Volume 61, pp. 66-91, 2024.

Software for limited memory restarted $l^p$-$l^q$ minimization methods using generalized Krylov subspaces

Alessandro Buccini and Lothar Reichel

Abstract

This paper describes software for the solution of finite-dimensional minimization problems with two terms, a fidelity term and a regularization term. The sum of the $p$-norm of the former and the $q$-norm of the latter is minimized, where $0 < p,q\leq 2$. We note that the “$p$-norm” is not a norm when $ 0< p < 1$, and similarly for the “$q$-norm”. This kind of minimization problems arises when solving linear discrete ill-posed problems, such as certain problems in image restoration. They also find applications in statistics. Recently, limited-memory restarted numerical methods that are well suited for the solution of large-scale minimization problems of this kind were described by the authors in [Adv. Comput. Math., 49 (2023), Art. 26]. These methods are based on the application of restarted generalized Krylov subspaces. This paper presents software for these solution methods.

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Key words

$\ell^p$-$\ell^q$ minimization, inverse problem, regression, iterative method

AMS subject classifications

65F10, 65R32, 90C26

Links to the cited ETNA articles

[21]	Vol. 38 (2011), pp. 233-257 Stefan Kindermann: Convergence analysis of minimization-based noise level-free parameter choice rules for linear ill-posed problems
[24]	Vol. 53 (2020), pp. 329-351 Alessandro Lanza, Monica Pragliola, and Fiorella Sgallari: Residual whiteness principle for parameter-free image restoration
[25]	Vol. 59 (2023), pp. 202-229 Alessandro Lanza, Monica Pragliola, and Fiorella Sgallari: Parameter-free restoration of piecewise smooth images

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