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Mathematics of Computation

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A diffusion generated method for orthogonal matrix-valued fields

Authors: Braxton Osting and Dong Wang
Journal: Math. Comp. 89 (2020), 515-550
MSC (2010): Primary 35K93, 35K05, 65M12
Published electronically: September 24, 2019
MathSciNet review: 4044441
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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 Merriman-Bence-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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Additional Information

Braxton Osting
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112

Dong Wang
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112

Keywords: Allen-Cahn equation, Ginzburg-Landau equation, Merriman-Bence-Osher (MBO) diffusion generated method, constrained harmonic map, orthogonal matrix-valued field
Received by editor(s): December 13, 2017
Received by editor(s) in revised form: August 1, 2018, March 27, 2019, and May 17, 2019
Published electronically: September 24, 2019
Additional Notes: The first author was partially supported by NSF DMS 16-19755 and 17-52202.
Article copyright: © Copyright 2019 American Mathematical Society