We aim for a solid theory of nonlocal models stimulating novel research directions, algorithms, and applications. New consistency results and error estimates
will make nonlocal methods more accessible and reliable in critical processing tasks. A key problem we plan to tackle is the open question to unify different theories of state of the art models for discrete respectively continuum nonlocal methods.
The main objective of work package 1 is to further investigate the theory of nonlocal methods; particularly in the variational and continuum setting. Part of this investigation will be the mathematical characterization of nonlocal operators by their respective nonlinear spectral decompositions. We will investigate the conditions for the existence of eigenfunctions (calibrable sets) with various restrictions on weights in the continuum and discrete settings. The nonlinear eigenvalue problems to be examined are induced by the nonlocal TV (NLTV) functional as well as additional nonlocal one-homogeneous convex functionals, such as nonlocalvariants of the total generalized variation. We will analyse nonlocal discs (which move at constant speed under the NLTV flow) to get new theoretical insights for applications such as denoising and segmentation. Relations to Cheeger sets will be explored, as well as possible extensions of the Rayleigh principle (which appear to fail in the general case in the nonlinear setting). The nonlocal Ginzburg-Landau equation will be studied, its relations in the limit to NLTV and its possible numerical uses, based on a Gamma convergence analysis. Finally, we are interested in the convergence of optimization algorithms onto local or global minima, depending on the convexity of the defined models.