Two-level preconditioners for regularized inverse problems. I: Theory
type: Item
instance of: scholarly article
title: Two-level preconditioners for regularized inverse problems. I: Theory
zbMATH Open document ID: 0941.65056
author: Martin Hanke
author: Curtis R. Vogel
published in: Numerische Mathematik
publication date: 2000-08-10T00:00:00Z
review text: 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.
reviewed by: Riho Lepp
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 47A50
Mathematics Subject Classification ID: 65J22
Mathematics Subject Classification ID: 65J20
zbMATH DE Number: 1355439
zbMATH Keywords: inverse problems
zbMATH Keywords: additive and multiplicative Schwarz preconditioners
zbMATH Keywords: regularization
zbMATH Keywords: operator equation
zbMATH Keywords: compact operator
zbMATH Keywords: Hilbert space
zbMATH Keywords: multigrid methods
zbMATH Keywords: condition number
MaRDI profile type: Publication
full work available at URL: s002110050455
OpenAlex ID: W2071485538
DOI: 10.1007/S002110050455
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description: scientific article; zbMATH DE number 1355439