Two-level preconditioners for regularized inverse problems. I: Theory
For the operator equation \({\mathcal A}u= b\) with \({\mathcal A}={\mathcal K}^*{\mathcal K}+\alpha{\mathcal L}\) the two Schwarz preconditioners (additive and multiplicative) are compared, where \({\mathcal K}\) is a compact operator in a Hilbert space and \({\mathcal L}\) is positive definite with closed range and \(\alpha>0\). The operators \({\mathcal K}^*{\mathcal K}\) and \({\mathcal L}\) have drastically different spectral properties, inherited by their discretizations what in turn accounts for the ineffectiveness of multigrid methods. In this paper the equation \({\mathcal A}u= b\) is decomposed using a \(2\times 2\) block matrix system and from this system two preconditioned matrices are defined: Jacobi-like additive and Gauss-Seidel-like multiplicative preconditioned Schwarz ones. It is shown that the additive Schwarz preconditioner significantly increases the condition number whereas the multiplicative one improves conditioning.
additive and multiplicative Schwarz preconditioners
scientific article; zbMATH DE number 1355439
scientific article