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

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