Inverse Problems for the Wave Equation

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。

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.