Dislocation structure of a grain boundary: Numerical result (left) and atomistic simulation result (right) Dynamics of a triple junction: Numerical result (left) and molecular dynamics result (right) Energy and Dynamics of Grain Boundaries Based on the Underlying Microstructures (PI: Prof. Yang XIANG) Grain boundaries are important interfaces between grains with different orientations in polycrystals, and their properties affect the mechanical and plastic behaviors of materials. Existing models for grain boundary dynamics neglect important microscopic structures and mechanisms, limiting their ability to describe novel phenomena such as stress-driven and shear-coupling motions. In this project, continuum models were developed to incorporate the underlying microstructure of line defects in grain boundaries, which include both grain boundary and line defect densities. These models also consider the stress field due to long-range elastic interaction between line defects. Efficient numerical methods were developed to handle the computation of nonlocal and singular stress field. Continuum models were also developed for the dynamics of triple junctions in the grain boundary network microstructure of polycrystalline materials. Geometric Landscape Analysis of Some Non-Convex Optimizations (PI: Prof. Jianfeng CAI) Non-convex optimization is a powerful tool for solving scientific and engineering problems, including low-rank approximation and deep neural network training. Simple algorithms like alternating minimization and gradient descent often work well for non-convex problems despite possible local minima. Recent research has shown that many non-convex functions arising in high-dimensional data analysis have no poor local minima, allowing efficient and effective solutions with a theoretical guarantee. In this project, the landscape of non-convex optimization arising from phase retrieval, a fundamental problem in imaging techniques, was investigated. The results reveal that a family of non-convex optimizations for phase retrieval can have a benign landscape, meaning that all local minima are global and all other critical points have negative directional curvatures. This allows for the development of new efficient algorithms with a theoretical guarantee for solving phase retrieval problems. SCIENTIFIC COMPUTING AND DATA SCIENCE 18
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