Inverse Problems for the Wave Equation

波动方程的反问题

基本信息

  • 批准号:
    1615616
  • 负责人:
  • 金额:
    $ 15.15万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2016
  • 资助国家:
    美国
  • 起止时间:
    2016-07-01 至 2021-06-30
  • 项目状态:
    已结题

项目摘要

Many measurements in science, engineering, medicine, and business call for determining properties of the interior of an object without physical probing or other destructive testing. Areas in which such noninvasive testing plays a major role range from medical imaging to oil and gas prospecting. In these situations, objects are probed by noninvasive methods utilizing sound waves, electromagnetic radiation, electrical currents, or other means. The expectation is that the interior of the object will affect the waves or currents, and the body response will provide a window into its properties so that one can construct an image of the interior of the object. The goal of this research project is to study the mathematics behind such imaging techniques and advance the rigorous foundations on which this field is based. Results have the potential for important advances in imaging technology.This project will encompass a study of three inverse problems for hyperbolic partial differential equations. The first two problems are associated with a hyperbolic operator in three space (and one time) dimensions, with the operator consisting of the wave operator plus a compactly-supported potential, where the potential is supported, say, in a ball of radius one. For the inverse back-scattering problem, one considers the solution of the hyperbolic initial value problem corresponding to a point source located on the unit sphere. The goal is the recovery of the potential from the back-scattering data consisting of the value of the solution measured at the source location, for all time. To do this, a study of the injectivity of the map from the potential to the backscattering data will be performed. For the second problem, one considers the solution of the hyperbolic initial value problem with the source placed at the origin, and the goal is the determination of the potential from the trace of the solution on the unit sphere, for all time. For the third problem, one considers the solution of the standard wave equation with zero initial position function and arbitrary initial velocity function. The goal of this part of the project is to recover the velocity function from the trace of the solution on the light cone, the cone where time equals the magnitude of the position.
科学、工程、医学和商业中的许多测量要求在没有物理探测或其他破坏性测试的情况下确定物体内部的属性。从医学成像到石油和天然气勘探,这种非侵入性测试发挥着重要作用的领域。在这些情况下,通过利用声波、电磁辐射、电流或其他手段的非侵入性方法来探测对象。期望的是,物体的内部会影响波或电流,身体的反应将提供一个窗口,以了解其属性,以便人们可以构建物体内部的图像。该研究项目的目标是研究此类成像技术背后的数学,并推进该领域所基于的严格基础。本计画将研究三个双曲型偏微分方程式之反问题。前两个问题与三维空间(和一维时间)中的双曲算子有关,该算子由波算子加上紧支势组成,其中该势被支撑在半径为1的球中。对于后向散射反问题,考虑了对应于位于单位球上的点源的双曲初值问题的解。我们的目标是恢复的潜力从后向散射数据组成的值的解决方案在源位置测量,为所有的时间。要做到这一点,将执行从电位到后向散射数据的映射的注入性的研究。对于第二个问题,我们考虑的双曲型初值问题的解决方案与源放在原点,和目标是确定潜在的单位球上的解决方案的轨迹,为所有的时间。对于第三个问题,考虑具有零初始位置函数和任意初始速度函数的标准波动方程的解。这个项目的这一部分的目标是从光锥上的解的轨迹中恢复速度函数,光锥上的时间等于位置的大小。

项目成果

期刊论文数量(0)
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会议论文数量(0)
专利数量(0)

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Rakesh Rakesh其他文献

Innovative Approaches for Characterizing Chlorantraniliprole and Its Metabolites in Soil, Water and Plants
表征土壤、水和植物中氯虫苯甲酰胺及其代谢物的创新方法
Smart Systems and IoT: Innovations in Computing
智能系统和物联网:计算创新
  • DOI:
    10.1007/978-981-13-8406-6
  • 发表时间:
    2020
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Arun K. Somani;Rajveer Singh;Ankit Mundra;S. Srivastava;Vivek Kumar Verma;.. .. .. .. .. D. Kumar;Nehal Patel;Radhika Patel;Jenny Kasudiya;Ankit Bhavsar;Harshal A. Arolkar;Tigilu Mitiku;M. S. Manshahia;Rutba Mufti;Kartike Khatri;Sumit Bhardwaj;Punit Gupta;Pankaj Kumar;Sidhartha Barui;Deepanwita Das;Mangala N. Sumedh;Sneha Srinivasan;S. Basavaraju;Nidhi Gangrade;Nirmal Choudhary;K. K. Bharadwaj;Abdul Rehman;Nitin Khan;Rakesh Rakesh;Matam;Dinesh Siddhant Goswami;Singh Shekhawat;Neetu Faujdar;Nitin Rakesh;P. Rohatgi;Karan Gupta;G. Chauhan;Y. Meena;Nidhi Gupta;Deepak Vaswani;Kuldeep Singh;Sakar Gupta;Sunita Gupta;Amit Deepak Soni;Kumar Behera;Dheeraj Sharma;M. Aslam;Shivendra Yadav;Nirav Bhatt;Amit Thakkar;Nikita Bhatt;Purvi Prajapati;Neeru Meena;Buddha Singh;Laxmi Chaudhary;Deepak Kumar
  • 通讯作者:
    Deepak Kumar

Rakesh Rakesh的其他文献

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{{ truncateString('Rakesh Rakesh', 18)}}的其他基金

The inverse backscattering problem and the inverse fixed angle scattering problem
逆后向散射问题和逆固定角散射问题
  • 批准号:
    2307800
  • 财政年份:
    2023
  • 资助金额:
    $ 15.15万
  • 项目类别:
    Standard Grant
Hyperbolic Inverse Problems
双曲反问题
  • 批准号:
    1908391
  • 财政年份:
    2019
  • 资助金额:
    $ 15.15万
  • 项目类别:
    Standard Grant
Formally determined inverse problems for hyperbolic PDEs
双曲偏微分方程的正式确定的反问题
  • 批准号:
    1312708
  • 财政年份:
    2013
  • 资助金额:
    $ 15.15万
  • 项目类别:
    Standard Grant
Inversion from Time Domain Backscattering Data for the Wave Equation
时域后向散射数据反演波动方程
  • 批准号:
    0907909
  • 财政年份:
    2009
  • 资助金额:
    $ 15.15万
  • 项目类别:
    Standard Grant

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