Adaptive Regularisation

自适应正则化

• 批准号: EP/P000835/1
• 负责人: Tristan Pryer
• 金额: $ 12.89万
• 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: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FP000835%2F1
• 关键词: Adaptive; Regularisation

Many physical phenomena can be modelled using differential equations. However, in general, mathematicians are not able to solve these analytically. For example, we know a given fluid can be modelled well using a Navier-Stokes equation, but we cannot solve the equation exactly, so we cannot predict what the fluid does over time. Hence to gain some knowledge on how the fluid is behaving we often turn to numerical approximations. Therefore we must design a scheme which can be run on a computer to simulate what our fluid does. Having access to "good" numerical approximations is very important; in particular, it is important to be able to quantify how accurate the numerical approximation is. This quantification allows us to determine whether to trust the simulation we generate.A posteriori error analysis is used to assess the accuracy of a given numerical approximation. It allows us to know when and where the simulation misbehaves and gives us the option to correct it by "adapting" the numerical scheme. This is called an adaptive procedure. Adaptive procedures allow us to make the simulation more efficient, in terms of computational time, allowing for more complex simulations to be carried out faster.One of the research aims of this project is to propose an alternative methodology to tackle the cases when a posteriori analysis fails. For example, when a jet's speed exceeds the sound barrier, shock waves form. Mathematically these are discontinuities in the underlying medium. This phenomena is exceptionally difficult to simulate and the subject of much research. In particular, the a posteriori analysis, our assessment of the simulation, does not provide any useful information.Another aim of this research is to lay the groundwork towards an application in the area of "data assimilation". Data assimilation is a technique useful when observations are available at specific points in time. Perhaps you are studying the evolution of a hurricane and have access to air pressure from certain weather monitoring stations at certain times. The mathematical model which is derived can then be updated based on these observations at the times they are observed. Data assimilation is a systematic way to provide such updates, and it allows for accurate prediction of how the hurricane evolves based on what has happened. But how are these incorporated into the numerical simulation? Current methodologies enforce that the mathematical model agrees with the observations on average.The numerical schemes developed in this project will develop the foundations for the design of simulations where the observations can be incorporated into the mathematical model in a "pointwise" sense, rather than on average. This is extremely important and will aid, among other applications, the development of more accurate weather prediction software.

许多物理现象可以用微分方程式来模拟。然而，一般而言，数学家不能解析地解决这些问题。例如，我们知道使用Navier-Stokes方程可以很好地模拟给定的流体，但我们无法准确地求解该方程，因此我们无法预测该流体随时间的变化。因此，为了获得一些关于流体行为的知识，我们经常求助于数值近似。因此，我们必须设计一种可以在计算机上运行的方案来模拟我们的流体的行为。获得“良好的”数值近似是非常重要的；尤其是能够量化数值近似的精度是非常重要的。这种量化使我们能够确定是否信任我们生成的模拟。后验误差分析用于评估给定数值近似的准确性。它允许我们知道模拟何时何地发生错误，并给我们提供了通过“调整”数值格式来纠正它的选项。这被称为自适应过程。自适应程序使我们能够在计算时间方面提高模拟的效率，从而允许更复杂的模拟更快地进行。本项目的研究目标之一是提出一种替代方法来处理后验分析失败的情况。例如，当喷气式飞机的速度超过音障时，就会形成冲击波。从数学上讲，这些都是底层介质中的不连续。这种现象特别难模拟，也是许多研究的主题。特别是，后验分析，我们对模拟的评估，并没有提供任何有用的信息。这项研究的另一个目的是为在“数据同化”领域的应用奠定基础。当在特定时间点有观测数据时，数据同化是一种有用的技术。也许你正在研究飓风的演变，并在特定时间获得特定天气监测站的气压。然后，可以在观测到这些观测的同时，基于这些观测来更新推导出的数学模型。数据同化是一种提供此类更新的系统方法，它可以根据所发生的情况准确预测飓风的演变。但这些是如何融入数值模拟的呢？目前的方法要求数学模型与平均观测值一致。本项目中开发的数值方案将为模拟设计奠定基础，其中观测值可以“逐点”地合并到数学模型中，而不是平均。这是极其重要的，除了其他应用外，还将有助于开发更准确的天气预报软件。

项目成果

Adaptive non-hierarchical Galerkin methods for parabolic problems with application to moving mesh and virtual element methods

• DOI: 10.1142/s0218202521500172
• 发表时间: 2020-05
• 期刊: ArXiv
• 作者: A. Cangiani;E. Georgoulis;Oliver J. Sutton

A posteriori error estimates for the virtual element method.

• DOI: 10.1007/s00211-017-0891-9
• 发表时间: 2017
• 期刊: Numerische mathematik
• 影响因子: 2.1
• 作者: Cangiani A;Georgoulis EH;Pryer T;Sutton OJ
• 通讯作者: Sutton OJ

The r-Hunter-Saxton equation, smooth and singular solutions and their approximation

r-Hunter-Saxton 方程、光滑解和奇异解及其近似

• DOI: 10.1088/1361-6544/abab4d
• 发表时间: 2020
• 期刊: Nonlinearity
• 影响因子: 1.7
• 作者: Cotter C

The r -Hunter-Saxton equation, smooth and singular solutions and their approximation

r -Hunter-Saxton 方程、光滑解和奇异解及其近似

• DOI: 10.17863/cam.59052
• 发表时间: 2020
• 作者: Cotter C

Adaptive modelling of variably saturated seepage problems

可变饱和渗流问题的自适应建模

• DOI: 10.1093/qjmam/hbab001
• 发表时间: 2021
• 期刊: The Quarterly Journal of Mechanics and Applied Mathematics
• 作者: Ashby B