Inverse Problems for the Wave Equation

科学、工程、医学和商业中的许多测量要求在不进行物理探测或其他破坏性测试的情况下确定对象内部的特性。这种非侵入性检测在从医学成像到石油和天然气勘探等领域发挥着重要作用。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。人们的期望是，物体的内部会影响波浪或水流，而身体的反应将提供一个了解其性质的窗口，以便人们可以构建物体内部的图像。这项研究项目的目标是研究这种成像技术背后的数学原理，并推进这一领域所基于的严格基础。这一结果有可能在成像技术方面取得重大进展。这个项目将包括对双曲型偏微分方程反问题的研究。前两个问题与三个空间(和一个时间)维的双曲算子有关，该算子由波动算子加上紧支的势组成，其中势被支撑在一个半径为1的球中。对于反向散射问题，考虑了单位球面上点源对应的双曲初值问题的解。目标是从后向散射数据中恢复电位，该数据包括在震源位置测量的所有时间的溶液的值。为此，将研究从位势到后向散射数据的映射的注入性。对于第二个问题，考虑了源位于原点的双曲初值问题的解，目标是由解在单位球面上的迹确定势。对于第三个问题，考虑了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。这部分项目的目标是从光锥上溶液的轨迹恢复速度函数，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.