The Carleman Contraction Principle for Three-Dimensional Phased and Phaseless Inverse Scattering

仅从外部测量识别区域内完全被遮挡的未知目标是数学家，物理学家和工程师之间具有挑战性的跨学科研究工作。该领域的重要应用包括无损检测，例如检查产品内部的裂缝，地震勘探，例如寻找海底石油，安全，例如识别掩埋的杀伤人员爆炸装置等。目前，广泛使用的基于优化的方法，这样的任务需要一些先验知识的真正的解决方案，这并不总是可用的，它们的计算成本是昂贵的。该项目旨在开发新的，理论上合理的技术，旨在快速提供可靠的解决方案，用于估计独立于初始估计的遮挡目标。本科生和研究生都将通过参与这项研究进行培训。在这个项目中，识别未知目标的数据将通过测量的散射波收集介质中。 在这项工作中开发的方法将能够处理相位可能或可能无法测量的数据，后者发生在许多具有挑战性的情况下，例如纳米成像的情况下，入射波的波长特别小。新的方法结合了联合收割机的Carleman估计，压缩映射原理，和一个代数公式，以恢复丢失的相位时，需要的。压缩映射原理的使用将保证新方法的关键优势：放松对初始猜测的要求和低廉的计算成本。这些新的方法将被应用到偏微分方程，包括三维亥姆霍兹方程和麦克斯韦的系统，管理波的传播。该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

The identification from only external measurements of unknown targets that are fully occluded inside a region is a challenging interdisciplinary research effort among mathematicians, physicists, and engineers. Important applications of this field include nondestructive testing, for example, checking for cracks inside a product, seismic exploration, for example, searching for oil under the sea, security, for example, the identification of buried anti-personnel explosive devices, and others. Current, widely used optimization-based methods for such tasks require some a priori knowledge about the true solution, which is not always available, and their computational cost is expensive. This project aims to develop new, theoretically sound techniques that are designed to quickly deliver reliable solutions for estimating occluded targets that are independent of the initial estimates. Both undergraduate and graduate students will be trained through involvement in this research. In this project, data for identifying unknown targets inside a medium will be collected through a measured scattering wave. The methodology to be developed in this work will be able to handle data where the phase may or may not be able to be measured, the latter which occurs in many challenging situations, such as the case of nano imaging, where the wavelength of the incident wave is particularly small. The new methods combine Carleman estimates, the contraction mapping principle, and an algebraic formula to retrieve the lost phase when needed. The use of the contraction mapping principle will guarantee key strengths of the new approach: relaxed requirements on the initial guesses and inexpensive computational cost. These new methods will be applied to partial differential equations that govern wave propagation, including the three-dimensional Helmholtz equation and Maxwell’s system. The new methods will be tested on both simulated and experimental data.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.