PhD Comprehensive Exam | Alexey Smirnov, Novel approaches to total variation discretization in multiple dimensions and its applications

MS Teams (please email amgrad@uwaterloo.ca for the meeting link)

Candidate

Alexey Smirnov | Applied Mathematics, University of Waterloo

Title

Novel approaches to total variation discretization in multiple dimensions and its applications

Abstract

Total variation is an important tool in the analysis of nonlinear stability and convergence of numerical solutions of one-dimensional scalar conservation laws. However, it proved to be challenging to extend this approach to two-dimensional problems. More than 35 years ago, Goodman and LeVeque have shown that total variation diminishing (TVD) solutions of two-dimensional hyperbolic equations are at most first-order accurate. This issue has attracted a lot of attention in recent years as it remains a deep open question in the theory of hyperbolic conservation laws. In this talk, we challenge the long-standing assertion of Goodman and LeVeque. We focus on a special approach to TV discretization resulting from a full discretization of the semi-discrete Raviart-Thomas TV. We numerically establish that limited solutions of two-dimensional hyperbolic equations obtained with the discontinuous Galerkin method are TVD in means when the new definition of TV is used.

Total variation (TV) is also used as a regularization tool in imaging problems, which is motivated by its ability to recover sharp discontinuities and preserve edges. We consider the state of the art TV-minimization-based imaging algorithm (ASD-POCS) for limited angle computed. We investigate how a change in the TV discretization approach impacts the algorithm's stability to formulate an improved version of the ASD-POCS algorithm, with the aim of suppressing image artifacts due to or limited angular sampling. Finally, we demonstrate numerically that an improved algorithm has a higher capability to preserve edge details when compared to the conventional one.