Center for                           Nonlinear Analysis CNA Home People Seminars Publications Workshops and Conferences CNA Working Groups CNA Comments Form Summer Schools Summer Undergraduate Institute PIRE Cooperation Graduate Topics Courses SIAM Chapter Seminar Positions Contact Publication 16-CNA-027 One dimensional fractional order TGV: Gamma-convergence and bi-level training scheme Elisa DavoliDepartment of Mathematics University of Vienna Oskar-Morgenstern-Platz 1 1090 Vienna, Austriaelisa.davoli@univie.ac.at Pan LiuCambridge Image Analysis Department of Applied Mathematics and Theoretical Physics University of Cambridge, UKpanliu.0923@maths.cam.ac.ukAbstract: New fractional $r$-order seminorms, $TGV^r$, $r\in \mathbb{R}$, $r\geq 1$, are proposed in the one-dimensional (1D) setting, as a generalization of the integer order $TGV^k$-seminorms, $k\in\mathbb{N}$. The fractional $r$-order $TGV^r$-seminorms are shown to be intermediate between the integer order $TGV^k$-seminorms. A bilevel training scheme is proposed, where under a box constraint a simultaneous optimization with respect to parameters and order of derivation is performed. Existence of solutions to the bilevel training scheme is proved by $\Gamma$-convergence. Finally, the numerical landscape of the cost function associated to the bilevel training scheme is discussed for two numerical examples.Get the paper in its entirety as  16-CNA-027.pdf