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abstract:
We consider an optimization problem of the type
\begin{equation}
\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}
\end{equation}
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. |
