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

三维相相和无相逆散射的卡尔曼收缩原理

• 批准号: 2208159
• 负责人: Loc Nguyen
• 金额: $ 24.66万
• 依托单位: University of North Carolina at Charlotte
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2208159&HistoricalAwards=false
• 关键词: Carleman; Contraction; Principle; Three; Dimensional

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.

仅通过外部测量来识别区域内完全被遮挡的未知目标对于数学家、物理学家和工程师来说是一项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纹）、地震勘探（例如寻找海底石油）、安全（例如识别埋藏的杀伤人员爆炸装置）等。目前，针对此类任务广泛使用的基于优化的方法需要一些关于真实解决方案的先验知识，而这些知识并不总是可用，而且它们的计算成本非常昂贵。该项目旨在开发理论上合理的新技术，旨在快速提供可靠的解决方案来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接受培训。在该项目中，将通过测量的散射波收集用于识别介质内未知目标的数据。 这项工作中开发的方法将能够处理可能无法测量相位的数据，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，其中入射波的波长特别小。新方法结合了卡尔曼估计、收缩映射原理和代数公式，以在需要时恢复丢失的相位。收缩映射原理的使用将保证新方法的关键优势：放宽对初始猜测的要求和廉价的计算成本。这些新方法将应用于控制波传播的偏微分方程，包括三维亥姆霍兹方程和麦克斯韦系统。新方法将在模拟和实验数据上进行测试。该奖项反映了 NSF 的法定使命，并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，被认为值得支持。

项目成果

Numerical differentiation by the polynomial-exponential basis

• DOI: 10.48550/arxiv.2304.05909
• 发表时间: 2023-04
• 期刊: ArXiv
• 作者: P. M. Nguyen;T. Le;L. Nguyen;M. Klibanov

The Carleman-Newton method to globally reconstruct the initial condition for nonlinear parabolic equations

• DOI: 10.1016/j.cam.2024.115827
• 发表时间: 2024-02
• 期刊: Journal of Computational and Applied Mathematics
• 影响因子: 2.4
• 作者: Anuj Abhishek;Thuy T. Le;Loc H. Nguyen;Taufiquar Khan

The Carleman Contraction Mapping Method for Quasilinear Elliptic Equations with Over-determined Boundary Data

• DOI: 10.1007/s40306-023-00500-w
• 发表时间: 2022-03
• 期刊: Acta Mathematica Vietnamica
• 影响因子: 0.5
• 作者: L. Nguyen

A Carleman-based numerical method for quasilinear elliptic equations with over-determined boundary data and applications

• DOI: 10.1016/j.camwa.2022.08.032
• 发表时间: 2021-08
• 期刊: Comput. Math. Appl.
• 作者: T. Le;L. Nguyen;H. Tran

Reconstructing a space-dependent source term via the quasi-reversibility method

• DOI: 10.48550/arxiv.2210.09112
• 发表时间: 2022-10
• 期刊: ArXiv
• 作者: L. Nguyen;Huong T. Vu