An MBO scheme for minimizing the graph Ohta-Kawasaki functional

TL;DR

This paper introduces a graph-based MBO scheme for minimizing the Ohta-Kawasaki functional, enabling pattern formation modeling on graphs with theoretical convergence guarantees and practical numerical minimization capabilities.

Contribution

It extends PDE-inspired graph methods to pattern formation models and establishes convergence, comparison principles, and numerical minimization for the graph Ohta-Kawasaki functional.

Findings

Lyapunov functionals $ ext{Γ}$-converge to the Ohta-Kawasaki functional.

Existence of graphs where the scheme reduces to a standard MBO scheme.

Numerical computation of approximate minimizers of the functional.

Abstract

We study a graph based version of the Ohta-Kawasaki functional, which was originally introduced in a continuum setting to model pattern formation in diblock copolymer melts and has been studied extensively as a paradigmatic example of a variational model for pattern formation. Graph based problems inspired by partial differential equations (PDEs) and varational methods have been the subject of many recent papers in the mathematical literature, because of their applications in areas such as image processing and data classification. This paper extends the area of PDE inspired graph based problems to pattern forming models, while continuing in the tradition of recent papers in the field. We introduce a mass conserving Merriman-Bence-Osher (MBO) scheme for minimizing the graph Ohta-Kawasaki functional with a mass constraint. We present three main results: (1) the Lyapunov functionals associated with this MBO scheme $\Gamma$-converge to the Ohta-Kawasaki functional (which includes the standard graph based MBO scheme and total variation as a special case); (2) there is a class of graphs on which the Ohta-Kawasaki MBO scheme corresponds to a standard MBO scheme on a transformed graph and for which generalized comparison principles hold; (3) this MBO scheme allows for the numerical computation of (approximate) minimizers of the graph Ohta-Kawasaki functional with a mass constraint.