Inverse Problems and Imaging with Nonlinear Physics

本项目旨在研究物理中非线性效应起重要作用的成像和反演问题。非线性物理既可以帮助我们对仅用线性物理无法成像的物体进行成像，也可以阻止我们对没有这种非线性效应时能够成像的物体进行成像。在任何一种情况下，了解这种非线性效应的影响将有助于我们提高底层成像模式的质量。该项目的目标是进行详细的数学和数值分析，以表征非线性对这种情况下图像重建的影响。本项目的主要研究工作包括但不限于：(i)开发通过非线性色散介质成像的高效重建算法；（ii）发展光学成像双光子吸收和二次谐波产生的数学理论和计算方法；（三）非线性吸收定量光声分析及相应的计算图像重建算法。在这个项目中发展的数学思想有望为与非线性偏微分方程相关的类似成像和反问题的未来研究提供见解。该研究也有望对实际有用的非线性介质成像方法的发展以及给定介质的非线性物理性质成像产生影响。该项目包括一个综合的教育组成部分，旨在培养高级本科生、硕士和博士研究生。该奖项反映了美国国家科学基金会的法定使命，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。

This project aims at studying imaging and inverse problems where nonlinear effects in physics play significant roles. Nonlinear physics could either help us image objects that could not be imaged with only linear physics or prevent us from imaging objects that we would have been able to image in the absence of such nonlinear effects. In either case, understanding the impact of such nonlinear effects would help us improve the quality of the underlying imaging modalities. The goal of the project is to perform detailed mathematical and numerical analysis to characterize the effects of nonlinearity on image reconstructions in such cases.The main research efforts in this project include, but are not limited to: (i) developing efficient reconstruction algorithms for imaging through nonlinear dispersive media; (ii) developing mathematical theory and computational methods for imaging two-photon absorption and second harmonic generation in optics; and (iii) analyzing quantitative photoacoustics with nonlinear absorption and developing corresponding computational image reconstruction algorithms. The mathematical ideas developed in this project are expected to provide insights on future studies of similar imaging and inverse problems related to nonlinear partial differential equations. The research is also expected to have impact on the development of practically useful methods for imaging through nonlinear media as well as imaging nonlinear physical properties of given media. This project involves an integrated educational component that aims at training advanced undergraduate, master and PhD students.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.