  This paper considers the application of reconfigurable intelligent surfaces
(RISs) (a.k.a. intelligent reflecting surfaces (IRSs)) to assist multiuser
multiple-input multiple-output (MIMO) uplink transmission from several
multi-antenna user terminals (UTs) to a multi-antenna base station (BS). For
reducing the signaling overhead, only partial channel state information (CSI),
including the instantaneous CSI between the RIS and the BS as well as the
slowly varying statistical CSI between the UTs and the RIS, is exploited in our
investigation. In particular, an optimization framework is proposed for jointly
designing the transmit covariance matrices of the UTs and the RIS phase shift
matrix to maximize the system global energy efficiency (GEE) with partial CSI.
We first obtain closed-form solutions for the eigenvectors of the optimal
transmit covariance matrices of the UTs. Then, to facilitate the design of the
transmit power allocation matrices and the RIS phase shifts, we derive an
asymptotically deterministic equivalent of the objective function with the aid
of random matrix theory. We further propose a suboptimal algorithm to tackle
the GEE maximization problem with guaranteed convergence, capitalizing on the
approaches of alternating optimization, fractional programming, and sequential
optimization. Numerical results substantiate the effectiveness of the proposed
approach as well as the considerable GEE gains provided by the RIS-assisted
transmission scheme over the traditional baselines.
