Past Event: Oden Institute Seminar

A generalized MBO diffusion generated method for constrained harmonic maps

Braxton Osting, Department of Mathematics, University of Utah

Abstract

A variety of tasks in inverse problems and data analysis can be formulated as the variational problem of minimizing the Dirichlet energy of a function that takes values in a certain submanifold and possibly satisfies additional constraints. These additional constraints may be used to enforce fidelity to data or other structural constraints arising in the particular problem considered. I'll present a generalization of the Merriman-Bence-Osher (MBO) method for minimizing such a functional. I’ll give examples of how this method can be used for the geometry processing task of generating quadrilateral meshes, finding Dirichlet partitions, and constructing smooth orthogonal matrix valued functions. For this last problem, I'll prove the stability of the method by introducing an appropriate Lyapunov function, generalizing a result of Esedoglu and Otto to matrix-valued functions. I'll also state a convergence result for the method. I’ll conclude with some applications in inverse problems for manifold-valued data. This is joint work with Dong Wang, Ryan Viertel, and Todd Reeb.