Hyperbolic Inverse Problems

双曲反问题

• 批准号: 1908391
• 负责人: Rakesh Rakesh
• 金额: $ 11.62万
• 依托单位: University of Delaware
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-15 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1908391&HistoricalAwards=false
• 关键词: Hyperbolic; Inverse; Problems

In fields such as oil and gas prospecting, mapping the interior of a planet, or medical imaging, one determines properties of the interior of an object such as the location of oil/gas deposits in the interior of the earth, characterize the interior composition of a planet, or determine if an interior lump in the body is cancerous or not. Since drilling or cutting is often expensive or unfeasible in these situations, these objects are probed by non-invasive methods such as sound waves generated on the boundary of the object. The expectation is that the interior composition of the object will influence the incoming waves and the response wave, also measured only on the boundary of the object, provides a mathematical window into the interior of the object. The principal investigator (PI) will study the mathematics behind this imaging technique. Further, the PI will train graduate students and postdocs in this type of mathematics through mini-courses, seminars, personal conversations and workshops. Some of these students and postdocs will use these skills to solve practical problems for companies exploring for oil, building imaging devices, or involved in remote sensing.Problems like those described above, but with over-determined data, where the unknown function depends on fewer variables than the data, have received a lot of attention. The PI focuses on the less studied formally determined problems where the unknown function depends on the same number of variables as the data. Such problems, in two or more space dimensions, are harder but very useful in situations where data acquisition is difficult, and their investigation is one of the important challenges in the field. This project will study the following problems in three space dimensions: the fixed angle scattering problem, the backscattering problem, the point source problem, and the incoming spherical wave problem. Recently, using an adaptation of the Bukhgeim-Klibanov method, the PI and his collaborators proved stability for the fixed angle scattering problem for coefficients which are even in one of the variables and proved uniqueness for the problem of recovering a coefficient given data from the point source problem as well as the incoming spherical wave problem. This project aims at further adapting the Bukhgeim-Klibanov method to use Robbiano-Tataru type Carleman estimates and unique continuation arguments to tackle the proposed problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

在油气和天然气勘探，映射行星内部或医学成像等领域中，人们确定了物体内部的特性，例如油/天然气沉积物在地球内部的位置，表征行星的内部组成，或者确定体内的内部肿块是体内的。由于在这些情况下钻孔或切割通常是昂贵或不可行的，因此这些对象是通过非侵入性方法（例如在物体边界上产生的声波）探测的。期望的是，对象的内部组成将影响传入波，并且仅在对象边界上测量的响应波提供了一个数学窗口，可以进入对象内部。首席研究员（PI）将研究这种成像技术背后的数学。此外，PI将通过迷你演奏，研讨会，个人对话和讲习班来培训这种数学的研究生和博士后。这些学生和博士后中的一些将使用这些技能来解决用于探索石油，构建成像设备或参与遥感的公司的实际问题。如上所述的问题，但具有过度确定的数据，其中未知功能取决于比数据少的变量，而不是数据的较少。 PI专注于未知函数取决于与数据相同数量的变量数量的正式确定的问题。在两个或多个空间维度中，此类问题在数据获取困难的情况下非常困难，但非常有用，并且它们的调查是该领域的重要挑战之一。该项目将在三个空间维度中研究以下问题：固定角度散射问题，反向散射问题，点源问题和传入的球形波问题。最近，使用Bukhgeim-Klibanov方法的改编，PI及其合作者证明了系数的固定角度散射问题的稳定性，这些系数甚至是一个变量之一，并且证明了从点源问题中恢复了系数给定数据的问题，以及来自点源问题的唯一性。该项目旨在进一步调整Bukhgeim-Klibanov方法，以使用Robbiano-Tataru型卡尔曼型估计和独特的延续论证来解决拟议的问题。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子的智力和更广泛的影响来通过评估来进行评估的支持。

项目成果

Point sources and stability for an inverse problem for a hyperbolic PDE with space and time dependent coefficients

具有空间和时间相关系数的双曲偏微分方程反问题的点源和稳定性

• DOI: 10.1016/j.jde.2022.10.025
• 发表时间: 2023
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: Krishnan, Venkateswaran P.;Rakesh;Senapati, Soumen
• 通讯作者: Senapati, Soumen

Fixed Angle Inverse Scattering for Almost Symmetric or Controlled Perturbations

• DOI: 10.1137/20m1319309
• 发表时间: 2019-05
• 期刊: SIAM J. Math. Anal.
• 作者: Rakesh;M. Salo

Stability for a Formally Determined Inverse Problem for a Hyperbolic PDE with Space and Time Dependent Coefficients

具有空间和时间相关系数的双曲偏微分方程形式确定反问题的稳定性

• DOI: 10.1137/21m1400596
• 发表时间: 2021
• 期刊: SIAM Journal on Mathematical Analysis
• 影响因子: 2
• 作者: Krishnan, Venkateswaran P.;Rakesh, Rakesh;Senapati, Soumen
• 通讯作者: Senapati, Soumen