Variational Methods for Materials and Imaging

这个项目的目标是追求对物理和技术应用中出现的非线性现象的数学上的严格理解，从材料科学中的不稳定性分析到计算机视觉中的图像分析。该项目将为认识当代数学领域的应用分析领域的下一代领导者提供研究培训机会，这些领域突出了数学科学与计算机科学、工程和物理科学之间的跨学科挑战。该项目的两个主要主题是材料变分问题和成像变分问题。统一这些主题的是，潜在能量涉及具有不连续可容许场的空间中的高阶导数，多尺度相互作用，体能和表面能竞争，并且通常预期性质的简并性占优势。这些阻碍了人们对数学理论的理解，并要求引入创新的数学工具。该项目将为理解材料的各个方面提供数学基础，包括材料缺陷(位错)、外延、微磁性和磁弹性材料以及复合材料(均质化)。分析工具与现代多层次学习方案(机器学习)将用于成像，以解决图像去噪和边缘检测、重新着色、图像分割和配准问题。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。

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.