α -Firmly nonexpansive operators on metric spaces
2022 | journal article; research paper. A publication with affiliation to the University of Göttingen.
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- Authors
- Bërdëllima, Arian; Lauster, Florian; Luke, D. Russell
- Abstract
- We extend to p-uniformly convex spaces tools from the analysis of fixed point iterations in linear spaces. This study is restricted to an appropriate generalization of single-valued, pointwise averaged mappings. Our main contribution is establishing a calculus for these mappings in p-uniformly convex spaces, showing in particular how the property is preserved under compositions and convex combinations. This is of central importance to splitting algorithms that are built by such convex combinations and compositions, and reduces the convergence analysis to simply verifying that the individual components have the required regularity pointwise at fixed points of the splitting algorithms. Our convergence analysis differs from what can be found in the previous literature in that the regularity assumptions are only with respect to fixed points. Indeed we show that, if the fixed point mapping is pointwise nonexpansive at all cluster points, then these cluster points are in fact fixed points, and convergence of the sequence follows. Additionally, we provide a quantitative convergence analysis built on the notion of gauge metric subregularity, which we show is necessary for quantifiable convergence estimates. This allows one for the first time to prove convergence of a tremendous variety of splitting algorithms in spaces with curvature bounded from above.
- Issue Date
- 2022
- Journal
- Journal of Fixed Point Theory and Applications
- Project
- SFB 1456: Mathematik des Experiments: Die Herausforderung indirekter Messungen in den Naturwissenschaften
SFB 1456 | Cluster B | B01: Mathematics of atomic orbital tomography
SFB 1456 | Cluster C | C02: Stochastic computed tomography: theory and algorithms for single-shot X-FEL imaging - Organization
- Institut für Numerische und Angewandte Mathematik
- Working Group
- RG Luke (Continuous Optimization, Variational Analysis and Inverse Problems)
- ISSN
- 1661-7738; 1661-7746
- Language
- English