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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 $\alpha\ge 0$, $\beta>0$, $K$ 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.

49K27Optimal control problems in abstract spaces (optimality conditions)
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