Integration of Randomized Methods and Fast and Reliable Matrix Computations

随机方法与快速可靠的矩阵计算的集成

基本信息

批准号：
2111007
负责人：
Jianlin Xia
金额：
$ 27万
依托单位：
Purdue University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2111007&HistoricalAwards=false
关键词：
Integration Randomized Methods Fast Reliable

项目摘要

In modern scientific computing, engineering simulations, and data analysis, the complexity of numerical problems and the scale of data sizes pose unique challenges to matrix computations. The demand for efficiency and reliability continues to grow, and in the meantime, researchers increasingly desire algorithms that are convenient to use. Randomized algorithms are not only convenient and fast to apply, but also powerful in the sense that they can sometimes extract surprisingly useful information from challenging situations that are otherwise very difficult to handle. This project will integrate a wide variety of randomized techniques and a sequence of novel matrix algorithms so as to build a comprehensive framework for fast, reliable, and flexible randomized matrix computations. The project will bridge the gap between convenient randomized ideas and various challenging numerical tasks. Innovative randomized methods will be designed to extract valuable information in numerical computations and also to guide algorithm design and parameter tuning. The project will help effectively make randomized strategies more widely accessible to broader scientific communities. It will help introduce novel randomized algorithms into various numerical analysis fields and can also significantly improve the efficiency and reliability of many practical computational tasks in application fields such as data science, image processing, geosciences, and engineering. The research results will be widely disseminated via multiple channels. The project will give students a nice platform to gain knowledge in different areas such as numerical analysis, statistics, and data analysis. Relevant course materials will be developed. Open-source software packages will be designed.The research will seamlessly integrate a wide variety of randomized techniques and a sequence of novel matrix algorithms. The research will result in a series of novel randomized methods for computations such as low-rank approximation, data-sparse preconditioning, and eigenvalue solution. Rigorous theories will be given to understand the effectiveness of the proposed methods and also to establish connections between randomized algorithms and various challenging numerical tasks. Unlike the usual compromise between efficiency and reliability in many randomized strategies, the integration of multiple randomized techniques and novel matrix computations can achieve the combined benefits of flexibility, convenience, efficiency, and also reliability. The studies can further help uncover intrinsic matrix properties that can be used to design robust numerical algorithms for handling difficult situations. The new analysis and randomized methods will help make matrix computations better meet the rapidly emerging challenges of modern computational tasks.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.

在现代科学计算，工程模拟和数据分析中，数值问题的复杂性和数据规模对矩阵计算构成了独特的挑战。对效率和可靠性的需求不断增长，与此同时，研究人员越来越希望使用方便使用的算法。随机算法不仅是方便且快速应用的，而且在某种意义上，它们有时可以从挑战性的情况下提取出令人惊讶的有用信息，而这些信息原本很难处理。该项目将集成多种随机技术和一系列新型矩阵算法，从而为快速，可靠和灵活的随机矩阵计算构建综合框架。该项目将弥合方便的随机思想和各种具有挑战性的数字任务之间的差距。创新的随机方法将旨在在数值计算中提取有价值的信息，并指导算法设计和参数调整。该项目将有助于使更广泛的科学社区更广泛地使随机策略更广泛地访问。它将有助于将新颖的随机算法引入各种数值分析字段，还可以显着提高应用程序领域中许多实用计算任务的效率和可靠性，例如数据科学，图像处理，地球科学和工程。研究结果将通过多个通道广泛传播。该项目将为学生提供一个很好的平台，以在不同领域（例如数值分析，统计和数据分析）中获得知识。相关课程材料将开发。将设计开源软件包。研究将无缝整合多种随机技术和一系列新型矩阵算法。这项研究将导致一系列新型的随机方法，用于计算，例如低级别近似，数据范围的预处理和特征值解决方案。将有严格的理论了解拟议方法的有效性，并在随机算法和各种具有挑战性的数值任务之间建立联系。与许多随机策略中效率和可靠性之间的通常折衷不同，多种随机技术和新型矩阵计算的整合可以实现灵活性，便利性，效率以及可靠性的综合益处。这些研究可以进一步有助于发现固有的矩阵特性，这些特性可用于设计用于处理困难情况的强大数值算法。新的分析和随机方法将有助于使矩阵计算更好地应对现代计算任务的快速促进的挑战。该奖项反映了NSF的法定任务，并且使用基金会的知识分子优点和更广泛的影响审查标准，认为值得通过评估来获得支持。