The Master of Science in Scientific Computing

The Program

The Departments of Mathematics and Computer Science at the Courant Institute of Mathematical Sciences, New York University, offer a new Master's degree in scientific computing. The program, which began in the fall 1995 semester, is designed to provide a broad yet rigorous training in areas related to scientific computing, including modern computing tools and methods, and numerical and mathematical analysis as arises in various applications. Further emphasis is placed upon data visualization, graphical user interfaces and UNIX tools, as well as exploring application areas.

The program accommodates both full-time and part-time students, with most courses scheduled to meet in the evening. It is a self-contained terminal master's program, providing a complete set of employment skills in a field where the need is greater than the supply.

Scientific Computing: Overview

Scientific computing is an indispensable part of almost all scientific investigation and technological development at universities, government laboratories, and within the private sector. Typically a scientific computing team consists of several people trained in some branch of mathematics, science, or engineering. What is often lacking is expertise in modern computing tools such as visualization, UNIX tools, and workstation environments. This program is designed to satisfy these needs, without omitting the basic training in numerical analysis and computer science, or the development of skills in mathematical modeling. Graduates of this program will qualify for jobs in research and development, where their skills and experience will complement those of more traditionally trained Ph.D.s. The program is also open to students who will go on to pursue doctoral studies in either department.

The M.S. program in Scientific Computing focuses on the mathematics and computer science related to advanced computer modeling. While the material is in mathematics and computer science, the program is similar in structure to terminal master's programs in engineering, where classroom training is combined with practical experience. The coursework spans the range from mathematical background and fundamental algorithms to such modern practical topics as data visualization and software tools. Students will be encouraged to explore specific application areas such as financial modeling, statistics, fluid mechanics and finite elements, or biomedical modeling. The program culminates in a master's project which serves to integrate the classroom material.

Admission Requirements

Students accepted into the program should have taken at least three semesters of calculus, as well as linear algebra (or its equivalent, e.g. econometrics, or through work experience). Experience with programming in a high-level language, not necessarily through coursework, is also required. Advanced calculus, differential equations, and coursework in data structures are desirable. A strong background in linear algebra is of particular importance, with mastery of the following subject materials:

Gaussian elimination, existence of solutions to matrix equations, matrix rank, determinant, and inverse.
Vector spaces, linear (in)dependence, bases, vector norms, inner products, orthogonality.
Eigenvalues and eigenvectors, diagonalization and similarity transformations, special matrices: normal, symmetric, Hermitian, orthogonal, and unitary matrices.

Students with promise but an inadequate mathematics background will be asked to take the course MATH-GA 1002 Multivariable Calculus and/or an undergraduate linear algebra course before being considered for admission to the program. (Note that the course MATH-GA 1002 Multivariable Calculus will not carry credit toward completion of this master's degree.)

The deadlines for application to the program is June 1 for the fall term and November 1 for the spring term. The program admits students on both a full-time and a part-time basis. Applications are accepted on-line by the Graduate School of Arts and Science. Please visit the Graduate School admissions web page at http://gsas.nyu.edu/page/grad.admissionsapplication.

For any questions contact us at:

Office of Admissions and Student Affairs

Department of Mathematics

Courant Institute of Mathematical Sciences

251 Mercer Street

New York, NY 10012-1185

Tel. (212) 998-3238

Fax (212) 995-4121

E-Mail: admissions@math.nyu.edu

E-Mail: arnon@cims.nyu.edu

Web page: http://www.math.nyu.edu

Degree Requirements

A candidate for a master's degree in scientific computing must fulfill two degree requirements:

30 points of course credit (10 courses) comprised of:

4 core courses (12 points) in mathematics;
4 core courses (12 points) in computer science;
2 elective courses (6 points); and,

6 points of credit toward a computational master's thesis.

Core Courses

The following are the four core courses in mathematics:

MATH-GA 2010 Numerical Methods I (fall term)
MATH-GA 2020 Numerical Methods II (spring term)
MATH-GA 2701 Methods of Applied Mathematics (fall term)
MATH-GA 2702 Fluid Dynamics (spring term)

The following are the four core courses in computer science:

CSCI-GA 2110 Programming Languages (fall, spring and summer terms)
CSCI-GA 1170 Fundamental Algorithms (fall, spring and summer terms)
CSCI-GA 3033 Open Source Tools (fall)
CSCI-GA 2270 Computer Graphics (varies from year to year)

With approval of the Director of the Program, students with a sufficient background may be able to waive certain core courses. Should any core course not be available for any reason, a re-arrangement of the curriculum can be discussed with the Director of the Program.

Elective Courses

The two elective courses may be taken in mathematics or computer science, and are subject to approval by the student's faculty advisor or the Director of the Program. A list of elective courses in mathematics and in computer science follows below. Note that students are strongly encouraged to take some courses in specific application areas, and will thus be allowed - with departmental approval - to take courses not on the list of electives, possibly in other NYU departments such as the Department of Chemistry, or in the Stern School of Business.

The following are suggested elective courses in mathematics:

MATH-GA 1410, 1420 Introduction to Mathematical Analysis I, II
MATH-GA 2011, 2012 Selected Topics in Numerical Analysis
MATH-GA 2030 Advanced Numerical Analysis: Computational Fluid Dynamics
MATH-GA 2031 Advanced Numerical Analysis: Nonlinear Optimization
MATH-GA 2032 Advanced Numerical Analysis: Initial Value Problems for DEs
MATH-GA 2040 Advanced Numerical Analysis: Finite Element Methods
MATH-GA 2045 Computational Methods for Finance
MATH-GA 2110, 2120 Linear Algebra I, II (only with permission of the Program Director)
MATH-GA 2450, 2460 Complex Variables I, II
MATH-GA 2470 Ordinary Differential Equations
MATH-GA 2490 Partial Differential Equations
MATH-GA 2781 Mathematical Modeling*
MATH-GA 2791 Derivative Securities
MATH-GA 2792 Continuous Time Finance
MATH-GA 2887 Magnetofluid Dynamics*
MATH-GA 2901 Basic Probability

* These courses are offered less frequently.

The following are suggested elective courses in computer science:

CSCI-GA 1171 Advanced Fundamental Algorithms*
CSCI-GA 2243 High Performance Computer Architecture
CSCI-GA 2250 Design of Operating Systems
CSCI-GA 2274 Advanced Computer Graphics*
CSCI-GA 2340 Elements of Discrete Mathematics
CSCI-GA 2631 Distributed Computing
CSCI-GA 2750 Nonlinear Optimization
CSCI-GA 2280 User Interfaces*
CSCI-GA 3033 Data Visualization*
CSCI-GA 3130 Honors Compilers and Computer Languages

* These courses are offered less frequently.

In addition, six points of coursework, designated as MATH-GA 3771, 3772, 3773, 3774 Independent Study, will be awarded for a required computational master's thesis project.

The Departments of Mathematics and Computer Science publish annual course description brochures, outlining the specific courses to be given that year. Students should consult these course offerings to determine the availability of desired courses.

The Computational Master Thesis Project

The master's thesis project would normally be undertaken in the final year of study. It would be completed under the supervision of a faculty member and the project would have to be approved by the Director of the Program.

The master's thesis need not be as original or substantial as a Ph.D. thesis, but it should include several elements:

it should involve a substantial scientific computation;
it should use modern techniques of software development;
it should make good use of computer graphics or visualization facilities.

Some recent thesis titles have been:

“Investigation of Computational and Visualization Methods for the Incompressible Navier-Stokes Equation,” Langston, Matthew Harper

“Branching and Capping of Femur End,” Lord, Dan

“Spiral Waves in a Reaction-Diffusion System,” Mao, Yiwen

“Axisymmetric Acoustic Scattering by Interpolation,” Meyer, Perrin

“Novel Sampling Algorithms for Biomolecular Simulations,” Minary, Peter

“Region Explorer: Software for Region of Interest Analysis of FMRI Data,” Pasley, Brian

“Solar System Simulation with 3-D Visualization,” Tumolo, Greg

Computing Facilities

The Courant Institute has a network of workstations run by systems administrators available for graduate training and coursework. All graduate students are given accounts for the duration of their studies. NYU also runs a high- performance computing center with both shared memory and distributed memory computers.

Sample Program

The sample program below is appropriate for full-time students entering the program in the fall term. Because of the structure of the program, it may be difficult for students who enter in the spring term to follow a program of full-time study and finish the program in three semesters. Students should discuss such problems with the Director of the Program.

Year I, Fall Term

Numerical Methods I

Fundamental Algorithms I

Methods of Applied Mathematics

Programming Languages

Year I, Spring Term

Numerical Methods II

Computer Graphics

Fluid Dynamics

Open Source Tools

Year II, Fall Term

Elective

Independent Study (Thesis Project)

Faculty

Many members of the faculty of both the Department of Mathematics and the Department of Computer Science have research interests bearing on scientific computing. The list includes:

Marsha J. Berger. B.S. 1974, SUNY/Binghamton; M.S. 1978, Ph.D. 1982, Stanford. Research interests: computational fluid dynamics, adaptive mesh refinement, parallel computing.

Yu Chen. B.S. 1982, Tsinghua; M.S. 1988, Ph.D. 1991, Yale. Research Interests: numerical scattering theory, ill-posed problems, scientific computing.

Aleksandar Donev. B.S. 2001, Michigan State; Ph.D. 1991, Yale. Research interests: multi-scale methods, fluctuating hydrodynamics, coarse-grained particle methods, jamming and packing.

Davi Geiger. B.S. 1980, Pontifica (Brazil); Ph.D. 1990, MIT. Research interests: computer vision, information theory, medical imaging and neuroscience.

Jonathan B. Goodman. B.S. 1977, MIT; Ph.D. 1982, Stanford. Research interests: numerical analysis, fluid dynamics, computational physics, partial differential equations.

Leslie Greengard. B.A. 1979, Wesleyan; M.S. 1987, Yale School of Medicine; Ph.D. 1987, Yale. Research interests: scientific computing, fast algorithms, potential theory.

Peter D. Lax. (Emeritus) B.A. 1947, Ph.D. 1949, NYU. Research interests: fluid dynamics, partial differential equations, computation.

Andrew Majda. B.S. 1970, M.S. 1971, Ph.D. 1973, Stanford. Research interests: modern applied mathematics, atmosphere/ocean science, turbulence, statistical physics.

David W. McLaughlin. B.S. 1966, Creighton; M.S. 1969, Ph.D. 1971, Indiana. Research interests: applied mathematics, nonlinear wave equations; neural science.

Bhubaneswar Mishra. B.S. 1980, India Institute of Technology, Kharagpur; M.S. 1982, Ph.D. 1985, Carnegie-Mellon. Research interests: robotics, mathematical and theoretical computer science.

Michael L. Overton. B.S. 1974, British Columbia; M.S. 1977, Ph.D. 1979, Stanford. Research interests: numerical linear algebra, optimization, linear and semidefinite programming.

Kenneth Perlin. B.A. 1979, Harvard; M.S. 1984, Ph.D. 1986, NYU. Research interests: computer graphics, simulation, computer-human interfaces, multimedia.

Charles S. Peskin. B.A. 1968, Harvard; Ph.D. 1972, Yeshiva. Research interests: physiology, fluid dynamics, numerical methods.

Aaditya V. Rangan. B.A. 1999, Dartmouth; Ph.D. 2003, Berkeley. Research interests: large-scale scientific modeling of physical, biological and neurobiological phenomena.

Weiqing Ren. B.S. 1994, Nanjing; Ph.D. 2002 NYU. Research Interests: applied mathematics, scientific computing, multiscale modeling of fluids.

Tamar Schlick. B.S. 1982, Wayne State; M.S. 1984, Ph.D. 1987, NYU. Research interests: mathematical biology, numerical analysis, computational chemistry.

Michael J. Shelley. B.S. 1981, Colorado; M.S. 1984, Ph.D. 1985, Arizona. Research interests: scientific computation, fluid dynamics, neuroscience.

Eero Simoncelli. B.A. 1984, Harvard; M.S. 1988, Ph.D. 1993, MIT. Research interests: image processing, computational neuroscience, computer vision.

Esteban Tabak. Bach. 1988, Buenos Aires, Ph.D. 1992, MIT. Research interests: fluid dynamics, conservation laws.

Mark Tygert. B.A. 2001, Princeton; Ph.D. 2004, Yale. Research interersts: computational science and engineering, particularly numerical analysis.

Olof B. Widlund. C.E. 1960, Tekn. L. 1964, Technology Institute, Stockholm; Ph.D. 1966, Uppsala. Research interests: numerical analysis, partial differential equations, parallel computing.

Margaret H. Wright. B.S. 1964, M.S. 1965, Ph.D. 1976, Stanford. Research interests: mathematical optimization, numerical methods, nonlinear programming.

Denis Zorin. B.S. 1991, Moscow Institute of Physics and Technology; M.S. 1993, Ohio State; Ph.D. 1997, Caltech. Research interests: computer graphics, geometric modeling, subdivision surfaces, multiresolution surface representations, perceptually based methods for computer graphics.

Academic Standards

To continue registering for courses in the Department of Mathematics, a student must be in good academic standing, fulfilling the following requirements:

Students must maintain an average of B or better in their first 12 credits. Students who fail to achieve this will not be permitted to continue in the program.
Students cannot obtain a master's degree unless they have maintained an overall average of at least B. Students in danger of failing to meet this requirement will receive a warning letter from the Department.
Students will be allowed no more than four no-credit grades, withdrawals, or unresolved incomplete grades during their academic tenure, and no more than two such grades in the first six courses for which they have registered.

Credit will be given for up to two core courses taken elsewhere, subject to the normal GSAS restrictions on transfer of credit and the approval of the Program Coordinator. At least 30 course points must be taken at New York University.

For further administrative information please contact:

            Tamar Arnon
            arnon@cims.nyu.edu
            Tel. 212 998-3257

For further academic information please contact:

            Professor Mark Tygert, Program Director
            tygert@cims.nyu.edu

Revised Fall 2010