# Regularization of linear least squares problems by total bounded variation

• Volume: 2, page 359-376
• ISSN: 1292-8119

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## Abstract

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We consider the problem : (P) Minimize ${\lambda }_{2}$ over u ∈ K ∩ X, where α≥ 0, β > 0, K is a closed convex subset of L2(Ω), and the last additive term denotes the BV-seminorm of u, T is a linear operator from L2 ∩ BV into the observation space Y. We formulate necessary optimality conditions for (P). Then we show that (P) admits, for given regularization parameters α and β, solutions which depend in a stable manner on the data z. Finally we study the asymptotic behavior when α = β → 0. The regularized solutions ûβ of (P) converge to the L2 ∩ BV minimal norm solution of the unregularized problem. The rate of convergence is β½ when the minimum-norm solution û is smooth enough.

## How to cite

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Chavent, G., and Kunisch, K.. "Regularization of linear least squares problems by total bounded variation." ESAIM: Control, Optimisation and Calculus of Variations 2 (2010): 359-376. <http://eudml.org/doc/116554>.

@article{Chavent2010,
abstract = { We consider the problem : (P) Minimize $\lambda _\{2\}$ over u ∈ K ∩ X, where α≥ 0, β > 0, K is a closed convex subset of L2(Ω), and the last additive term denotes the BV-seminorm of u, T is a linear operator from L2 ∩ BV into the observation space Y. We formulate necessary optimality conditions for (P). Then we show that (P) admits, for given regularization parameters α and β, solutions which depend in a stable manner on the data z. Finally we study the asymptotic behavior when α = β → 0. The regularized solutions ûβ of (P) converge to the L2 ∩ BV minimal norm solution of the unregularized problem. The rate of convergence is β½ when the minimum-norm solution û is smooth enough. },
author = {Chavent, G., Kunisch, K.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Ill-posed inverse problems / regularization / bounded variation seminorm.; asymptotic analysis; functions of bounded variation; regularization of linear least-squares problems; necessary optimality conditions},
language = {eng},
month = {3},
pages = {359-376},
publisher = {EDP Sciences},
title = {Regularization of linear least squares problems by total bounded variation},
url = {http://eudml.org/doc/116554},
volume = {2},
year = {2010},
}

TY - JOUR
AU - Chavent, G.
AU - Kunisch, K.
TI - Regularization of linear least squares problems by total bounded variation
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 2
SP - 359
EP - 376
AB - We consider the problem : (P) Minimize $\lambda _{2}$ over u ∈ K ∩ X, where α≥ 0, β > 0, K is a closed convex subset of L2(Ω), and the last additive term denotes the BV-seminorm of u, T is a linear operator from L2 ∩ BV into the observation space Y. We formulate necessary optimality conditions for (P). Then we show that (P) admits, for given regularization parameters α and β, solutions which depend in a stable manner on the data z. Finally we study the asymptotic behavior when α = β → 0. The regularized solutions ûβ of (P) converge to the L2 ∩ BV minimal norm solution of the unregularized problem. The rate of convergence is β½ when the minimum-norm solution û is smooth enough.
LA - eng
KW - Ill-posed inverse problems / regularization / bounded variation seminorm.; asymptotic analysis; functions of bounded variation; regularization of linear least-squares problems; necessary optimality conditions
UR - http://eudml.org/doc/116554
ER -

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