  The solution of parameter-dependent linear systems, by classical methods,
leads to an arithmetic effort that grows exponentially in the number of
parameters. This renders the multigrid method, which has a well understood
convergence theory, infeasible. A parameter-dependent representation, e.g., a
low-rank tensor format, can avoid this exponential dependence, but in these it
is unknown how to calculate the inverse directly within the representation. The
combination of these representations with the multigrid method requires a
parameter-dependent version of the classical multigrid theory and a
parameter-dependent representation of the linear system, the smoother, the
prolongation and the restriction. A derived parameter-dependent version of the
smoothing property, fulfilled by parameter-dependent versions of the Richardson
and Jacobi methods, together with the approximation property prove the
convergence of the multigrid method for arbitrary parameter-dependent
representations. For a model problem low-rank tensor formats represent the
parameter-dependent linear system, prolongation and restriction. The smoother,
a damped Jacobi method, is directly approximated in the low-rank tensor format
by using exponential sums. Proving the smoothing property for this
approximation guarantees the convergence of the parameter-dependent method.
Numerical experiments for the parameter-dependent model problem, with bounded
parameter value range, indicate a grid size independent convergence rate.
