Faculty

Ralph Byers
Weizhang Huang
Erik Van Vleck
Hongguo Xu

Current and Former Graduate Students

Eric Barth
Steve Bond
Karen Braman
John Emenyu
Jason Frank
Jason Hibbeler
Mary Horning
Xiang Ji
Andrew Kappen
Uma Kolandai
Xusheng Lang
Sokun Men
Stephane Petti
David Ratner
Pat Roberts
Andrew Singh
Tong Wang
Chou Yang
Yan Zhu

Numerical Analysis Course Offerings

Math 581 - Numerical Methods

Math 780 - Numerical Linear Algebra

Math/EECS 781 - Numerical Analysis I

Math 782 - Numerical Analysis II

Math 783 - Applied Numerical Methods for Partial Differential Equations

Math 796 - Multigrid Methods

Math 796 - Advanced Numerical methods for Partial Differential Equations

Math 881 - Advanced Numerical Linear Algebra

Math 882 - Advanced Numerical Differential Equations

Math 996 - Parallel and Serial Numerical Methods and Analysis for Large Scale Problems

Math 996 - Computational Dynamical Systems

Faculty Research

Faculty

Ralph Byers
Homepage

Ph.D. 1983: Cornell University

Research Interests:
Dr. Byers' area of interest is in numerical linear algebra including problems of large scale or problems with special structure. Much (but not all) of his work is in the area of
computational control. He is interested in developing practical, accurate, and efficient numerical methods and in understanding the effects of errors and uncertainties on
computations and on mathematical models.
Data are typically corrupted by errors. There are rounding errors, measurement errors, modeling errors, truncation errors, equipment wears, and noise. Condition numbers
quantify the sensitivity of a computational problem to data perturbations. In addition to conditioning and accuracy, often it is essential that some qualitative property such as
stability or controllability be preserved. A continuing interest of Dr. Byers is in finding computable measures of the robustness of such qualitative properties to perturbations in data.
Another line of research is the development of algorithms for numerical linear algebra problems with special structure. This line of research has both theoretical and practical
motivation. These algorithms often rest on elegant mathematical foundations and are less expensive and less sensitive to rounding errors than conventional methods. The state
of the art of these problems has progressed from special cases of practical interest to an understanding of a large class of structured problems.

Dr. Byers' Recent Publications

Significant Publications:

1. Karen Braman, RB, Roy Mathias, The Multi-Shift QR
Algorithm Part I: Maintaining Well Focused Shifts and
Level 3 Performance,SIAM Journal on Matrix Analysis and Applications,
Volume 23, pages 929--947, 2002.

2. Karen Braman, RB, Roy Mathias, The Multi-Shift QR Algorithm
Part II: Aggressive Early Deflation, SIAM Journal on
Matrix Analysis and Applications, Volume 23, pages 948--973, 2002.

3. RB, C. He, and V. Mehrmann, Where is the Nearest
Non-regular Pencil? Linear Algebra and Its Applications,
285(1998), 81--105.

4. A. Bunse-Gerstner, RB, and V. Mehrmann, A Chart of
Numerical Methods for Structured Eigenvalue Problems, SIAM
Journal of Matrix Analysis and Its Applications, 13, 419--453,
1992.

5. RB, A Bisection Method for Measuring the Distance of
a Stable Matrix to the Unstable Matrices, SIAM Journal on
Scientific and Statistical Computing, 9, 875-881, 1988.

6. RB, A Hamiltonian QR Algorithm, SIAM Journal of Scientific
and Statistical Computing, 7, 212--229, 1986.

Weizhang Huang
Homepage

Ph.D. 1989: Chinese Academy of Sciences, Beijing

Research Interests: The emphasis of Dr. Huang's research is in the areas of
numerical analysis, scientific computing, and geometric
integration. Most of his research contributions have been
made in analysis, development, and application of numerical
methods for solving partial differential equations. Topics
include adaptive mesh generation and movement, spectral
methods, geometric integration of Hamiltonian systems, and
their applications to engineering and other real world
problems.

5 Recent Publications:

1. W. Huang and W. Sun,
Variational mesh adaptation II: Error estimates and monitor
functions, J. Comput. Phys. (In press)

2. W. Cao, R. Carretero-Gonzalez, W. Huang, and R. D. Russell,
Variational mesh adaptation methods for axisymmetrical problems,
SIAM J. Numer. Anal. (In press)

3. W. Cao, W. Huang, and R. D. Russell,
Approaches for generating moving adaptive meshes: location
versus velocity, Appl. Numer. Math. (In press)

4. W. Huang, X. Zhan, and L. Zheng,
Adaptive moving mesh methods for simulating one-dimensional
groundwater problems with sharp moving fronts,
Int. J. Numer. Meth. Eng. 54 (2002), 1579 -- 1603.

5. W. Cao, W. Huang, and R. Russell,
A moving mesh method based on the geometric conservation law,
SIAM J. Sci. Comput. 24 (2002), 118 -- 142.

5 Other Publications:

1. W. Huang,
Variational mesh adaptation: isotropy and equidistribution
J. Comput. Phys. 174 (2001), 903 -- 924.

2. W. Huang,
Practical aspects of formulation and solution of moving
mesh partial differential equations,
J. Comput. Phys. 171 (2001), 753 -- 775.

3. W. Cao, W. Huang, and R.D. Russell,
An error indicator monitor function for an r-adaptive
finite-element method,
J. Comput. Phys. 170 (2001), 871 -- 892.

4. W. Huang and R. D. Russell,
Adaptive mesh movement -- the MMPDE approach and its applications,
J. Comput. Appl. Math. 128 (2001), 383 -- 398.

5. W. Cao, W. Huang, and R.D. Russell,
Comparison of two-dimensional r-adaptive finite element methods
using various error indicators,
Math. Comput. Simulation 56 (2001), 127 -- 143.

Dr. Huang's Publications

Erik S. Van Vleck
Homepage

Ph.D. 1991: Georgia Institute of Technology

Research Interests: The focus of Dr. Van Vleck's research is on issues in computational differential equations.
His interest is in
the analysis and computation of spectra for nonautonomous linear homogeneous
differential equations, analysis and computation of lattice differential
equations, in particular traveling wave solutions, and in bringing together our
study of spectra as a tool for the existence and stability of traveling waves.
This research has as a central theme the combination of
dynamical systems and numerical analysis ideas, and our interests are
in the development and analysis of efficient, accurate numerical techniques
that are useful for the computation and analysis of dynamical systems.

5 Recent Publications:

1. C.E. Elmer and E.S. Van Vleck,
``Spatially Discrete FitzHugh-Nagumo Equations,''
(2005) SIAM J. Appld. Math. 65 pp. 1153--1174.

2. L. Dieci and E.S. Van Vleck,
``On the Error in Computing Lyapunov Exponents by QR Methods,''
(2005) Numer. Math. 101 pp. 619--642.

3. M.D. Bateman and E.S. Van Vleck,
``Traveling Wave Solutions to a Coupled System of Spatially Discrete Nagumo Equations,''
(2006) SIAM J. Appld. Math. 66 pp. 945--976.

4. W. Liu and E.S. Van Vleck,
``Turning Points and Traveling Waves in FitzHugh-Nagumo Type Equations,''
(2006) J. Diff. Eqn. 225 pp. 381--410.

5. L. Dieci and E.S. Van Vleck,
``Perturbation Theory for the Approximation of Lyapunov Exponents by QR Methods,''
(2006) J. Dyn. Diff. Eqn. 18 pp. 815--840.

5 Other Publications:

1. L. Dieci, R.D. Russell and E.S. Van Vleck,
``Unitary Integrators and Applications to Continuous Orthonormalization
Techniques,'' (1994) SIAM J. Numer. Anal. 31 pp. 261-281.

2. J.W. Cahn, S.N. Chow and E.S. Van Vleck,
``Spatially Discrete Nonlinear Diffusion Equations,''
(1995) Rocky Mount. J. Math. 25 pp. 87-118.

3. J.W. Cahn, J. Mallet-Paret and E.S. Van Vleck,
``Traveling Wave Solutions for Systems of ODEs on a Two-Dimensional
Spatial Lattice,'' (1999) SIAM J. Appld. Math. 59 pp. 455--493.

4. C.E. Elmer and E.S. Van Vleck,
``A Variant of Newton's Method for
the Computation of Traveling Waves
of Bistable Differential-Difference Equations,''
(2002) J. Dyn. Diff. Eqn. 14 pp. 493--517.

5. L. Dieci and E.S. Van Vleck,
``Lyapunov Spectral Intervals: Theory and Computation,''
(2003) SIAM J. Numer. Anal. 40 pp. 516--542.

Dr. Van Vleck's Publications and
Preprints.

Hongguo Xu
Homepage

Ph.D. 1991: Fudan University, Shanghai, China.

Research Interests: Dr. Xu's research interests include numerical linear algebra, scientific
computing, matrix theory, perturbation analysis, and their application in
science and engineering. Currently he is studying the numerical problems
about matrices with certain special structures. These matrices arise
in many applications. The matrix structures usually reflect the physical
principle of the underlying application, and they are also inherited
by the solutions. His goal is to develop the numerical methods that make use
of matrix structures and that produce numerical solutions with the required
structures, so that the numerical results are computed accurately and
quickly, and are meaningful to the underlying application. He is also
developing the matrix theory for helping to understand the numerical
behavior of structured matrices.

Dr. Xu's Publications