Variational Data Assimilation via Calculus of Variations in L^infinity

The project is based upon the following mathematical intuition. Suppose we are given a system of ordinary differential equations and some output of the form of partially observation data. We think of the pair vector field - observation operator as a model for some real-world dynamical process. The objective of variational data assimilation is to find a vector function so that approximately satisfies the model so that the output approximately matches the observations. This problem is ill-posed and over-determined, and hence in general an exact may not have any solution or might have infinitely-many solutions. The standard approach is to construct approximate solutions via the classical calculus of variations by minimising a certain error functional which measures the average quadratic error of the deviations. Throughout this project, an alternative approach will be pursued, based on the recently developed field of vector-valued calculus of variations in the space L^inifity. The idea is that rather than minimising the average error using least squares, one can minimise instead the maximum error. This is very challenging, but the huge advantage is that by minimising the maximum (i.e. in L^infinity), any "spikes" of huge error deviations are excluded, as opposed to minimisation of quadratic average.

The method of minimisation of the maximum provides much more realistic models
when compared to the case of integrals, where instead the average is minimised. By applying this approach to Weather forecasting, Oceanography and Atmospheric chemistry this may lead to much more accurate, precise and realistic predictions.

The area of vector-valued Calculus of Variations in L^infinity is still very much under development, but for the scalar case there is a successful sophisticated theory. The general concept of Calculus of Variations in L^infinity already has extensive applications to: L^infity problems relate to Lipschitz Extensions, Quasiconformal maps, Game Theory, Control Theory, Inverse Problems, Partial Differential Equation constrained optimisation & Data Assimilation.

The aims of this project are to exploit the existing concepts to make further progress in the general theory of vector-valued calculus of variations in the space L^infinity. The project will be composed of proving theoretical deductions that are illustrated with fundamental examples. We will specialise to functionals arising from variational data assimilation and make a direct comparison between the minimising the average error and minimising the maximum error. Our intentions are to develop the underlying framework of vectorial variational data assimilation for real world dynamical processes.

Efforts towards the project goals will be in an organised, focused and disciplined fashion.
There will be consistent progress monitoring order to obtain the desired goals. Initially the supervisor will meet regularly supporting the student's ideas through discussion and demonstration. As the project progresses the student will routinely present their work for critical evaluation from the supervisor. During the process an analysis of the project will be made at each stage of the development and any changes will be amended with the appropriate course of action.