Regularization of inverse problems by an approximate matrix-function technique

Cipolla, Stefano, Donatelli, Marco and Durastante, Fabio (2021) Regularization of inverse problems by an approximate matrix-function technique. Numerical Algorithms, 88 (3), 1275-1308. (doi:10.1007/s11075-021-01076-y).

Record type: Article

Abstract

In this work, we introduce and investigate a class of matrix-free regularization techniques for discrete linear ill-posed problems based on the approximate computation of a special matrix-function. In order to produce a regularized solution, the proposed strategy employs a regular approximation of the Heavyside step function computed into a small Krylov subspace. This particular feature allows our proposal to be independent from the structure of the underlying matrix. If on the one hand, the use of the Heavyside step function prevents the amplification of the noise by suitably filtering the responsible components of the spectrum of the discretization matrix, on the other hand, it permits the correct reconstruction of the signal inverting the remaining part of the spectrum. Numerical tests on a gallery of standard benchmark problems are included to prove the efficacy of our approach even for problems affected by a high level of noise.

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Accepted/In Press date: 20 January 2021

e-pub ahead of print date: 11 April 2021

Published date: November 2021

Additional Information: Funding Information: Open Access funding provided by Università degli Studi di Padova within the CRUI-CARE Agreement. This work has been partially funded by the 2019 GNCS-INDAM Project “Tecniche innovative e parallele per sistemi lineari e non lineari di grandi dimensioni, funzioni ed equazioni matriciali ed applicazioni”. .

Keywords: Krylov methods, Matrix function, Regularization