# T:A:L:K:S

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 title: Regularization results for inverse problems with sparsity functional name: Wachsmuth first name: Gerd location/conference: cssip10 PREPRINT-link: http://www.tu-chemnitz.de/mathematik/part_dgl/publications/Wachsmuth_Wachsmuth__Convergence_and_Regularization_Results_for_Optimal_Control_Problems_with_Sparsity_Functional.pdf PRESENTATION-link: http://www.dfg-spp1324.de/nuhagtools/event/dateien/talks_cssip/wachsmuth.pdf abstract: We consider an optimization problem of the type \left\{ \begin{aligned} \text{Minimize}\quad & F(u) = \frac12 \| \mathcal{S} u - z_\delta \|_{H}^2 + \alpha \, \| u \|_{L^2(\Omega)}^2 + \beta \, \| u \|_{L^1(\Omega)} \\ \text{such that}\quad & u \in U_\textup{ad} \subset L^2(\Omega). \end{aligned} \right.\tag{$\textbf{P}_{\alpha,\delta}$} \label{eq:P} Here, $\Omega \subset \mathbb{R}^n$ is a bounded domain, $H$ is some Hilbert space, $\mathcal{S} \in \mathcal{L}(L^2(\Omega), H)$ compact (e.g.\ the solution operator of an elliptic partial differential equation), $\alpha > 0$, and $\delta, \beta \ge 0$. The problem~\eqref{eq:P} can be interpreted as an inverse problem as well as an optimal control problem. Let us denote the solution with $u_{\alpha,\delta}$. The estimate $\| u_{\alpha,0} - u_{\alpha,\delta}\|_{L^2(\Omega)} \le \delta \, \alpha^{-1/2}$ for the error due to the noise level $\delta$ is well known for $\beta = 0$ and the proof can be extended to the case $\beta > 0$. A typical way to estimate the regularization error as $\alpha \searrow 0$ is via a source condition, e.g.\ $u_{0,0} = \mathcal{S}^\star w$ with some $w \in H$. This yields $\| u_{\alpha,0} - u_{0,0} \|_{L^2(\Omega)} \le C \, \alpha^{1/2}$. But if pointwise constraints are present ($U_\textup{ad} = \{ u \in L^2(\Omega) : u_a \le u \le u_b \}$), $u_{0,0}$ often is bang-bang, i.e.\ $u_{0,0}(x) \in \{u_a, 0, u_b\}$ a.e.\ in $\Omega$. Hence, $u_{0,0} \\notin H^1(\Omega)$ and by $\operatorname{range}(\mathcal{S}^\star) \subset H^1(\Omega)$ a source condition with $\mathcal{S^\star}$ can not hold. In this talk we present a new technique for deriving rates of the regularization error using a combination of a source condition and a regularity assumption on the adjoint variable $p_{0,0} = \mathcal{S}^\star(z_0 - \mathcal{S} u_{0,0})$. If the measure of $\big\{ \big| |p_{0,0}| - \beta \big| \le \varepsilon \big\} \le C \, \varepsilon$ for all $\varepsilon \ge 0$ it is possible to show $\| u_{\alpha,0} - u_{0,0} \|_{L^2(\Omega)} \le C \, \alpha^{1/2}$ without a source condition. We present examples showing that the error rates are sharp.