## 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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### References:

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