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.

计算模型中的数据同化是自然环境中预测科学的一项非常重要的任务。特别是对于天气预报和心血管流量等生物流问题，必须使用测量数据来完成模型。通常，可用数据与模拟物理现象的偏微分方程不兼容。问题是不适定的。在模型和测量误差的某种温和假设下，我们仍然可以将模型与数据一起使用来获得计算预测，通常使用吉洪诺夫正则化来控制由于不适定特征而导致的不稳定性。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，在物理偏微分方程模型的约束下，在我们的例子中，寻找一种解决方案，使测量结果之间的差异最小化，或者所谓的背景状态（如果存在）由偏微分方程表示。 3DVAR 和 4DVAR 之间的区别在于，3DVAR 数据同化时间演化没有被考虑在内。因此，它仅适用于静止问题或数据“快照”的重复同化，然后进行演化。在 4DVAR 中，数据预计分布在时空中，并且所有时空数据都用于生成同化解。-- 尽管有关于使用 3DVAR/4DVAR 数据同化主题的重要文献，但似乎没有严格的数值分析对于二维或三维问题（对于一维空间的例外情况，请参见 [JBFS15]），结合 (a) 建模误差的解的影响； (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正则化是最佳的，以及在给定一组测量数据的情况下可以合理地近似哪种数量。通常，计算方法的趋势是从低阶方法发展到高分辨率方法。我们的目标是设计和分析用于变分数据同化问题的高分辨率方法。