Calculus of Variations and Geometric Measure Theory

E. Davoli - P. Liu

One dimensional fractional order TGV: Gamma-convergence and bi-level training scheme

created by davoli on 15 Dec 2016
modified on 04 Sep 2020


Published Paper

Inserted: 15 dec 2016
Last Updated: 4 sep 2020

Journal: Communications in Mathematical Sciences
Year: 2017


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.