Applied Numerical Mathematics
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In this work the Schur complement is analyzed and applied for preconditioning symmetric matrices. In particular we eliminate the in modulo extreme eigenvalues by computing the Schur complement with respect to a coarse subspace, which captures the corresponding eigenvectors. The potential of this method is presented as an application in state estimation, where we employ a multiscale approach based on a wavelet-Galerkin discretization. The hierarchy of arising optimization problems is solved in the framework of nested iterations. Due to the wavelet approach, for all scales we can use one coarse adapted subspace in the Schur complement method. Typically this provides not only a reduced condition number but also large gaps in the arising eigenvalue distribution, which is favorable for several iterative methods. Numerical investigations show the enormous reduction in the condition number, the necessary cg-iteration steps and, consequently, in the computation time.