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Vectorial Calculus of Variations in L-infinity, generalised solutions for fully nonlinear PDE systems and applications to Data Assimilation

L-无穷变分的矢量微积分、全非线性偏微分方程系统的广义解及其在数据同化中的应用

基本信息

批准号：
EP/N017412/1
负责人：
Nikos Katzourakis
金额：
$ 12.62万
依托单位：
University of Reading
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2016
资助国家：
英国
起止时间：
2016 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FN017412%2F1
关键词：
Vectorial Calculus Variations infinity generalised

项目摘要

Finding the extremal values of some physically meaningful, quantifiable entity is a ubiquitous problem of great importance in science. From antiquity, when the problem might have been to find the perimeter that enclosed the largest area of land, to the most sophisticated application nowadays, a complete solution to such problems always opens large horizons for applications and is intrinsically interesting to mathematicians, as it translates usually into hard and often technical questions. But the answers impact applications and everyday life, as in the example above.In particular, in classical Calculus of Variations one seeks to minimise a functional defined on a class of maps, typically such functionals are integrals and model some "energy". The extrema of these functionals satisfy a certain system of PDE (Partial Differential Equations) known as the Euler-Lagrange equations. In the early 1960s G. Aronsson initiated the study of functionals which are instead defined as a maximum. Except for the intrinsic mathematical interest connected to geometric problems, minimising the "max" of an energy provides more realistic models as opposed to the classical case of the "average" energy. "Calculus of Variations in L-infinity", as this area is known today, has undergone huge development since. However, until recently the theory was restricted exclusively to the scalar case and to first order variational problems (involving minimisation of the map and its first derivatives). In the early 2010s the PI pioneered the study of vectorial L-infinity problems for maps valued in higher-dimensional spaces and involving perhaps higher order derivatives. The vectorial case is of interest to a large number of real-world applications. The main reason that hindered the development of the vector case was the absence of the appropriate analytic framework: the new complicated equations possess singular "solutions" and a theory is needed in order to make rigorous sense and to be studied effectively. The problem is that standard PDE approaches based on either duality/integration-by-parts or on the maximum principle do not apply. In particular, the systems arising are non-divergence, highly nonlinear, degenerate and with discontinuous coefficients. The situation is analogous to that the mathematical community faced in the 1910s when attempting to understand and make rigorous sense of the "Dirac Delta" which arose in Quantum Theory. The development of the theory of "generalised functions" allowed the understanding of fundamental physical phenomena.Motivated by the newly discovered equations, the PI very recently proposed a novel theory of "generalised solutions" for fully nonlinear PDE systems of any order which allows for discontinuous solutions and coefficients. This approach is duality-free and relies on the probabilistic interpretation of those derivatives which do not exist classically. Our theory is a nonlinear alternative to distributions compatible with all existing approaches. In this setting, the PI has recently begun studying successfully certain cases of the L-infinity equations. The proposed research will continue the study of L-infinity variational problems and of their equations in the proper analytic framework. We are interested in developing new mathematical tools in order to study 1st and 2nd order variational problems and the associated PDE systems. A further particular focus will be to apply our results to models of variational Data Assimilation in Earth sciences and in weather forecasting. Mathematically, Data Assimilation faces problems which are not exactly solvable and instead one tries to minimise an "error" which describes the deviation of approximate solutions from being the exact solution we would like to have. By replacing the standard models currently used with their "max" counterparts, we could obtain better predictions: spikes of large errors are at the outset excluded when minimising the maximum.

寻找一些具有物理意义、可量化实体的极值是科学中普遍存在的非常重要的问题。从古代，当问题可能是找到包围最大土地面积的周长时，到当今最复杂的应用，此类问题的完整解决方案总是为应用打开广阔的视野，并且本质上对数学家来说很有趣，因为它通常转化为困难且往往是技术问题。但答案会影响应用程序和日常生活，如上面的示例所示。特别是，在经典变分法中，人们试图最小化在一类映射上定义的函数，通常此类函数是积分并模拟一些“能量”。这些泛函的极值满足特定的 PDE（偏微分方程）系统，称为欧拉-拉格朗日方程。 20 世纪 60 年代初，G. Aronsson 发起了泛函研究，泛函被定义为最大值。除了与几何问题相关的内在数学兴趣之外，与“平均”能量的经典情况相反，最小化能量的“最大值”提供了更现实的模型。今天这个领域被称为“L-无穷变分微积分”，此后经历了巨大的发展。然而，直到最近，该理论还仅限于标量情况和一阶变分问题（涉及映射及其一阶导数的最小化）。在 2010 年代初，PI 率先研究了在高维空间中估值并可能涉及高阶导数的地图的矢量 L-无穷大问题。矢量情况引起了大量实际应用的兴趣。阻碍矢量情况发展的主要原因是缺乏适当的分析框架：新的复杂方程具有奇异的“解”，需要一种理论才能具有严格的意义并进行有效的研究。问题是基于对偶/分部积分或最大值原理的标准偏微分方程方法不适用。特别是，所产生的系统是非发散的、高度非线性的、简并的且具有不连续系数。这种情况类似于 1910 年代数学界在试图理解和严格理解量子理论中出现的“狄拉克三角洲”时所面临的情况。 “广义函数”理论的发展使人们能够理解基本的物理现象。在新发现的方程的推动下，PI 最近提出了一种新的“广义解”理论，适用于任何阶的完全非线性 PDE 系统，允许不连续解和系数。这种方法是无对偶性的，并且依赖于那些经典不存在的导数的概率解释。我们的理论是与所有现有方法兼容的分布的非线性替代方案。在这种情况下，PI 最近开始成功研究 L-无穷方程的某些情况。拟议的研究将继续在适当的分析框架中研究 L-无穷变分问题及其方程。我们有兴趣开发新的数学工具来研究一阶和二阶变分问题以及相关的偏微分方程系统。进一步特别关注的是将我们的结果应用于地球科学和天气预报中的变分数据同化模型。从数学上讲，数据同化面临着无法完全解决的问题，而是试图最小化“误差”，该“误差”描述了近似解与我们想要的精确解之间的偏差。通过用“最大”对应模型替换当前使用的标准模型，我们可以获得更好的预测：在最小化最大值时，一开始就排除了大误差的峰值。