Smooth Solutions to Linear Inequalities, Constrained Sobolev interpolation, and Trace Problems on Domains

在工程、科学、社会学、经济学、工业甚至医学等领域中，求解带参数的线性不等式组是非常重要的(例如为了提高疗效和安全性而设计的最优药物组合)。约束内插的高效算法可以应用于Twitter数据等大数据的分析。这一努力将有助于促进跨学科研究和改善当前的计算基础设施。将为博士后、研究生和本科生提供研究机会。将讨论线性不等系统的光滑解。文中还将讨论如何利用光滑函数和任意区域上的延拓问题来实现约束插补的有效算法。将要开发的方法将围绕共同的数学主题，如Calderon-Zygmund分解、良好分离的对和凸优化。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Solving a system of linear inequalities with parameters is essential in various applications, such as engineering, science, sociology, economics, industry, and even medicine (such as the optimal combination of drugs for efficacy and safety). The efficient algorithms on constrained interpolation can be applied to analyze big data such as Twitter data. This endeavor will help promote interdisciplinary research and improve the current computing infrastructure. Research opportunities will be provided for postdocs, graduate students, and undergraduate students.Smooth solutions to systems of linear inequalities will be addressed. Attention will also be paid to efficient algorithms for constrained interpolation by smooth functions and extension questions on arbitrary domains. The methods to be developed will revolve around common mathematical themes, such as Calderon-Zygmund decomposition, well-separated pairs, and convex optimization.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.