Inverse Scattering Using Data-Driven Reduced-Order Models

波在雷达、超声波或地球物理勘探等许多应用中用于成像。本计画主要是针对奈米光子学中的周期性色散结构，发展一种新的以降阶模型为基础的逆散射与成像方法。降阶模型在动力系统领域得到了广泛的应用，但是在逆问题中的应用却非常有限。该项目中的逆散射问题是针对典型的设置制定的，其中数据由传感器阵列收集，这些传感器阵列用波探测未知的非均匀介质并测量返回。这种新的方法不是直接问“哪种散射介质可能导致测量的波返回”的问题，而是首先从测量数据构建降阶模型。降阶模型被定义为保持波传播的物理特性。该研究的动机是在材料的无损评估，使用机载雷达估计海冰厚度，医学成像，石油勘探和纳米光子学的应用。开放源码软件将在本项目的范围内提供。本项目寻求逆散射和降阶建模的新方向。它展示了如何将波传播视为由包含介质信息的线性算子驱动的动力系统，该线性算子由声波或电磁波方程中的未知系数建模。动态系统框架允许连接到降阶建模技术，这需要进行调整，以便在逆散射中是有用的。该项目针对以下具体问题追求这一目标：（1）如何以像素扫描方式控制波以改善聚焦和随后的成像？(2)如何构造和使用ROM来定量估计控制波动方程中的系数？(3)如何在有损介质中反转？后一个问题是连接到一个重要的问题，在降阶建模，有关的端口哈密尔顿动力系统的测量其传递函数的识别。该项目还寻求为色散、周期性纳米光子结构开发模型驱动ROM，以加速设计优化和参数反演。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Waves are used to image in many applications like radar, ultrasound or geophysical exploration. This project is concerned with developing a novel reduced-order model based methodology for inverse scattering and for imaging of periodic, dispersive structures in nano-photonics. Reduced-order models are extensively used in the dynamical-systems community; however, their use in inverse problems has been limited. The inverse scattering problems in this project are formulated for typical setups, where data are gathered by arrays of sensors that probe an unknown heterogeneous medium with waves and measure the returns. Rather than asking the question "which scattering medium could have caused the measured wave returns" directly, this novel methodology first constructs a reduced-order model from the measured data. The reduced-order model is defined to preserve the physics of wave propagation. The research is motivated by applications in nondestructive evaluation of materials, estimation of sea ice thickness using airborne radar, medical imaging, oil exploration and nano-photonics. Open source software will be provided in the scope of this project This project seeks new directions in inverse scattering and reduced-order modeling. It shows how to view wave propagation as a dynamical system driven by a linear operator that contains information about the medium, modeled by unknown coefficients in the acoustic or electromagnetic wave equation. The dynamical system framework allows the connection to reduced-order modeling techniques, which need to be adapted in order to be useful in inverse scattering. The project pursues this goal for the following specific questions: (1) How can one control waves for improved focusing and subsequent improved imaging, in a pixel scanning fashion? (2) How can one construct and use a ROM for quantitative estimation of the coefficients in the governing wave equations? (3) How can one invert in lossy media? The latter question is connected to an important problem in reduced-order modeling, concerned with the identification of port-Hamiltonian dynamical systems from measurements of their transfer function. This project also seeks to develop model-driven ROMs for dispersive, periodic nano-photonic structures, for speeding up design optimization and parametric inversion.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.