CAREER: Mathematical Modeling from Data to Insights and Beyond

该项目将为数据驱动的应用开发分析和计算工具。特别是，分析工具将有望为如何比目前的实践更有效地获取数据提供理论指导。为了从数据中检索有用的信息，将研究数值方法，重点是保证收敛和算法加速。由于与数据科学和信息技术合作者的密切互动，研究者将确保所提议的研究的实用性，从而产生真正的影响。研究者还将致力于数据科学领域的各种推广活动。例如，她将发起一个由学生、教师和领域专家组成的本地网络，以发展数学与工业之间的紧密联系，并拓宽数学专业学生的就业机会。这一举措将对整个数学科学界产生积极影响。此外，她还将与德克萨斯大学达拉斯分校的多元化奖学金项目合作，倡导将数学建模融入K-16教育，以接触数学/科学教师。该项目解决了从数据中提取洞察力和在“大数据”时代培养下一代的重要问题。该研究侧重于从有限数量的测量中恢复信号/图像，其中“有限”是指可以采集或传输的数据量受到技术或经济限制的事实。当数据不足时，通常需要来自应用程序领域的额外信息来构建数学模型，然后使用数值方法。在这个项目中要探索的问题包括：(1)从数据中提取见解的过程有多困难？(2)建立数学模型应如何考虑合理的假设？(3)如何设计一种高效的算法来寻找模型解？更重要的是，将引入从见解到数据的反馈循环，即(4)如何改进数据获取，使信息变得更容易检索？由于这些问题模拟了数学建模的标准程序，因此本研究提供了大量的说明性示例，以丰富数学建模的教育。事实上，这个职业奖的教育目标之一是提倡将数学建模融入到K-16教育中，以便学生在早期培养解决问题的能力。此外，拟议的研究需要与商业、工业和政府（BIG）领域的专家密切互动，这些领域是现实世界问题的来源。这一要求有助于实现另一个教育目标，即通过为学生提供解决大问题的成功方法和大劳动力技能的充分培训，促进大就业。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。

This project will develop both analytical and computational tools for data-driven applications. In particular, analytical tools will hold great promise to provide theoretical guidance on how to acquire data more efficiently than current practices. To retrieve useful information from data, numerical methods will be investigated with emphasis on guaranteed convergence and algorithmic acceleration. Thanks to close interactions with collaborators in data science and information technology, the investigator will ensure the practicability of the proposed research, leading to a real impact. The investigator will also devote herself to various outreach activities in the field of data science. For example, she will initiate a local network of students, faculty members, and domain experts to develop close ties between mathematics and industry as well as to broaden career opportunities for mathematics students. This initiative will have a positive impact on the entire mathematical sciences community. In addition, she will advocate for the integration of mathematical modeling into K-16 education by collaborating with The University of Texas at Dallas Diversity Scholarship Program to reach out to mathematics/sciences teachers.This project addresses important issues in extracting insights from data and training the next generation in the "big data" era. The research focuses on signal/image recovery from a limited number of measurements, in which "limited" refers to the fact that the amount of data that can be taken or transmitted is limited by technical or economic constraints. When data is insufficient, one often requires additional information from the application domain to build a mathematical model, followed by numerical methods. Questions to be explored in this project include: (1) how difficult is the process of extracting insights from data? (2) how should reasonable assumptions be taken into account to build a mathematical model? (3) how should an efficient algorithm be designed to find a model solution? More importantly, a feedback loop from insights to data will be introduced, i.e., (4) how to improve upon data acquisition so that information becomes easier to retrieve? As these questions mimic the standard procedure in mathematical modeling, the proposed research provides a plethora of illustrative examples to enrich the education of mathematical modeling. In fact, one of this CAREER award's educational objectives is to advocate the integration of mathematical modeling into K-16 education so that students will develop problem-solving skills in early ages. In addition, the proposed research requires close interactions with domain experts in business, industry, and government (BIG), where real-world problems come from. This requirement helps to fulfill another educational objective, that is, to promote BIG employment by providing adequate training for students in successful approaches to BIG problems together with BIG workforce skills.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.