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