A modularized algorithmic framework for interface related optimization problems using characteristic functions

TL;DR

This paper introduces a modular algorithmic framework for interface-related optimization problems using characteristic functions, demonstrating convergence and effectiveness through numerical experiments.

Contribution

It presents a novel modular framework for solving interface optimization problems with convergence guarantees, applicable to various fields.

Findings

Convergence of the proposed iterative scheme to local minimizers.

Effective numerical results demonstrating the framework's applicability.

The framework's versatility across different interface-related problems.

Abstract

In this paper, we consider the algorithms and convergence for a general optimization problem, which has a wide range of applications in image segmentation, topology optimization, flow network formulation, and surface reconstruction. In particular, the problem focuses on interface related optimization problems where the interface is implicitly described by characteristic functions of the corresponding domains. Under such representation and discretization, the problem is then formulated into a discretized optimization problem where the objective function is concave with respect to characteristic functions and convex with respect to state variables. We show that under such structure, the iterative scheme based on alternative minimization can converge to a local minimizer. Extensive numerical examples are performed to support the theory.