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   Abstract This paper addresses the approximation of fractional harmonic maps. Besides a unit-length constraint, one has to tackle the difficulty of nonlocality. We establish weak compactness results for critical points of the fractional Dirichlet energy on unit-length vector fields. We devise and analyze numerical methods for the approximation of various partial differential equations related to fractional harmonic maps. The compactness results imply the convergence of numerical approximations. Numerical examples on spin chain dynamics and point defects are presented to demonstrate the effectiveness of the proposed methods. 

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   Wang, Yu; Guo, Minghao; Solomon, Justin (July 2023, ACM Transactions on Graphics)

   Computation of injective (or inversion-free) maps is a key task in geometry processing, physical simulation, and shape optimization. Despite being a longstanding problem, it remains challenging due to its highly nonconvex and combinatoric nature. We propose computation ofvariational quasi-harmonic mapsto obtain smooth inversion-free maps. Our work is built on a key observation about inversion-free maps: A planar map is a diffeomorphism if and only if it is quasi-harmonic and satisfies a special Cauchy boundary condition. We hence equate the inversion-free mapping problem to an optimal control problem derived from our theoretical result, in which we search in the space of parameters that define an elliptic PDE. We show that this problem can be solved by minimizing within a family of functionals. Similarly, our discretized functionals admit exactly injective maps as the minimizers, empirically producing inversion-free discrete maps of triangle meshes. We design efficient numerical procedures for our problem that prioritize robust convergence paths. Experiments show that on challenging examples our methods can achieve up to orders of magnitude improvement over state-of-the-art, in terms of speed or quality. Moreover, we demonstrate how to optimize a generic energy in our framework while restricting to quasi-harmonic maps. 

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   Ouyang, Charles (January 2023, Geometry & Topology)