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

计算模型中数据的同化是预测科学在自然环境中的一个非常重要的任务。特别是对于天气预报和生物流量问题，如心血管流量，必须使用测量数据来完成模型。通常可用的数据与模拟物理现象的偏微分方程不相容。这个问题是不适定的。然而，在对模型和测量误差的某些温和假设下，人们可以将模型与数据一起使用以获得计算预测，通常使用吉洪诺夫正则化来控制由于不适定特性引起的不稳定性。两个重要的工具是3DVAR和4dvar。这些是变分数据同化方法，总的来说，在物理pde模型的约束下，在偏微分方程表示的情况下，寻找最小化解决方案与测量或所谓的背景状态之间的差异的一些范数。3DVAR和4DVAR的区别在于3DVAR的数据同化不考虑时间演化。因此，它只适用于平稳问题或数据“快照”的重复同化，然后是进化。在4DVAR中，期望数据在时空中分布，并使用所有时空数据来产生同化解。—尽管有关于使用3DVAR/4DVAR进行数据同化的重要文献，但结合对(a)建模误差解决方案的影响，似乎没有对二维或三维问题（对于一个空间维度的例外，参见[JBFS15]）进行严格的数值分析；(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具有挑战性的情况下，对上述点（a-d）的近似解的影响提供严格的估计。这种估计的推导将清楚地表明哪种类型的正则化是最佳的，以及给定一组测量数据可以合理地近似哪种数量。计算方法有从低阶方法向高分辨率方法发展的典型趋势。我们的目标是设计和分析这种高分辨率的方法来解决变分数据同化问题。

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.