A Unified Framework for Total Variation Regularized Optimization in Fluid Dynamics and Related Physical Systems

TL;DR

This paper introduces a unified optimization framework incorporating Total Variation regularization for physical systems like fluid dynamics, providing insights into solution properties and demonstrating effectiveness on key equations such as Navier-Stokes and Maxwell.

Contribution

It develops a generalized convex optimization approach with TV regularization for various physical systems, enhancing solution analysis and convergence understanding.

Findings

Convexity analysis of the energy functional.

Application to Navier-Stokes, Boltzmann, and Maxwell equations.

Insights into solution space and convergence for convection-dominated problems.

Abstract

An optimization framework is presented for minimizing the energy functional developed around a generalized equation governing physical systems such as fluid dynamics, particle transport, phase transition, and other related systems. The convexity of the energy functional is investigated to derive the necessary conditions for a smooth and global optimum solution. Furthermore, the Total Variation (TV) regularization term is introduced to gain insights into the solution space and convergence analysis of convection-dominated problems. We demonstrate the practical application of our method by applying it to some selected examples such as the Boltzmann, Navier-Stokes, and Maxwell equations.