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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.

寻找一些物理意义上的，可量化的实体的极值是一个普遍存在的问题，在科学中非常重要。从古代，当问题可能是找到周长，包围最大面积的土地，以最复杂的应用程序，今天，一个完整的解决方案，这样的问题总是打开大视野的应用和本质上是有趣的数学家，因为它通常转化为困难，往往是技术问题。但是这些答案会影响应用和日常生活，就像上面的例子一样。特别是，在经典的变分法中，人们试图最小化定义在一类映射上的泛函，通常这样的泛函是积分并模拟一些“能量”。这些泛函的极值满足称为欧拉-拉格朗日方程的偏微分方程组。60年代初，G.阿龙松开始研究泛函，而不是定义为一个最大的。除了与几何问题相关的内在数学兴趣之外，最小化能量的“最大值”提供了更现实的模型，而不是“平均”能量的经典情况。“L-无穷大变分法”，因为这一领域是众所周知的今天，经历了巨大的发展以来。然而，直到最近的理论仅限于标量的情况下，一阶变分问题（涉及最小化的地图和其第一导数）。在2010年代早期，PI开创了高维空间中向量L无穷大问题的研究，可能涉及高阶导数。矢量的情况是大量的现实世界的应用程序的兴趣。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”，需要一个理论来进行严格的意义和有效的研究。问题是，标准的PDE方法的基础上，无论是对偶/积分的部分或最大值原则不适用。特别地，所产生的系统是非发散的，高度非线性的，退化的和不连续的系数。这种情况类似于20世纪10年代数学界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。受新发现的方程的启发，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，该系统允许不连续的解和系数。这种方法是无对偶的，依赖于那些经典上不存在的衍生物的概率解释。我们的理论是一个非线性的替代与所有现有的方法兼容的分布。在这种情况下，PI最近开始成功地研究L-无穷方程的某些情况。拟议的研究将继续研究L-无穷变分问题及其方程在适当的分析框架。我们有兴趣开发新的数学工具，以研究第一和第二阶变分问题和相关的偏微分方程系统。进一步特别关注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。在数学上，数据同化面临的问题是不完全可解决的，而是试图尽量减少一个“错误”，它描述了近似解决方案的偏差，从我们想要的精确解决方案。通过将目前使用的标准模型替换为“最大”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尖峰。