论文标题
从单个测量中的波动方程的时间相关系数的全球恢复
Global recovery of a time-dependent coefficient for the wave equation from a single measurement
论文作者
论文摘要
我们考虑从对波动方程限制到小子集的限制，恢复未知时间依赖的电势函数的正式确定的反问题，但要受单个外部源。我们表明，通过使用放置在时空结构域外部的单个来源，然后在时空域外的小社区中测量溶液，可以确定问题的潜在函数，直到对问题的自然障碍。该方法基于考虑光线的密集集合并构建一个源函数，该源功能结合了可数的源集合，每个源都会在集合中的光线附近生成波数据包。我们表明，测量与该单个源相对应的溶液同时确定沿集合中所有光线射线的灯光转换。结果是从光线变换的注射率遵循的。我们的证明还提供了重建算法。
We consider the formally determined inverse problem of recovering an unknown time-dependent potential function from the knowledge of the restriction of the solution of the wave equation to a small subset, subject to a single external source. We show that one can determine the potential function, up to the natural obstruction for the problem, by using a single source placed in the exterior of the spacetime domain and subsequently measuring the solution in a small neighborhood outside of the spacetime domain. The approach is based on considering a dense collection of light rays and constructing a source function that combines a countable collection of sources that each generates a wave packet near a light ray in the collection. We show that measuring the solution corresponding to that single source simultaneously determines the light ray transform along all the light rays in the collection. The result then follows from injectivity of the light ray transform. Our proof also provides a reconstruction algorithm.
