Approximate Singular Value Expansions and Solutions of Ill-Posed Problems

该项目的主要重点是开发用于解决大规模反问题的新颖有效的计算算法。与本研究计划的数学发展相关的各种问题的例子出现在（i）从无创程序采集的数据进行医学图像重建和（ii）测量多孔介质的渗透性，这对于预测地下流体和污染物的流动和运输非常重要。 在计算成本和效率方面，用于解决这些问题的改进方法具有显著的社会影响。应用数学的学生将接受正在开发的技术培训，来自代表性不足群体的研究生和本科生将在本项目中获得支持。因此，该项目有助于培养下一代数学科学领域的广大学生。首席研究员将扩展和增强线性代数技术，这是大规模逆问题求解器中固有的。该项目的具体目标包括：（i）过采样迭代Krylov算法的数学和计算分析;（ii）大规模欠定问题的随机奇异值分解估计的混合预处理的开发和分析;以及（iii）评估将这种大规模和欠采样问题的反演所需的多种正则化类型结合在一起的技术。该项目的目标是认识到，一个大规模的问题的重要信息包含在通过降维获得的占主导地位的谱子空间。因此，该研究汇集了使用最新线性代数技术求解的基本方程组的分辨率和秩估计研究，并为数据和模型压缩提供了理论上合理的下采样。该奖项反映了NSF的法定使命，并被认为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来支持。

The primary focus of this project is on the development of novel and efficient computational algorithms for the solution of large scale inverse problems. Examples of the kinds of problems that are relevant for the planned mathematical developments for this study arise in (i) medical image reconstruction from data acquired without invasive procedures and (ii) measurements of the permeability of a porous medium which is of great importance for predicting flow and transport of fluids and contaminants in the subsurface. Improved approaches for the solution of such problems, both in terms of computational cost and efficiency, have significant societal impact. Students in applied mathematics will be trained in the techniques that are being developed and graduate and undergraduate students from underrepresented groups will be supported for their roles in this project. As such, the project contributes to the training of the next generation of a broad group of students in the mathematical sciences. The principal investigator will extend and enhance the linear algebra techniques that are inherent within the solvers for the large scale inverse problems. Specific goals for the project include the (i) mathematical and computational analysis of oversampled iterative Krylov algorithms; (ii) the development and analysis of hybrid preconditioning of randomized singular value decomposition estimates for large scale under-determined problems, and (iii) assessment of techniques that incorporate multiple regularization types that are required for the inversion of such large scale and under sampled problems. The encompassing goal of this project is the recognition that significant information of a large scale problem is contained within the dominant spectral subspace obtained via dimension reduction. Thus the research brings together studies of resolution and rank estimation for the underlying systems of equations that are solved using the latest linear algebra techniques and provides theoretically-justified down-sampling for data and model compression.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.