Data driven inversion methods and image reconstruction for nonlinear media

该项目涉及开发根据外部测量确定内部材料参数的新方法，重点是雷达应用、医学成像和光学设计。在新的算法中，一个关键的创新在于利用数据本身来生成一个紧凑的波传播模型，从而实现高效的图像重建。这些算法的适用性将扩展到处理大数据集和噪声、复杂媒体以及源和接收器未配置的场景-这是现有简化模型方法的一个重大限制。此外，还将开发新的方法来重建具有非线性波传播特性的材料。通过降低计算成本，该项目中开发的方法将能够在大规模情况下改进成像。在这项研究的过程中，将对几名学生进行培训，并将促进研究中多样性和包容性的计划整合到学生招生中。这项研究将通过改进一些最有希望的直接倒置方法来克服图像重建的挑战。这包括(I)将数据驱动的降阶模型成像方法扩展到非常一般的数据集、具有非局部算子的模型和传输问题，同时建立严格的基础；(Ii)开发用于在基础正演模型是非线性的情况下确定系数的新的反演方法；(Iii)在混合方法中成像一般的线性和非线性各向异性。新的降阶模型方法将被开发来处理大数据集和噪声数据集，其中源和接收器没有配置的问题，以及与色散的成像问题。逆Born级数将适用于处理正向问题是非线性的反演，其应用将依赖于对正向级数的描述，并将涉及收敛分析。具有一般各向异性磁感应成像的磁声层析成像传输方程系统的适定性将使用仔细定义的流入边界条件来确定。该奖项反映了NSF的法定使命，并已通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project involves the development of new methods for determining internal material parameters from external measurements, with a focus on radar applications, medical imaging, and optical design. In the new algorithms, a key innovation lies in using the data itself to generate a compact model for wave propagation, resulting in highly efficient image reconstruction. The applicability of these algorithms will be expanded to handle large and noisy data sets, complex media, and scenarios where sources and receivers are not collocated—a significant limitation in existing reduced model approaches. Furthermore, novel methodologies will be developed for the reconstruction of materials exhibiting nonlinear wave propagation characteristics. By reducing computational costs, the methods developed in this project will enable improved imaging in large-scale situations. Several students will be trained over the course of this research, and plans to promote diversity and inclusion in research are integrated into student recruitment.This research will overcome challenges to image reconstruction by improving some of the most promising direct methods for inversion. This includes (i) expanding data driven reduced order model imaging methods to very general classes of data sets, models with nonlocal operators and transmission problems, while at the same time establishing a rigorous foundation, (ii) developing novel inversion methods for coefficient determination when the underlying forward model is nonlinear and (iii) imaging general linear and nonlinear anisotropy in the hybrid method Magnetoacoustic Tomography with Magnetic Induction. The novel reduced order model approaches will be developed to handle large and noisy data sets, problems in which source and receiver are not collocated, and imaging problems with dispersion. The inverse Born series will be adapted to handle inversion where the forward problem is nonlinear, and its application will rely on the description of the forward series and will involve a convergence analysis. The well-posedness of the system of transport equations for Magnetoacoustic Tomography with Magnetic Induction imaging of general anisotropy will be established using carefully defined inflow boundary conditions.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.