Variational Data Assimilation via Calculus of Variations in L^infinity

该项目基于以下数学直觉。假设我们有一个常微分方程组和一些部分观测数据形式的输出。我们把对矢量场观测算子看作是现实世界动力学过程的一个模型。变分数据同化的目的是找到一个近似满足模型的向量函数，使输出近似匹配观测值。这个问题是不适定的和过度确定的，因此一般来说，一个精确的问题可能没有任何解，或者可能有无限多个解。标准的方法是通过最小化某个误差泛函来构造近似解，该泛函测量偏差的平均二次误差。在整个项目中，将采用另一种方法，基于最近发展的空间L^无穷大变分的向量值演算领域。其思想是，与其使用最小二乘最小化平均误差，不如最小化最大误差。这是非常具有挑战性的，但巨大的优势在于，通过最小化最大值（即在L^∞中），任何巨大误差偏差的“尖峰”都被排除在外，而不是最小化二次平均值。与积分的情况相比，最大值的最小化方法提供了更现实的模型，而积分的平均值是最小化的。通过将这种方法应用于天气预报、海洋学和大气化学，这可能会导致更准确、精确和现实的预测。L无穷向量值变分学的研究领域还处于发展阶段，但对于标量情况已经有了一个成功的成熟理论。L^∞变分法的一般概念已经广泛应用于：与Lipschitz扩展、拟共形映射、博弈论、控制论、反问题、偏微分方程约束优化和数据同化相关的L^∞问题。本课题的目的是利用现有的概念，在L^∞空间中向量值变分的一般理论中取得进一步的进展。该项目将由证明理论推论组成，并以基本实例加以说明。我们将专门研究由变分数据同化引起的函数，并在最小化平均误差和最小化最大误差之间进行直接比较。我们的目的是为现实世界的动态过程开发向量变分数据同化的基本框架。实现项目目标的努力将以有组织、有重点和有纪律的方式进行。将有一致的进度监控，以实现预期的目标。最初，导师将定期会面，通过讨论和演示来支持学生的想法。随着项目的进展，学生们将定期展示他们的作品，以获得导师的批判性评价。在此过程中，将在开发的每个阶段对项目进行分析，并根据适当的行动方针对任何更改进行修改。

The project is based upon the following mathematical intuition. Suppose we are given a system of ordinary differential equations and some output of the form of partially observation data. We think of the pair vector field - observation operator as a model for some real-world dynamical process. The objective of variational data assimilation is to find a vector function so that approximately satisfies the model so that the output approximately matches the observations. This problem is ill-posed and over-determined, and hence in general an exact may not have any solution or might have infinitely-many solutions. The standard approach is to construct approximate solutions via the classical calculus of variations by minimising a certain error functional which measures the average quadratic error of the deviations. Throughout this project, an alternative approach will be pursued, based on the recently developed field of vector-valued calculus of variations in the space L^inifity. The idea is that rather than minimising the average error using least squares, one can minimise instead the maximum error. This is very challenging, but the huge advantage is that by minimising the maximum (i.e. in L^infinity), any "spikes" of huge error deviations are excluded, as opposed to minimisation of quadratic average. The method of minimisation of the maximum provides much more realistic modelswhen compared to the case of integrals, where instead the average is minimised. By applying this approach to Weather forecasting, Oceanography and Atmospheric chemistry this may lead to much more accurate, precise and realistic predictions. The area of vector-valued Calculus of Variations in L^infinity is still very much under development, but for the scalar case there is a successful sophisticated theory. The general concept of Calculus of Variations in L^infinity already has extensive applications to: L^infity problems relate to Lipschitz Extensions, Quasiconformal maps, Game Theory, Control Theory, Inverse Problems, Partial Differential Equation constrained optimisation & Data Assimilation.The aims of this project are to exploit the existing concepts to make further progress in the general theory of vector-valued calculus of variations in the space L^infinity. The project will be composed of proving theoretical deductions that are illustrated with fundamental examples. We will specialise to functionals arising from variational data assimilation and make a direct comparison between the minimising the average error and minimising the maximum error. Our intentions are to develop the underlying framework of vectorial variational data assimilation for real world dynamical processes.Efforts towards the project goals will be in an organised, focused and disciplined fashion. There will be consistent progress monitoring order to obtain the desired goals. Initially the supervisor will meet regularly supporting the student's ideas through discussion and demonstration. As the project progresses the student will routinely present their work for critical evaluation from the supervisor. During the process an analysis of the project will be made at each stage of the development and any changes will be amended with the appropriate course of action.