# Regularization of linear least squares problems by total bounded variation

ESAIM: Control, Optimisation and Calculus of Variations (2010)

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

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topChavent, 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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