  The performance of projection-based model order reduction methods for solving
parameter-dependent systems of equations highly depends on the properties of
the operator, which can be improved by preconditioning. In this paper we
present strategies to construct a parameter-dependent preconditioner by an
interpolation of operator's inverse. The interpolation is obtained by
minimizing a discrepancy between the (preconditioned) operator and the matrix
defining the metric of interest. The discrepancy measure is chosen such that
its minimization can be efficiently performed online for each parameter value
by the solution of a small least-squares problem. Furthermore, we show how to
tune the discrepancy measure for improving the quality of Petrov-Galerkin
projection or residual-based error estimation. This paper also addresses
preconditioning for the randomized model order reduction methods from
[Balabanov and Nouy 2019, Part I]. Our methodology can be readily used for
efficient and stable solution of ill-conditioned parametric systems and an
effective error estimation/certification without the need to estimate expensive
stability constants. The proposed approach involves heavy computations in both
offline and online stages that are circumvented by random sketching. The norms
of high-dimensional matrices and vectors are estimated by l2-norms of their
low-dimensional images, called sketches, through random embeddings. For this we
extend the framework from [Balabanov and Nouy 2019, Part I] to random
embeddings of operators.
