Variational Methods for Materials and Imaging

Carnegie Mellon University, Pittsburgh PA

Investigators

Abstract

The objective of this project is the pursuit of a mathematically rigorous understanding of emerging nonlinear phenomena in physical and technological applications, ranging from the analysis of instabilities in materials science to image analysis in computer vision. The project will provide research training opportunities to the next generation of leaders in applied analysis cognizant of contemporary mathematical areas that underscore interdisciplinary challenges at the interface of mathematical sciences with computer science, engineering, and physical sciences. The two main themes of this project are Variational Problems for Materials and Variational Problems for Imaging. What unifies these topics is that underlying energies involve higher order derivatives in spaces with discontinuous admissible fields, multiple scales interact, bulk and surface energies compete, and degeneracy of usually expected properties prevail. These prevent the use of well understood mathematical theories and require the introduction of innovative mathematical tools. The project will provide a mathematical foundation for the understanding of aspects of materials, including materials defects (dislocations), epitaxy, micromagnetic and magnetoelastic materials, and composite materials (homogenization). Analytical tools combined with contemporary multilevel learning schemes (machine learning) will be used in imaging to address denoising of images and edge detection, recolorization, image segmentation and registration. 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.