COMPUTATIONAL MATHEMATICS

• Prerequisites
• Examinations
• Course Descriptions
• Faculty

Computational Mathematics is concerned with all questions related to the computer solution of problems from the natural, social and engineering sciences. The research areas of the computational mathematics faculty include numerical modeling of ordinary and partial differential equations, optimization, approximation, data mining using linear algebra concepts, inverse problems, image processing, and all algebraic aspects of computing. There are many opportunities for interdisciplinary research in fields including the many areas of engineering, geology, biology, atmospheric sciences, and computer science. The group has had, and continues to have, research funding from a number of reseach agencies over recent years, including the NSF mathematics programs, NSF interdisciplinary grants, NIH, the Arizona Department of Health Services, and the Sloan Foundation. Students entering the area of computational mathematics will be expected to become active participants in these funded research programs and can anticipate some portion of their study at ASU to be covered by research positions.

Prerequisites

In addition to the general departmental prerequisites, proficiency in a high-level computer language is required.

Course Work

All computational mathematics students must take courses in numerical numerical linear algebra and numerical solution of differential equations. This is met by preferably completing the MAT 520-530 sequence of courses upon arrival to the PhD program. Students who receive PhD qualifying passes in MAT 423-425 are not required to take MAT 520-530, but should ensure that they understand the material in those courses for their continued coursework.

Ph.D. students must take five additional 500-level computational mathematics courses and actively participate in the computational and applied mathematics seminar during at least three semesters. At least two seminar presentations are expected during this time. Additional course work is selected in consultation with a supervisory committee from the department. Students with an interdisciplinary area will have at least one committee member from outside the department to ensure breadth and depth in the required courses.

Examinations

All computational mathematics students must take the qualifying examination in numerical analysis (MAT 423-425). Incoming students with strong backgrounds in numerical analysis are instead encouraged to take MAT 520 and MAT 530. A grade of A(A-) indicates a doctoral level of competency. See below for more details. The second qualifying exam necessary for Ph. D. students will be approved by the student's advisory committee. Comprehensive exams must include at least one area from computational mathematics, such as spectral methods, optimization, numerical solution of partial differential equations, iterative methods and advanced techniques for ordinary differential equations.

Qualifying Examination in Computational Mathematics

The Computational Math group of the department has modified, with approval of the departmental graduate committee (Feb 26,2007), the mechanisms by which students can obtaining PhD qualifying status in the area of Computational Mathematics:

1. Students with a strong background in mathematics are encouraged to take graduate entry level courses, MAT520 and MAT 530. A grade of A(A-) in each class, indicates doctoral level competency in the two areas of computational mathematics which have been assessed by qualifying exams in Numerical analysis I ( MAT423) and Numerical Analysis II ( MAT 425), respectively. Students who achieve a grade of B will be assumed to have satisfied qualifier competency at the Masters level for these two areas of study.
2. Students with lesser background will still be able to take the senior level courses MAT423 and MAT425 and sit for the qualifying exam, as per standard departmental procedures.
3. Courses MAT 423 and MAT520 will be offered every Fall semester, and at the same time, so as to facilitate movement of students between the courses, dependent on their initial skill levels. Likewise, MAT425 and MAT 530 will be taught each Spring semester at the same time.
4. The noted modification includes a complete revision of course descriptions for courses in computational mathematics, and major syllabus changes for the entry level courses MAT 520 and MAT530. (see below)

Course Descriptions

Senior Level for Math Students

M MAT 425 Numerical Analysis II. (3) spring
Analysis of and algorithms for numerical interpolation, integration, and differentiation. Numerical solution of ordinary and partial differential equations, introductory level. Applications.
Prerequisites: both MAT 274 (or 275) and fluency in computer programming or only instructor approval. MAT 371 recommended. General Studies: CS

Graduate Level Courses :
Introductory Computational Techniques:

M MAT 530 Comp. Math.: Differential Eqns. (3)
Full TITLE: Computational Mathematics: Differential Equations spring In depth analysis of and algorithms for numerical interpolation, integration, and differentiation. Numerical solution of ordinary and partial differential equations. Applications.
Prerequisites: both MAT 371 and MAT 275 or instructor approval.

Graduate Level Courses :
Advanced Computational Techniques - could be used for students' Comprehensives in Computational Mathematics

M MAT 531 Adv. Comp. Math.: Diff. Systems. (3)
Full Title: Advanced Computational Mathematics: Differential Systems fall Runge-Kutta, linear multistep methods, consistency, stability,convergence, error estimation,adaptive stepsize control, stiff and differential-algebraic problems, general linear methods, applications in science and engineering.
Prerequisites: MAT 530, or instructor approval.

Graduate Level Courses :
Topics Based

M MAT 524 Adv. Scientific Computation (3)
Full Title: Advanced Scientific Computation selected semesters Selected topics from current research in scientific computation, including computational differential equations, optimization, signal processing, image processing. Programming and project based. Lecture, lab. Fee.
Prerequisites: both MAT 520 and 530 or MAT 423 and 425 or only instructor approval.

M MAT 533 Comp. Ell. Para. PDEs. (3)
Full Title: Computational Elliptic and Parabolic Partial Differential Equations. fall Parabolic and elliptic equations, finite difference, finite element methods, stability, consistency, convergence, practical aspects, applications, software.
Prerequisites: MAT 530 or 425 or instructor approval.

M MAT 534 Comp. Hyp. Cons. Laws (3)
Full Title: Computational Hyperbolic Conservation Laws spring Numerical solutions of hyperbolic PDEs, finite difference methods, well-posedness, stability, consistency, convergence, adaptive grids; Maxwell's equations, elastic wave propagation; Navier-Stokes.
Prerequisites: MAT 530 or 425 or instructor approval.

M MAT 535 Numerical Spectral Methods (3) selected semesters
Fourier and orthogonal polynomial spectral methods, Galerkin, collocation, Tau-methods, global approximation properties, stability; convergence; solutions for linear, nonlinear systems.
Prerequisites: MAT 530 or 425 or instructor approval.

Carl Gardner - Professor, Ph.D., M.I.T., 1981
Anne Gelb - Professor, PhD, Division of Applied Mathematics, Brown University, 1996.
Zdzislaw Jackiewicz - Professor, Ph.D., Gdansk (Poland), 1980
Hans Mittelmann - Professor, Ph.D., Darmstadt (Germany), 1973
Rosemary Renaut - Professor, Ph.D., Cambridge (U.K.), 1985
Christian Ringhofer - Professor, Ph.D., Vienna (Austria), 1981
Bruno Welfert - Associate Professor, Ph.D., California - San Diego, 1990