A generalized MBO diffusion generated motion for orthogonal matrix-valued fields

TL;DR

This paper introduces a generalized MBO diffusion method for finding stationary points of the Dirichlet energy in orthogonal matrix-valued fields, extending stability and convergence analysis with numerical experiments on surfaces.

Contribution

It develops a new diffusion-generated motion algorithm for orthogonal matrix fields, with proven stability, convergence, and practical implementation details.

Findings

The method effectively finds local minimizers of the energy.

The algorithm is unconditionally stable and converges in finite steps.

Numerical experiments reveal classical and new phenomena on surfaces.

Abstract

We consider the problem of finding stationary points of the Dirichlet energy for orthogonal matrix-valued fields. Following the Ginzburg-Landau approach, this energy is relaxed by penalizing the matrix-valued field when it does not take orthogonal matrix values. A generalization of the MBO diffusion generated motion is introduced that effectively finds local minimizers of this energy by iterating two steps until convergence. In the first step, as in the original method, the current matrix-valued field is evolved by the diffusion equation. In the second step, the field is pointwise reassigned to the closest orthogonal matrix, which can be computed via the singular value decomposition. We extend the Lyapunov function of Esedoglu and Otto to show that the method is non-increasing on iterates and hence, unconditionally stable. We also prove that spatially discretized iterates converge to a stationary solution in a finite number of iterations. The algorithm is implemented using the closest point method and non-uniform fast Fourier transform. We conclude with several numerical experiments on flat tori and closed surfaces, which, unsurprisingly, exhibit classical behavior from the Allen-Cahn and complex Ginzburg Landau equations, but also new phenomena.