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Ionel Navon's Math Page

Ionel Michael Navon
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Ph.D., University of the Witwatersrand, Johannesburg, Republic of South Africa, 1979

Detailed description of research

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.