Two-level preconditioners for regularized inverse problems. I: Theory. (English) Zbl 0941.65056

Hanke, Martin; Vogel, Curtis R.

Numer. Math. 83, No. 3, 385-402 (1999).

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.

Reviewer: R.Lepp (Tallinn)

MSC:

65J10	Numerical solutions to equations with linear operators
47A50	Equations and inequalities involving linear operators, with vector unknowns
65J22	Numerical solution to inverse problems in abstract spaces
65J20	Numerical solutions of ill-posed problems in abstract spaces; regularization

Keywords:

inverse problems; additive and multiplicative Schwarz preconditioners; regularization; operator equation; compact operator; Hilbert space; multigrid methods; condition number