Inverse problems for hyperbolic partial differential equations

双曲偏微分方程的反问题

• 批准号: EP/P01593X/1
• 负责人: Lauri Oksanen
• 金额: $ 80.21万
• 依托单位: University College London
• 依托单位国家: 英国
• 项目类别: Fellowship
• 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%2FP01593X%2F1
• 关键词: Inverse; problems; hyperbolic; partial; differential

A typical inverse boundary value problem arises when an object is probed by sending waves to it in order to determine some properties of the object from the measured responses. The proposed research focuses on the case where the measurements are modelled using the acoustic wave equation. The acoustic case can be seen as a prototype case for several geophysical imaging problems. These problems can be local, for example, when the goal is to detect geological deposits, or global when the goal is to chart the internal structure of the Earth. The aim of the proposed research is to significantly extend the mathematical theory of inverse boundary value problems and to bridge the gap between this theory and practical applications in geophysical imaging. The main theoretical objective is to show that the inverse boundary value problem for the wave equation has a unique solution in the fully coordinate invariant context. The main practical objective is to develop a new computational method for inverse boundary value problems. The presently used computational methods in geophysical imaging are not based on the mathematical theory of inverse boundary value problems. Bridging the gap between the theory and applications has a potential for a high impact on the latter, and the resulting physical understanding will also guide further mathematical study.The research brings together ideas from analysis of partial differential equations, differential geometry, and numerical analysis.

当通过向对象发送波来探测对象以便从测量的响应确定对象的某些属性时，就会出现典型的逆边值问题。拟议的研究重点是使用声波方程对测量进行建模的情况。声学案例可以被视为多个地球物理成像问题的原型案例。这些问题可以是局部的，例如，当目标是探测地质沉积物时，或者是全球性的，当目标是绘制地球的内部结构时。本研究的目的是显着扩展逆边值问题的数学理论，并弥合该理论与地球物理成像实际应用之间的差距。主要理论目标是证明波动方程的反边值问题在全坐标不变的情况下具有唯一的解。主要的实际目标是开发一种新的逆边值问题的计算方法。目前地球物理成像中使用的计算方法并不是基于反边值问题的数学理论。弥合理论与应用之间的差距可能会对后者产生重大影响，由此产生的物理理解也将指导进一步的数学研究。该研究汇集了偏微分方程分析、微分几何和数值分析的思想。

项目成果

Spacetime finite element methods for control problems subject to the wave equation

波动方程控制问题的时空有限元方法

• DOI: 10.1051/cocv/2023028
• 发表时间: 2023
• 期刊: Control, Optimisation and Calculus of Variations
• 作者: Burman E

Primal dual mixed finite element methods for the elliptic Cauchy problem

椭圆柯西问题的原始对偶混合有限元方法

• DOI: 10.48550/arxiv.1712.10172
• 发表时间: 2017
• 作者: Burman E

Stability estimate for scalar image velocimetry

标量图像测速的稳定性估计

• DOI: 10.1515/jiip-2020-0107
• 发表时间: 2023
• 期刊: Journal of Inverse and Ill-posed Problems
• 影响因子: 1.1
• 作者: Burman E

Space time stabilized finite element methods for a unique continuation problem subject to the wave equation

波动方程下唯一连续问题的时空稳定有限元方法

• DOI: 10.1051/m2an/2020062
• 发表时间: 2021
• 期刊: Mathematical Modelling and Numerical Analysis
• 作者: Burman E

Lorentzian Calderón problem under curvature bounds

曲率边界下的洛伦兹卡尔德隆问题

• DOI: 10.1007/s00222-022-01100-5
• 发表时间: 2022
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: Alexakis S