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

不可压缩流变分数据同化中离散化和建模误差的定量估计

基本信息

批准号：
EP/T033126/1
负责人：
Erik Burman
金额：
$ 63.64万
依托单位：
University College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FT033126%2F1
关键词：
Quantitative estimates discretisation modelling errors

项目摘要

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.

计算模型中数据的同化是自然环境中预测科学中非常重要的任务。尤其是在天气开发和生物流动问题之类的诸如鞘内血管流的情况下，必须使用测量的数据来完成theodel。通常，可用数据与对物理苯甲瘤建模的部分微分方程不兼容。这个问题是错误的。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合并来获得计算预测，通常使用Tikhonov正则化来控制由于座位的特征而导致的不稳定性。两个重要的工具是3DVAR和4DVAR。这些是变异数据同化方法，通过andlarge，在我们的情况下，在我们案例以部分微分方程为代表的情况下，通过且在物理PDE模型的约束下，将解决方案之间的某些差异或所谓背景状态之间的差异的某些规范。 3DVAR和4DVAR之间的区别在于，在3DVAR数据同化时间中未考虑。因此，它仅适用于固定问题或重复同化数据``快照''，然后进化。在4DVAR中，预计将在时空分布，并使用所有时间时间数据来产生同化的解决方案。-尽管使用3DVAR/4DVAR的重要文献进行了有关数据同化的重要文献，但对于两个或三维问题似乎没有严格的数值分析（对于一个空间减小中，请参见[JBfs15]的模型），该材料的模型是在[Jbfs15]中的模型。（b）偏微分方程的离散化；（c）由于正则化而引起的扰动；（d）测量数据的扰动。-本项目的目的是在挑战性的不可压缩流问题的情况下，对上面的点（A-d）的近似解决方案（A-D）的影响提供了严格的严格估计。此类估计的推导将清楚地表明正规化的何种典型是最佳的，并且在一组测量数据的情况下，可以合理地近似哪种数量。通常，计算方法中的趋势是从低阶方法演变为高分辨率方法的趋势。 theambition是设计和分析这种高分辨率方法的外界数据同化问题。