A multigrid method to solve large scale Sylvester equations

We consider the Sylvester equation AX - XB + C = 0, where the matrix C is an element of R-nxm is of low rank and the spectra of A is an element of R-nxn and B is an element of R-mxm are separated by a line. The solution X can be approximated in a data-sparse format, and we develop a multigrid algorithm that computes the solution in this format. For the multigrid method to work, we need a hierarchy of discretizations. Here the matrices A and B each stem from the discretization of a partial differential operator of elliptic type. The algorithm is of complexity O(n + m), or, more precisely, if the solution can be represented with (n + m)k data (k similar to log(n + m)), then the complexity of the algorithm is O(( n + m) k(2)).