一种求解物理驱动源反问题的采样框架
A Sampling Framework for Solving Physics-driven Inverse Source Problems
摘要：
偏微分方程是描述许多物理现象的核心。在许多应用中，通过传感器网络观察这些现象，目的是推断它们的基本属性。利用采样和近似理论中的某些结果，我们提出了一个新的框架来解决一类由线性偏微分方程控制的物理场的反源问题。具体来说，我们证明了未知的场源可以恢复从一个序列，所谓的，广义测量，通过使用多维频率估计技术。接下来，我们表明-物理驱动的领域-这一序列的广义测量可以通过计算传感器测量的线性加权和来估计;其中确切的权重（和）对应于那些重现多维指数，当用于线性组合翻译的特定原型函数相关的格林函数的基础领域。显式公式，然后推导出的序列的权重，映射传感器样本的确切序列的广义测量时，格林函数满足广义的Strang-Fix条件。否则，相同的映射产生广义测量的近似。基于这个新的框架，我们开发实用的，噪声鲁棒的，传感器网络的策略，解决逆源问题，然后提出数值模拟结果，以验证其性能。
英文摘要：
Partial differential equations are central to describing many physical phenomena. In many applications these phenomena are observed through a sensor network, with the aim of inferring their underlying properties. Leveraging from certain results in sampling and approximation theory, we present a new framework for solving a class of inverse source problems for physical fields governed by linear partial differential equations. Specifically, we demonstrate that the unknown field sources can be recovered from a sequence of, so called, generalised measurements by using multidimensional frequency estimation techniques. Next we show that---for physics-driven fields---this sequence of generalised measurements can be estimated by computing a linear weighted-sum of the sensor measurements; whereby the exact weights (of the sums) correspond to those that reproduce multidimensional exponentials, when used to linearly combine translates of a particular prototype function related to the Green's function of our underlying field. Explicit formulae are then derived for the sequence of weights, that map sensor samples to the exact sequence of generalised measurements when the Green's function satisfies the generalised Strang-Fix condition. Otherwise, the same mapping yields a close approximation of the generalised measurements. Based on this new framework we develop practical, noise robust, sensor network strategies for solving the inverse source problem, and then present numerical simulation results to verify their performance.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
