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Spectral element methods for fractional differential equations, with applications in applied analysis and medical imaging

分数阶微分方程的谱元方法，在应用分析和医学成像中的应用

基本信息

批准号：
EP/T022132/1
负责人：
Sheehan Olver
金额：
$ 72.94万
依托单位：
Imperial College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FT022132%2F1
关键词：
Spectral element methods fractional differential

项目摘要

Fractional differential equations are of increasing importance in a wide-range of applications, including medical imagining, collective behaviours, finance, image analysis, and elsewhere. These equations are challenging to solve numerically as they involve nonlocal interactions, which if tackled naively lead to dense discretisations that are too computationally difficult to solve, thereby limiting the scope of feasible numerical simulations. This project will develop a state-of-the-art spectral element method for simulating these models based on reducing the equations to highly structured linear systems, using a key observation that singular behaviour can be captured exactly in the development of numerical schemes. This will lead to faster and more accurate simulations facilitating progress in a wide range of applications. We will apply the results to challenging problems arising in applied analysis. Fractional differential equations arise in collective behaviour models such as swarming of animal species, cell movement by chemotaxis, granular media interaction and self-assembly of particles, and give important information about the equilibrium behaviour of such systems. These equations are difficult to solve numerically due to possible blow-up behaviour, where the model develops a singularity in finite time. The proposed scheme will allow for refinement near singularities to concentrate computational power in these difficult regions while keeping control on computational cost, allowing for high performance simulations. We will also tackle real world applications in medical imaging, including ultrasound imagining of the brain. Fractional differential equations have proven powerful tools in designing modern models that capture nonlocal behaviour caused by memory effects in the tissue. The developed spectral element method will facilitate more accurate simulations involving non-trivial geometries, for example ellipsoidal models of the skull, while avoiding inaccuracies in current schemes caused by sharp transitions between the skull and tissue.

分数阶微分方程在医学想象、集体行为、金融、图像分析和其他领域的广泛应用中具有越来越重要的作用。这些方程的数值求解具有挑战性，因为它们涉及非局部相互作用，如果简单地处理，会导致密集的离散化，计算上太难求解，从而限制了可行的数值模拟的范围。这个项目将开发一种最先进的谱元素方法来模拟这些模型，方法是将方程简化为高度结构的线性系统，使用一个关键观测数据，即在数值方案的开发中可以准确地捕获奇异行为。这将导致更快和更准确的模拟，促进在广泛的应用中取得进展。我们将把结果应用于应用分析中出现的具有挑战性的问题。分数阶微分方程组出现在动物种群聚集、细胞趋化运动、颗粒介质相互作用和颗粒自组装等集体行为模型中，并给出了关于这些系统的平衡行为的重要信息。由于可能的爆破行为，这些方程很难在数值上求解，其中模型在有限时间内发展出奇异性。所提出的方案将允许在奇点附近进行精化，以便在这些困难区域集中计算能力，同时保持对计算成本的控制，从而实现高性能的模拟。我们还将解决现实世界中医学成像的应用，包括大脑的超声波成像。分数阶微分方程组已被证明是设计现代模型的强大工具，这些模型捕捉由组织中的记忆效应引起的非局部行为。开发的谱元素方法将有助于更准确地模拟非平凡几何，例如头骨的椭球体模型，同时避免当前方案中由于头骨和组织之间的急剧过渡而导致的不准确。