  Mutual Coupling (MC) is an unavoidable feature in Reconfigurable Intelligent
Surfaces (RISs) with sub-wavelength inter-element spacing. Its inherent
presence naturally leads to non-local RIS structures, which can be efficiently
described via non-diagonal phase shift matrices. In this paper, we focus on
optimizing MC in RIS-assisted multi-user MIMO wireless communication systems.
We particularly formulate a novel problem to jointly optimize active and
passive beamforming as well as MC in a physically consistent manner. To
characterize MC, we deploy scattering parameters and propose a novel approach
to optimize them through an offline optimization method, rather than optimizing
MC on the fly. Our numerical results showcase that the system performance
increases with the proposed MC optimization, and this improvement is achievable
without the need for optimizing MC on-the-fly, which can be rather cumbersome.
