Steve Melczer USRA project page
Contents |
Introduction
The goal of the summer USRA is to keep the student motivated and engaged in learning new mathematics, as well as productive in an accessible research project. This can involve, learning theory not normally covered in the undergraduate curriculum, seeing how the theory applies in current applications, writing up a research notes or a report, and possibly even engaging in original research.
From the theory point of view, one option is to learn Convex Analysis, aiming towards Ekeland's Variational Principle which explains how minima of variational problems behave as the problem is perturbed. (See the Dover reprint of Ekeland and Teman) These perturbed minima are important in current variational problems, such as Compressive Sensing, and Image Processing. From the research point of view, solving Hamilton-Jacobi equations using Linear Programming, in different polyhedral norms will tie together the optimization theory, some coding in MATLAB, and form an introductions to topics mentioned above. It also relates to Homogenization of metric Hamilton-Jacobi Equations, which is a current hot topic with an Application to Cloaking.
Project Schedule and Options
- Learn enough convex analysis to understand necessary and sufficient conditions for a minimum of a non-smooth convex function.
- Learn subdifferential calculus, and apply it to the shrinkage problem in general norms
- Applications/Research
- Compute Solution of special cases of the shrinkage problem
- Learn and fill out Uzawa interpretation of Bregman iteration
- Apply Bregman iteration to the TV minimization problem and to TC minimization problem.
- Implement Linear Programming for Polyhedral Hamilton-Jacobi
- Learn Convex Analysis, and write up notes on Ekeland's Variational Principle, from Applied Nonlinear Analysis, by Aubin and Ekeland.
- Expand the Notes on norms and dual norms
Mid-June
- Learn basic algorithms for Linear Programming.
- Install and use some packages packages (Matlab, cvx, etc)
- write a crude interior point solver, using linear algebra in Matlab
- apply it to the structure systems coming from Finite Differences.
July 11th
- Work on Fast_Solver_for_Fully_Nonlinear
- Work on split bregman iteration, implement it for the TV model, then try it for the TC model see Generalized_Shrinkage