenvalues, eigenvalue localization regions, unitarily invariant norms, operator monotonicity and convexity, convexity of matrix functions, matrix means, matrix inequalities, perturbation theory, tensor products and linear matrix equations, Hadamard products, nonnegative matrices, positive and completely positive linear maps, positive-definite functions.
Instructor: Tropp.

ACM 206.
Topics in Computational Mathematics.
9 units (6-0-3):
second term.
Prerequisites: ACM 106ab.
Covers modern numerical methods. Possible topics include randomized matrix computations, numerical optimization, Monte Carlo methods. Not offered 2021-2022.
Instructor: Tropp.

ACM 210.
Numerical Methods for PDEs.
9 units (3-0-6):
third terms.
Prerequisites: ACM 11, 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 Not offered 2021-2022.

ACM/IDS 213.
Topics in Optimization.
9 units (3-0-6):
third term.
Prerequisites: ACM/IDS 104, CMS/ACM/IDS 113.
Material varies year-to-year. Example topics include discrete optimization, convex and computational algebraic geometry, numerical methods for large-scale optimization, and convex geometry.
Instructor: Qian.

ACM/IDS 216.
Markov Chains, Discrete Stochastic Processes and Applications.
9 units (3-0-6):
second term.
Prerequisites: ACM/EE/IDS 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 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.

ACM 217.
Advanced Topics in Stochastic Analysis.
9 units (3-0-6):
second term.
Prerequisites: ACM/CMS/EE/IDS 117.
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. 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. Not offered 2021-2022.
Instructor: Tropp.

Ae/ACM/ME 232 ab.
Computational Fluid Dynamics.
9 units (3-0-6):
first, second terms.
Prerequisites: Ae/APh/CE/ME 101 abc or equivalent; ACM 100 abc or equivalent.
Development and analysis of algorithms used in the solution of fluid mechanics problems. 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. Survey of finite difference, finite element, finite volume and spectral approximations for the numerical solution of the incompressible and compressible Euler and Navier-Stokes equations, including shock-capturing methods.
Instructors: Bae, Meiron.

ACM 256.
Special Topics in Applied Mathematics.
9 units (3-0-6):
first term.
Prerequisites: ACM/IDS 101 or equivalent.
Introduction to finite element methods. Development of the most commonly used method-continuous, piecewise-linear finite elements on triangles for scalar elliptic partial differential equations; practical (a posteriori) error estimation techniques and adaptive improvement; formulation of finite element methods, with a few concrete examples of important equations that are not adequately treated by continuous, piecewise-linear finite elements, together with choices of finite elements that are appropriate for those problems. Homogenization and optimal design. Topics covered include periodic homogenization, G- and H-convergence, Gamma-convergence, G-closure problems, bounds on effective properties, and optimal composites. Not offered 2021-2022.

ACM 257.
Special Topics in Financial Mathematics.
9 units (3-0-6):
third term.
Prerequisites: ACM 95/100 or instructor's permission.
A basic knowledge of probability and statistics as well as transform methods for solving PDEs is assumed. This course develops some of the techniques of stochastic calculus and applies them to the theory of financial asset modeling. The mathematical concepts/tools developed will include introductions to random walks, Brownian motion, quadratic variation, and Ito-calculus. Connections to PDEs will be made by Feynman-Kac theorems. Concepts of risk-neutral pricing and martingale representation are introduced in the pricing of options. Topics covered will be selected from standard options, exotic options, American derivative securities, term-structure models, and jump processes. Not offered 2021-2022.

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

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

### Please Note

The online version of the Caltech Catalog is provided as a convenience; however, the printed version is the only authoritative source of information about course offerings, option requirements, graduation requirements, and other important topics.