Approximate Singular Value Expansions and Solutions of Ill-Posed Problems

这个项目的主要重点是开发新的和有效的计算算法来解决大规模的反问题。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)从没有侵入性程序的数据中获得的医学图像重建和（ii）多孔介质渗透率的测量，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，解决这些问题的改进方法具有重大的社会影响。应用数学专业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得到支持，以发挥他们在该项目中的作用。因此，该项目有助于培养下一代广泛的数学科学学生群体。首席研究员将扩展和增强线性代数技术，这些技术是大型反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）针对大规模欠确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）对此类大规模和样本不足问题的反演所需的包含多种正则化类型的技术进行评估。该项目的总体目标是认识到大规模问题的重要信息包含在通过降维获得的主要谱子空间中。因此，该研究汇集了使用最新线性代数技术解决的方程底层系统的分辨率和秩估计的研究，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。

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.