Structured Condition Numbers of Structured Tikhonov Regularization Problem and their Estimations

TL;DR

This paper analyzes structured condition numbers for Tikhonov regularization with matrices like Toeplitz, Hankel, Vandermonde, and Cauchy, providing explicit formulas, estimation algorithms, and demonstrating their effectiveness through numerical tests.

Contribution

It introduces explicit structured condition numbers for Tikhonov regularization and develops fast algorithms for their estimation, improving perturbation analysis accuracy.

Findings

Structured condition numbers are smaller than unstructured ones.

Structured mixed condition numbers provide sharper bounds.

The proposed estimation algorithms are reliable and efficient.

Abstract

Both structured componentwise and structured normwise perturbation analysis of the Tikhonov regularization are presented. The structured matrices under consideration include: Toeplitz, Hankel, Vandermonde, and Cauchy matrices. Structured normwise, mixed and componentwise condition numbers for the Tikhonov regularization are introduced and their explicit expressions are derived. For the general linear structure, we prove the structured condition numbers are smaller than their corresponding unstructured counterparts based on the derived expressions. By means of the power method and small sample condition estimation, the fast condition estimation algorithms are proposed. Our estimation methods can be integrated into Tikhonov regularization algorithms that use the generalized singular value decomposition (GSVD). The structured condition numbers and perturbation bounds are tested on some numerical examples and compared with their unstructured counterparts. Our numerical examples demonstrate that the structured mixed condition numbers give sharper perturbation bounds than existing ones, and the proposed condition estimation algorithms are reliable.