**Finite Element Research**
The current research efforts of my students and myself as well as collaborators
on finite elements include: Implementation of a Numerov-Galerkin
technique for extracting higher accuracy from non-linear advective terms
in a shallow water equations model, involving enforcing of "a posteriori"
integral-invariants conservation via augmented Lagrangian constrained
minimization,and a two-stage Galerkin approach using high-order generalized
splines. The impact of these developments on long-term integration
of shallow-water equations is being studied. Current research work
centers on variable-resolution mesh with adaptive h-p mesh refinement using a
'posteriori' error estimates, applications of optimal control of distributed
parameters using adjoint models as well as porting f.e.m. codes to high performance parallel computer architectures using domain decomposition methods .
**Variational 4-D Data-Assimilation Methods **
The objective of this research effort which has been sponsored by NSF and NASA for
many years is to use the mathematical theory of optimal control of partial differential
equations for distributed parameter systems.
An application is variational data-assimilation methods which are very important to
observationally based geophysical sciences such as dynamic meteorology and
oceanography wherein remote sensing satellite data collected over a physical domain
and in a finite span of time must be incorporated into a self-consistent set of initial
conditions. The basic idea of these methods is to define a measure, called a cost
functional, of the misfit between the observations and the output of a model. The cost
functional is minimized subject to being constrained by the dynamical model by
finding control variable vector such that the corresponding solution minimizes the cost
functional and satisfies the model equations. Each minimization entails a forward
integration of the model equations and a backward integration of the adjoint model
equation. Computational cost of this method include solution of the adjoint model,
storage of intermediate model states up to the end of the assimilation period, and large
scale minimization of the gradient of the cost functional.
Innovative methods included development of a theory for efficient sensitivity analysis,
nonlinear parameter estimation and 4-D variational data assimilation for NASA and
NMC/NOAA operational 3-D models including a full physical package placing us at the
cutting edge of research in this field.
Relationship between Kaman Filter and 4-D variational data assimilation is now being investigated.
**Large-Scale Minimization**
Methods for large scale constrained and unconstrained minimization are
developed designed to mimic the behavior of variable-metric quasi-Newton
methods which have an enhanced convergence rate. In these memoryless methods
the Hessian matrix is updated but not stored. Current research work on
these methods includes tests of the bundle algorithm for minimization of
discontinuous functions and function separability to permit efficient parallelization.
Application of novel hybrid methods combining Hessian-free mehods with LBFGS
in large scale minimization in molecular dynamics and
4-D variational data assimilation as well as flow control are currently researched. |