Research Areas

## Numerical Optimization and Control,

## Numerical Solution of Ordinary and Partial Differential Equations,

## Mathematical Modeling and Analysis, and

## Numerical Linear Algebra; Parallel Computing.

## 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.

Numerical Optimization and ControlOther 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 onaggregation/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.