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Variational Data Assimilation via Calculus of Variations in L^infinity

通过 L^无穷变分演算进行变分数据同化

基本信息

批准号：
2272180
负责人：
金额：
--
依托单位：
University of Reading
依托单位国家：
英国
项目类别：
Studentship
财政年份：
2019
资助国家：
英国
起止时间：
2019 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=studentship-2272180
关键词：
Variational Data Assimilation via Calculus

项目摘要

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.

该项目基于以下数学直觉。假设我们得到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符看作是一些真实世界动力学过程的模型。变分同化的目标是找到一个向量函数，使其近似满足模式，从而使输出与观测值近似匹配。这个问题是不适定的和超定的，因此，一般而言，精确可能没有任何解，或者可能有无限多个解。标准方法是通过使某一误差泛函最小化来构造近似解，该泛函测量偏差的平均二次误差。在整个项目中，将寻求另一种方法，基于最近发展起来的L空间中的矢量值变分领域。其想法是，人们可以将最大误差最小化，而不是使用最小二乘法来最小化平均误差。这很有挑战性，但其巨大的优势在于，通过最小化最大值(即在L无穷大中)，排除了任何巨大误差偏差的“尖峰”，而不是最小化二次平均。与积分的情况相比，最小化最大值的方法提供了更真实的模型，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、更精确和更现实的预测。向量值变分在L无穷远中的应用还很不成熟，但对于标量情形，已经有了成熟的理论。L无穷维变分学的一般概念已经在与Lipschitz扩张、拟共形映射、博弈论、控制论、反问题、偏微分方程约束最优化和数据同化有关的微扰问题中得到了广泛的应用。本项目的目的是利用已有的概念来进一步发展L无穷空间向量值变分的一般理论。该项目将包括证明理论推论，并用基本的例子加以说明。我们将专门研究变分数据同化产生的泛函，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。我们的目的是为真实世界的动力过程开发矢量变分数据同化的基本框架。朝着项目目标的努力将以有组织、有重点和有纪律的方式进行。将有一致的进度监测顺序，以实现预期目标。最初，导师会定期开会，通过讨论和演示来支持学生的想法。随着项目的进展，学生将例行地将他们的作品提交给导师进行批判性评估。在这一过程中，将在开发的每个阶段对项目进行分析，并将根据适当的行动方针对任何变化进行修改。