Regularization of linear least squares problems by total bounded variation. (English) Zbl 0890.49010

Chavent, G.; Kunisch, K.

ESAIM, Control Optim. Calc. Var. 2, 359-376 (1997).

Summary: We consider the problem \[ \text{Minimize}\quad {1\over 2} |Tu-z|^2_Y+{\alpha\over 2} |u|^2_{L^2}+ \beta \int_\Omega|\nabla u|\text{ over }u\in K\cap X,\tag{P} \] where \(\alpha\geq 0\), \(\beta>0\), \(K\) is a closed convex subset of \(L^2(\Omega)\), and the last additive term denotes the BV-seminorm of \(u\), \(T\) is a linear operator from \(L^2\cap\text{BV}\) into the observation space \(Y\). We formulate necessary optimality conditions for (P). Then we show that (P) admits, for given regularization parameters \(\alpha\) and \(\beta\), solutions which depend in a stable manner on the data \(z\). Finally, we study the asymptotic behavior when \(\alpha= \beta\to 0\). The regularized solutions \(\widehat u_\beta\) of (P) converge to the \(L^2\cap \text{BV}\) minimal norm solution of the unregularized problem. The rate of convergence is \(\beta^{{1\over 2}}\) when the minimum-norm solution \(\widehat u\) is smooth enough.

MSC:

49K27

Optimality conditions for problems in abstract spaces

Keywords:

asymptotic analysis; functions of bounded variation; regularization of linear least-squares problems; necessary optimality conditions

References:

[1]	J.-P. Aubin, I. Ekeland: Applied NonlinearAnalysis, Wiley-Interscience, New York, 1984. Zbl0641.47066 MR749753 · Zbl 0641.47066
[2]	J. Baumeister: Stable Solutions of Inverse Problems, Vieweg, Braunschweig, 1987. Zbl0623.35008 MR889048 · Zbl 0623.35008
[3]	E. Casas, K. Kunisch, C. Pola: Regularization by functions of bounded variation and applications to image enhancement, preprint. MR1692383 · Zbl 0942.49014
[4]	G. Chavent, K. Kunisch: Convergence of Tikhonov regularization for constrained ill-posed inverse problems, Inverse Problems, 10 ( 1994), 63-76. Boston, 1985. Zbl0799.65061 MR1259438 · Zbl 0799.65061 · doi:10.1088/0266-5611/10/1/006
[5]	V. Girault, P. A. Raviart: Finite Elements, Methods for Navier-Stokes Equations, Springer, Berlin, 1984. Zbl0585.65077 MR851383 · Zbl 0585.65077
[6]	E. Giusti: Minimal Surfaces and Functions of Bounded Variation, Birkhäuser, Boston, 1984. Zbl0545.49018 MR775682 · Zbl 0545.49018
[7]	C. W. Groetsch: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, Boston, 1984. Zbl0545.65034 MR742928 · Zbl 0545.65034
[8]	A. K. Louis: Inverse und schlechtgestellte Probleme, Teubner, Stuttgart, 1989. Zbl0667.65045 MR1002946 · Zbl 0667.65045
[9]	V. G. Mazja: Sobolev Spaces, Springer, Berlin, 1985. MR817985
[10]	W. Rudin: Real and Complex Analysis, McGraw Hill, London, 1970. Zbl0925.00005 · Zbl 0925.00005
[11]	L. Rudin, S. Osher, E. Fatemi: Nonlinear total variation based noise removal algorithm, Physica D, 60 ( 1992), 259-268. Zbl0780.49028 · Zbl 0780.49028 · doi:10.1016/0167-2789(92)90242-F
[12]	R. Temam: Mathematical Problems in Plasticity, Gauthier-Villars, Kent, 1985. MR711964 · Zbl 0457.73017
[13]	C. Vogel, M. Oman: Iterative methods for total variation denoising, preprint. Zbl0847.65083 MR1375276 · Zbl 0847.65083 · doi:10.1137/0917016