  We propose a method for the construction of preconditioners of
parameter-dependent matrices for the solution of large systems of
parameter-dependent equations. The proposed method is an interpolation of the
matrix inverse based on a projection of the identity matrix with respect to the
Frobenius norm. Approximations of the Frobenius norm using random matrices are
introduced in order to handle large matrices. The resulting statistical
estimators of the Frobenius norm yield quasi-optimal projections that are
controlled with high probability. Strategies for the adaptive selection of
interpolation points are then proposed for different objectives in the context
of projection-based model order reduction methods: the improvement of
residual-based error estimators, the improvement of the projection on a given
reduced approximation space, or the recycling of computations for sampling
based model order reduction methods.
