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

计算模式中数据的同化是自然环境预测科学中一个非常重要的问题。特别是对于天气预报和生物流动问题，如心血管流动，必须使用测量数据来完成模型。通常情况下，可用的数据与模拟物理现象的偏微分方程不兼容.这个问题是不适定的。在某些温和的假设下，对模型和测量误差，人们仍然可以使用模型与数据一起获得计算预测，通常使用Tikhonov正则化来控制由于不适定特性而导致的不稳定性。两个重要的工具是3DVAR和4DVAR。这些都是变分资料同化方法，大体上，寻找一个解决方案，最小化测量的解决方案之间的差异，或所谓的背景状态的情况下，它存在，在物理偏微分方程模型的约束下，在我们的情况下表示的偏微分方程。3DVAR和4DVAR的区别在于3DVAR资料同化没有考虑时间演化。因此，它只适用于固定的问题或重复同化的数据“快照”，然后演变。在4DVAR中，预计数据将分布在时空中，所有时空数据都用于产生同化解。尽管关于利用3DVAR/4DVAR进行资料同化的重要文献，但似乎没有对二维或三维问题进行严格的数值分析（关于一维空间中的例外情况，见[JBFS 15]）结合对以下各项的解的影响：（a）建模误差;（B）偏微分方程的离散化;（c）正则化引起的扰动;（d）测量数据的扰动。本项目的目的是在不可压缩流动问题的挑战性情况下，对上述点（a-d）的近似解的影响提供精确的严格估计。这种估计的推导将清楚地表明哪种类型的调节是最佳的，以及在给定一组测量数据的情况下，什么样的量可以合理地近似。典型的趋势是在计算方法是发展从低阶方法的高分辨率方法。本文的目标是设计和分析这种高分辨率的变分同化方法。

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.