Identifying unknown differential equations from given discrete time dependent data is a challenging problem. A small amount of noise can make the recovery unstable; nonlinearity and differential equations with varying coefficients add complexity to the problem. We propose a new direction based on the fundamental idea of convergence analysis of numerical PDE schemes. The new algorithm is explored for data with non-periodic boundary conditions, noisy data and PDEs with varying coefficients. (Joint work with Wenjing Liao and Yingjie Liu.) Second part of the talk will present a new framework for lattice pattern representation and comparison, motivated by the problems of lattice mixture identification and grain irregularity detection. We define new scale and shape descriptors, which considerably reduce the size of equivalence classes of lattice bases. We construct the lattice space based on the equivalent descriptors and define a metric to accurately quantify the visual similarities and differences between lattices. (joint work with Yuchen He.) If time permits, this talk will include variational imaging models, and new developments in fast algorithms for higher order variational imaging models, such as Euler’s Elastica energy in imaging models.
Dr. Sung Ha Kang works in applied mathematics, specifically in variational approaches to image processing and related problems such as path optimization, fast algorithm design and data analysis. Following a PhD in Applied Mathematics from UCLA in 2002, she held a tenure track Assistant Professorship at University of Kentucky. In 2008 she joined Georgia Tech, where her research has been supported through NSF and the Simons Foundation. Currently as a professor of Mathematics, she holds leadership roles in the GT-Math and Applications Portal, and the Computational Science and Engineering graduate program.