A Multigrid Method to Solve Large Scale Sylvester Equations

Abstract

We consider the Sylvester equation $AX-XB+C=0$, where the matrix $C\in\mathbb{R}^{n\times m}$ is of low rank and the spectra of $A\in\mathbb{R}^{n\times n} $ and $B\in\mathbb{R}^{m\times m}$ 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 $\mathcal{O}(n+m)$, or, more precisely, if the solution can be represented with $(n+m)k$ data ($k\sim \log(n+m)$), then the complexity of the algorithm is $\mathcal{O}((n+m)k^{2})$.