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

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.