4.5 Article

A DIFFUSION GENERATED METHOD FOR ORTHOGONAL MATRIX-VALUED FIELDS

Journal

MATHEMATICS OF COMPUTATION
Volume 89, Issue 322, Pages 515-550

Publisher

AMER MATHEMATICAL SOC
DOI: 10.1090/mcom/3473

Keywords

Allen-Cahn equation; Ginzburg-Landau equation; Merriman-Bence-Osher (MBO) diffusion generated method; constrained harmonic map; orthogonal matrix-valued field

Funding

  1. NSF DMS [16-19755, 17-52202]

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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 MerrimanBence-Osher (MBO) diffusion generated method 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.

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