  In this paper we introduce an algorithm based on a sparse grid adaptive
refinement, for the approximation of the eigensolutions to parametric problems
arising from elliptic partial differential equations. In particular, we are
interested in detecting the crossing of the hypersurfaces describing the
eigenvalues as a function of the parameters. The a priori matching is followed
by an a posteriori verification, driven by a suitably defined error indicator.
At a given refinement level, a sparse grid approach is adopted for the
construction of the grid of the next level, by using the marking given by the a
posteriori indicator. Various numerical tests confirm the good performance of
the scheme.
