In function spaces \(X\) of (equivalence classes) of functions of the type \(f:\Omega\to\mathbb{R}\) with \(\Omega\subset\mathbb{R}^N\) such as Lebesgue spaces \(L^p(\Omega)\) and Sobolev spaces \(W_0^{1,p}(\Omega),W^{1,p}(\Omega)\) we often find sets of the of the type
\begin{equation*}
\mathbf{K}(X)=\{v\in X: |v(x)|\leq \alpha(x) \quad \text{for almost every } x\in\Omega\}
\end{equation*}
where \(\alpha:\Omega\to\mathbb{R}\) is some given function.
Suppose that the boundary of the set \(\Omega\) is smooth enough (at least Lipschitz), then we have the following density results
\begin{align*}
\overline{C^\infty(\Omega)}^{L^p(\Omega)}&=L^p(\Omega), &\overline{C_c^\infty(\Omega)}^{W_0^{1,p}(\Omega)}&=W_0^{1,p}(\Omega), &\text{and} & &\overline{C^\infty(\bar{\Omega})}^{W^{1,p}(\Omega)}&=W^{1,p}(\Omega).
\end{align*}
That is smooth functions on \(\Omega\), with compact support within \(\Omega\), and up to the boundary \(\bar{\Omega}\) are dense in \(L^p(\Omega)\), \(W_0^{1,p}(\Omega)\), and \(W^{1,p}(\Omega)\), respectively. However, in order to obtain the analogous results with contraints as for example
\begin{align*}
\overline{\mathbf{K}(W^{1,p}(\Omega))\cap C^\infty(\bar{\Omega})}^{W^{1,p}(\Omega)}&=\mathbf{K}(W^{1,p}(\Omega)),
\end{align*}
additional requirements on \(\alpha:\Omega\to\mathbb{R}\) are needed.
\(\mathbf{i)}\) There are irregular enough \(\alpha:\Omega\to\mathbb{R}\) that the above fail (see Hintermüller-Rautenberg-Rösel(2017))
\(\mathbf{ii)}\) In general, a dense continuous embedding \(Y\hookrightarrow X\) is not enough to guarantee
\begin{equation*}
\overline{\mathbf{K}\cap Y}^X=\mathbf{K},
\end{equation*}
for \(\mathbf{K}\subset X\).
Regularization of Optimization Problems
Sets of the type \(\mathbf{K}:=\mathbf{K}(X)\) commonly are constraints (on controls or state variables) in optimization problems, i.e., \begin{equation}\tag{$\mathrm{P}$}\label{P} \min_{u\in \mathbf{K}\subset X} J(u). \end{equation} Discretization or regularization approaches to solving \eqref{P} require to consider \begin{equation}\tag{$\mathrm{P}_\gamma$}\label{Pg} \min_{u\in Y\subset X} J(u)+\gamma R(u), \end{equation} where \(\gamma>0\) and \(R(u)=g(\|u\|_Y)\) for some \(g:\mathbb{R}^+_0\to \mathbb{R}^+_0\) strictly increasing. Under natural conditions on \(J\) that provide existence of minimizers, sufficient conditions for the sequence of minimizers \(u_\gamma\) to \eqref{Pg} to converge to a minimizer \(u^*\) of \eqref{P} are given by \begin{equation*} \overline{\mathbf{K}\cap Y}^X=\mathbf{K}. \end{equation*}![OA](ContentHP/Icons/openlock.png)
![OA](ContentHP/Icons/paper.png)
with M. Hintermüller, K. Papafitsoros, and H. Sun
arXiv preprint 2002.05614, 2020.
![OA](ContentHP/Icons/openlock.png)
![OA](ContentHP/Icons/paper.png)
with M. Hintermüller, and K. Papafitsoros.
arXiv preprint 1910.02751, 2019.
![OA](ContentHP/Icons/openlock.png)
![OA](ContentHP/Icons/paper.png)
with A. N. Ceretani.
Zeitschrift für angewandte Mathematik und Physik (ZAMP) 70: 14, 2019.
![OA](ContentHP/Icons/openlock.png)
![OA](ContentHP/Icons/paper.png)
with M. Hintermüller, and S. Rösel.
Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 473(2205), 2017.
![OA](ContentHP/Icons/openlock.png)
![OA](ContentHP/Icons/paper.png)
Part I: Modelling and theory
with M. Hintermüller.
Journal of Mathematical Imaging and Vision, 59(3), 498-514, 2017.
![OA](ContentHP/Icons/openlock.png)
![OA](ContentHP/Icons/paper.png)
Part II: Algorithm, its analysis and numerical tests.
with M. Hintermüller, T. Wu and A. Langer.
Journal of Mathematical Imaging and Vision, 59(3), 515-533, 2017.
![OA](ContentHP/Icons/openlock.png)
![OA](ContentHP/Icons/paper.png)
with M. Hintermüller, and K. Papafitsoros.
Journal of Mathematical Analysis and Applications, 454(2), pp. 891-935, 2017.
![OA](ContentHP/Icons/openlock.png)
![OA](ContentHP/Icons/paper.png)
with M. Hintermüller.
Journal of Mathematical Analysis and Applications, 426(1), pp. 585-593, 2015.