权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

The Unified Transform, Imaging and Asymptotics

统一变换、成像和渐进

基本信息

批准号：
EP/N006593/1
负责人：
Athanassios Fokas
金额：
$ 153.78万
依托单位：
University of Cambridge
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2015
资助国家：
英国
起止时间：
2015 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FN006593%2F1
关键词：
Unified Transform Imaging Asymptotics

项目摘要

A plethora of physical, chemical, biological and even social processes, can be modelled by mathematical equations. Many of these processes involve continuous change, and then the relevant equations take the form of differential equations. In models containing more than one variable, which is the great majority of situations, the relevant equations are called partial differential equations (PDEs). Given that these equations are instrumental in modelling the world around us, it is crucial that appropriate tools are developed for solving PDEs so that the associated models can be properly analysed. PDEs come into two broad categories: linear and non-linear. A general technique for solving linear PDEs was developed by the great French mathematician Fourier in the early 1800s. Non-linear PDEs are much more difficult to solve analytically. In 1997 the PI introduced a new method for solving a large class of non-linear PDEs. In an unexpected development, these results have motivated the development of a completely new method for solving linear PDEs in two dimensions. This is remarkable, since until then it was thought that the methods developed by Fourier and others in the 18th century could not be improved. This method is reviewed by three authors in the March 2014 issue of the Journal SIAM Review in the article titled "The Method of Fokas for solving linear PDEs", and in an accompanied editorial the importance of this method for solving linear PDEs is compared with the importance of the "Fosbury flop" in the high jump. The first part of this project involves completing the implementation of the above method to some important linear and non-linear problems in two dimensions, and then extending this method from 2 to 3 dimensions.Several medical imaging techniques, including Computed Tomography, Positron Emission Tomography (PET) and Single Photon Emission Computed Tomography (SPECT) are based on the solution of a particular class of mathematical problems, called inverse problems. In the second part of this project, new numerical and analytical techniques will be implemented for PET and SPECT.The Riemann function occurs in many different areas of mathematics. Several conjectures related to the Riemann function remain open, including the famous Riemann hypothesis and the Lindeloef hypothesis. The third part of the project involves the analysis of the asymptotics of the Riemann and related functions, which is expected to enhance our understanding of the relevant, most important mathematical structures.

大量的物理、化学、生物甚至社会过程都可以用数学方程来模拟。其中许多过程涉及连续变化，然后相关方程采用微分方程的形式。在大多数情况下，在包含多个变量的模型中，相关方程称为偏微分方程（PDEs）。考虑到这些方程对我们周围的世界建模很有帮助，开发适当的工具来解决偏微分方程是至关重要的，这样相关的模型才能得到适当的分析。偏微分方程分为两大类：线性和非线性。19世纪初，伟大的法国数学家傅立叶提出了求解线性偏微分方程的一般技术。非线性偏微分方程更难解析求解。1997年，PI引入了一种求解大型非线性偏微分方程的新方法。在一个意想不到的发展中，这些结果推动了求解二维线性偏微分方程的全新方法的发展。这是值得注意的，因为在此之前，人们一直认为傅里叶和其他人在18世纪开发的方法无法改进。三位作者在2014年3月的SIAM Review杂志上发表了一篇题为“求解线性偏微分方程的Fokas方法”的文章，对该方法进行了综述，并在一篇随附的社论中将该方法对求解线性偏微分方程的重要性与跳高中的“Fosbury跳高”的重要性进行了比较。本课题的第一部分是完成上述方法在二维上对一些重要的线性和非线性问题的实现，然后将该方法从二维扩展到三维。几种医学成像技术，包括计算机断层扫描，正电子发射断层扫描（PET）和单光子发射计算机断层扫描（SPECT）是基于解决一类特殊的数学问题，称为逆问题。在该项目的第二部分，新的数值和分析技术将用于PET和SPECT。黎曼函数出现在许多不同的数学领域。与黎曼函数有关的几个猜想仍然是开放的，包括著名的黎曼假设和林德洛夫假设。该项目的第三部分涉及黎曼和相关函数的渐近性分析，这有望提高我们对相关的，最重要的数学结构的理解。