Collaborative Research: Randomized Numerical Linear Algebra for Large Scale Inversion, Sparse Principal Component Analysis, and Applications

在许多科学应用中，如遗传学、地球物理学、生物信息学和医学，数据正在以不断增长的速度产生。对于这些应用程序和其他数据密集型应用程序，数据集的巨大规模以及不断增长的模型复杂性带来了基本的计算挑战。最先进的推理方法已经超出了它们的适用范围，迫切需要先进的数学、计算和统计工具来提取相关信息。本研究解决了大规模反问题求解方法的迫切需要。该项目将推进反问题的工具，这些工具将与随机数值线性代数和稀疏主成分分析的新方法相结合。该项目产生的扩展工具将有能力改变大规模反问题领域，并随后受益于各种各样的应用。大规模反演新方法的发展将显著推进当前解决方案在广泛应用中的应用，如机器学习、地球物理学和遗传学。该项目将研究反问题的先进迭代方法，随机化，草图方案，以及稀疏主成分分析方法。通过加速数值方法，提供理论收敛分析，并生产一个用户友好的软件包，更广泛的科学界将能够将这些先进的工具集成到他们的应用领域。该项目为学生提供了计算数学和应用数学方面的培训机会。这些措施包括实施一门新颖的跨机构研究生课程，将三个pi在逆问题、随机线性代数和数值优化方面的专业知识结合起来，为研究生提供更广泛的机会参与及时的研究项目；与美国各地的同龄人联系；扩大我们项目中学生的多样性。项目团队和领域专家之间的协作保证了所提出的算法和软件将对实际数据产生影响。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。

In many scientific applications such as genetics, geophysics, bioinformatics, and medicine, data are being generated at ever-increasing rates. For these, and other data-intensive applications, the massive size of the data sets, as well as the growing model complexities, present fundamental computational challenges. State-of-the-art inference methods have exceeded their limits of applicability and advanced mathematical, computational, and statistical tools are urgently needed to extract relevant information. This research addresses the urgent need to advance efficient methods for computing solutions of large-scale inverse problems. This project will advance tools from inverse problems which will be merged with novel approaches from randomized numerical linear algebra, and sparse principal component analysis. The expanded tools produced by this project will have the ability to transform the field of large-scale inverse problems and subsequently benefit a wide variety of applications.The development of novel approaches for large-scale inversion will significantly advance current solutions in a wide range of applications, such as machine learning, geophysics, and genetics. This project will investigate advanced iterative methods for inverse problems, randomization, sketching schemes, as well as methods for sparse principal component analysis. By accelerating numerical methods, providing theoretical convergence analysis, and producing a user-friendly software package, the broader scientific community will be able to integrate these advanced tools within their application areas. The project offers training opportunities for students in computational and applied mathematics. These include the implementation of a novel cross-institutional graduate course, merging the expertise of the three PIs in the topics of inverse problems, randomized linear algebra, and numerical optimization, to provide a broader opportunity for graduate students to engage in timely research projects; connect with their peers across the US; and expand the diversity pool of students in our programs. Collaborations between the project team and domain experts guarantee that the proposed algorithms and software will have an impact on real 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.