Convergence of spectral discretization for the flow of diffeomorphisms
<|reference_start|>Convergence of spectral discretization for the flow of diffeomorphisms: The Large Deformation Diffeomorphic Metric Mapping (LDDMM) or flow of diffeomorphism is a classical framework in the field of shape spaces and is widely applied in mathematical imaging and computational anatomy. Essentially, it equips a group of diffeomorphisms with a right-invariant Riemannian metric, which allows to compute (Riemannian) distances or interpolations between different deformations. The associated Euler--Lagrange equation of shortest interpolation paths is one of the standard examples of a partial differential equation that can be approached with Lie group theory (by interpreting it as a geodesic ordinary differential equation on the Lie group of diffeomorphisms). The particular group $\mathcal D^m$ of Sobolev diffeomorphisms is by now sufficiently understood to allow the analysis of geodesics and their numerical approximation. We prove convergence of a widely used Fourier-type space discretization of the geodesic equation. It is based on a new regularity estimate: We prove that geodesics in $\mathcal D^m$ preserve any higher order Sobolev regularity of their initial velocity.<|reference_end|>
@article{wirth2024convergence,
    title={Convergence of spectral discretization for the flow of diffeomorphisms},
    author={Benedikt Wirth},
    journal={arXiv preprint arXiv:2410.06788},
    year={2024},
    archivePrefix={arXiv},
    eprint={2410.06788},
    primaryClass={math.NA cs.NA}
}