Abstract
Tikhonov regularization of linear discrete ill-posed problems often is applied with a finite difference regularization operator that approximates a low-order derivative. These operators generally are represented by a banded rectangular matrix with fewer rows than columns. They therefore cannot be applied in iterative methods that are based on the Arnoldi process, which requires the regularization operator to be represented by a square matrix. This paper discusses two approaches to circumvent this difficulty: zero-padding the rectangular matrices to make them square and extending the rectangular matrix to a square circulant. We also describe how to combine these operators by weighted averaging and with orthogonal projection. Applications to Arnoldi and Lanczos bidiagonalization-based Tikhonov regularization, as well as to truncated iteration with a range-restricted minimal residual method, are presented.
Document Type
Article
Publication Date
5-7-2009
Funding Information
Supported in part by NSF under grant DMS-0411502.
Repository Citation
Reichel, Lothar and Ye, Qiang, "Simple Square Smoothing Regularization Operators" (2009). Mathematics Faculty Publications. 15.
https://uknowledge.uky.edu/math_facpub/15
Notes/Citation Information
Published in Electronic Transactions on Numerical Analysis, v. 33, p. 63-83.
Copyright © 2009, Kent State University.
The copyright holder has granted the permission for posting the article here.