Quantitative estimates of discretisation and modelling errors in variational data assimilation for incompressible flows

点击翻译按钮获取中文摘要

The assimilation of data in computational models is a very importanttask in predictive science in the natural environment. In particularfor weather forcasting and biological flow problems such ascardiovascular flows, measured data must be used to complete themodel. More often than not the available data is not compatible withthe partial differential equations modelling the physicalphenomenon. The problem is ill-posed. Under certain mild assumption onthe model and measurement errors one can nevertheless use the modeltogether with the data to obtain computational predictions, typicallyusing Tikhonov regularisation to control instabilities due to theill-posed character. Two important tools for this are 3DVAR and4DVAR. These are variational data assimilation methods that, by andlarge, look for a solution minimising some norm of the differencebetween the solution to the measurements, or to a so called backgroundstate in case it exists, under the constraint of the physical pde model, in our case represented by a partial differential equation. The difference between 3DVAR and 4DVAR is that in 3DVAR data assimilation time evolution is not accounted for. It is therefore applicable only to stationary problem or to repeated assimilation of data ``snapshots'' followed by evolution. In 4DVAR data is expected to be distributed in space time and all space time data is used to produce the assimilated solution.-- In spite of the important literature on the topic of data assimilation using 3DVAR/4DVAR there appears to be no rigorous numerical analysis for two or three dimensional problems (for an exception in one space dimension see [JBFS15]) combining the effect on the solution of (a) modelling errors; (b) discretisation of the partial differential equations; (c) perturbation due to regularisation; (d) perturbations of the measured data.-- The aim of the present project is to provide sharp rigorous estimates for the effect on the approximate solution of points (a-d) above in the challenging case of incompressible flow problems. The derivation of such estimates will give a clear indication on whattype of regularisations are optimal and also what kind of quantities can reasonably be approximated given a set of measured data. Typically the tendency in computational methodsis to evolve from low order approaches to high resolution methods. Theambition is to design and analyse such high resolution methods forvariational data assimilation problems.