Geometric Approximation and Variational Problems

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The proposed research program address problems in geometric numerical methods. Besides numerous engineering applications, accurate computational methods for approximation and estimation of geometric information can help understanding and saving lives. Numerical treatment of biomembrane problems is one example of application to life sciences. Approximation methods of manifold-valued data can be applied to diffusion tensor image data for reconstructing white matter structure of human brain; such techniques have shown promises in diagnosing psychiatric disorders. Due to the success in applications such as machine learning, signal processing, and control system, large scale numerical optimization is now considered as a key component of engineering, a patient study of optimization methods for specific geometric problems with practical relevance will contribute to the understanding of solving large scale optimization problems. The projects outlined in this research also provide interdisciplinary research and training opportunities for graduate students, and stimulate collaboration among computational mathematicians, engineers and scientists. The publicly available software implementation of our research results further facilitates such training and collaborations.A number of projects under the headline of "geometric approximation and variational problems". Extensions of the Wmincon software with applications to geometric variational problems in general relativity This work details a line of research related to geometric approximation and variational problems, including systematic studies of numerical solution of biomembranes and bilayer plate models from mechanical engineering, as well as approximation and analysis of geometric data. The projects will lead to a cross fertilization of geometry, optimization theory, computational mathematics, as well as application areas such as engineering simulation, processing of novel geometric signals, and geometric machine learning.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.