Research Areas
Numerical
Optimization and Control
Major emphasis is on the design, analysis
and implementation of fast algorithms for nonlinear equations and optimization
problems, especially for systems arising as discrete approximations to
infinite dimensional systems (e.g., integral equations, ordinary and partial
differential equations, optimality systems). CRSC researchers are currently
investigating Newton-like and quasi-Newton iterative methods and development
of preconditioners for both smooth and nonsmooth problems. Fast algorithms
with hybrid methods are being used in a multigrid approach; emphasis is
on systems of elliptic partial differential equations in higher than one
space dimension. In addition, hybrid algorithms employing scaled gradient
projection and Lagrangian augmentation (e.g., multiplier) techniques are
being developed for optimality systems arising in constrained optimal control
problems.
Other research topics being pursued
include finite element and spectral approximation methods for feedback
control (LQR/MinMax, H-infinity) systems for partial differential equations
(e.g., nonlinear equations of elasticity/viscoelasticity, Burgers equation
and the Navier-Stokes equations for fluid flow, Maxwell's equations for
electromagnetic fields, fluid/structure interactions including structural
acoustics) and techniques for parameter estimation and inverse problems
for elliptic, parabolic and hyperbolic systems arising in numerous
applications.
CRSC researchers for these topics
include H.T. Banks, J.C. Dunn, K. Ito, C.T. Kelley, I. Lauko, G. Pinter,
R.C. Smith, and H.T. Tran.
Numerical Solution of Ordinary
and Partial Differential Equations
Among the CRSC research topics in this
area are development of algorithms for solution and control of mixed
differential/algebraic
equations and general dynamical systems, algebraic and geometric methods
for solution of nonlinear ordinary differential equations, as well as mathematical
and computational aspects of continuous realization methods for nonlinear
ordinary differential equations and algebraic eigenvalue problems. Research
on partial differential equations includes methods for elasto-plastic deformations
in granular materials, parallel pseudo-spectral and spectral and finite
element methods for flow and transport equations, and existence techniques
and stability analysis for solitary waves along with numerical methods
for related wave phenomena.
Investigators with a primary interest
in these topics include S.L. Campbell, M.T. Chu, P. Gremaud, J.F. Selgrade,
M. Shearer, C.E. Siewert, M. Singer, J.S. Scroggs, R.C Smith and
R.E. White.
Mathematical Modeling and
Analysis
Investigations in modeling and analysis
of physical, biology, engineering and scientific systems are a major focus
for many CRSC researchers. Topics of current interest include reaction-diffusion
systems arising in biology, ecology and chemical engineering, modeling
of smart material structures, fluid/structure interaction problems, acoustics
and noise suppression, population dynamics and general biological systems
and the qualitative, quantitative and statistical (least squares, Bayesian)
aspects of modeling experimental data.
A majority of CRSC researchers have
interests in one or more of these topics.
Numerical
Linear Algebra; Parallel Computing
Research on both direct methods and
iterative methods is being pursued. Direct methods research includes algorithms
for oblique projection methods to solve large sparse unstructured systems
of linear equations. Iterative methods research focuses on
aggregation/disaggregation algorithms
for solving large scale nearly uncoupled systems of equations with special
emphasis on systems in which there are weakly linked clusters of strongly
coupled states such as arise in the computation of stationary probabilities
for finite-state Markov chains; methods for large scale stochastic matrices
arising in Markov Chains with special emphasis on those arising in queueing
network computations; circulant preconditioning schemes for Toeplitz least
squares computations with applications to image restoration (e.g., recovery
in blurred and noisy images).
Additional research goals are the
design of numerical control structures for the solution of large scale
problems: systems of linear equations, eigenvalue problems, and singular
value problems. Numerical control structures are the software components
that have the greatest influence on numerical accuracy. They include convergence
tests; stopping and deflation criteria; identification of problem instances
that require special treatment; as well as strategies for partitioning
the problem into smaller, parallel subproblems. The development of numerical
control structures requires new sensitivity analyses for special classes
of perturbations and results in highly accurate algorithms.
In this CRSC research, several investigators
place special emphasis on parallel algorithms for matrix systems to be
used on state-of-the-art parallel machines such as the Intel i860 hypercube
and the Kendall Square KSR1.
CRSC researchers with interests
in these areas include R.E. Funderlic, I. Ipsen, C.D. Meyer, W.J. Stewart
and P.J. Turinsky.
Center for Research in Scientific Computation
Box 8205
North Carolina State University
Raleigh, NC 27695-8205
Voice: (919) 515-5289
Fax: (919) 515-1636
Email: crscinfo@lists.ncsu.edu