A new numerical scheme for discrete constrained total variation flows and its convergence

In this paper, we propose a new numerical scheme for a spatially discrete model of total variation flows whose values are constrained to a Riemannian manifold. The difficulty of this problem is that the underlying function space is not convex; hence it is hard to calculate a minimizer of the functio...

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Bibliographic Details
Published in:Numerische Mathematik 2020-09, Vol.146 (1), p.181-217
Main Authors: Giga, Yoshikazu, Sakakibara, Koya, Taguchi, Kazutoshi, Uesaka, Masaaki
Format: Article
Language:English
Subjects:
Constraints
Function space
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Numerical Analysis
Numerical and Computational Physics
Riemann manifold
Simulation
Theoretical
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Summary:In this paper, we propose a new numerical scheme for a spatially discrete model of total variation flows whose values are constrained to a Riemannian manifold. The difficulty of this problem is that the underlying function space is not convex; hence it is hard to calculate a minimizer of the functional with the manifold constraint even if it exists. We overcome this difficulty by “localization technique” using the exponential map and prove a finite-time error estimate. Finally, we show a few numerical results for the target manifolds S 2 and SO (3).
ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-020-01134-y