11:20
Optimal control of the parabolic obstacle problem: numerical analysis
Dominik Hafemeyer | Technische Universität München | Germany
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Dominik Hafemeyer | Technische Universität München | Germany
We consider the numerical analysis of an optimal control problem with a parabolic obstacle problem as constraint. A parabolic obstacle problem, intuitively speaking, is a form of parabolic equation, e.g. heat equation, in which the state is not allowed to drop below a given obstacle. It is a special form of a variational inequality and appears for example in the Black-Scholes model in financial mathematics. Variational inequalities are challenging from the point of numerical analysis as they can exhibit non-smooth properties.
In our talk we first regularize the parabolic obstacle problem by a family of suitable semi-linear equations. Those semi-linear equations are then discretized using piecewise constant finite elements in time and piecewise linear finite elements in space. We present an a priori error estimate that shows that the regularization error and discretization error actually decouple making this a valid approximation strategy. The a priori error estimate is numerically validated to be sharp.
It is then relatively straightforward to transfer those results to an optimal control problem (with quadratic tracking terms and quadratic control penalty) where state and control are related via the parabolic obstacle problem. We will present a priori error estimates for the semi discrete and fully discrete problem and provide error estimates for the control and state under appropriate second order conditions on the optimal control of the unregularized, undiscretized problem.
11:40
Numerical Analysis for the Optimal Control of simplified mechanical damage processes
M. Sc. Marita Holtmannspötter | University of Duisburg-Essen | Germany
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M. Sc. Marita Holtmannspötter | University of Duisburg-Essen | Germany
Prof. Dr. Arnd Rösch
Prof. Dr. Boris Vexler
In this talk we investigate a priori error estimates for the space-time Galerkin finite element discretization of an optimal control problem governed by a simplified damage model. The model equations are of a special structure as the state equation consists of a coupled PDE-ODE system. One challenge for the derivation of error estimates arises from low regularity properties of solutions provided by this system. The state equation is discretized by a piecewise constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. For the discretization of the control we employ the same discretization technique which turns out to be equivalent to a variational discretization approach. We provide error estimates both for the discretization of the state equation as well as for the optimal control. Numerical experiments are added to illustrate the proven rates of convergence.
12:00
Iterated total variation regularization with finite element methods for reconstruction the source term in elliptic systems
Dr. Tran Nhan Tam Quyen | University of Goettingen | Germany
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Dr. Tran Nhan Tam Quyen | University of Goettingen | Germany
Prof. Michael Hinze | Universität Hamburg | Germany
In this talk we would like to present the problem of recovering the source $f$ in the elliptic system
\begin{equation*}
\begin{aligned}
-\nabla \cdot \big(\alpha \nabla u \big) + \beta u &= f \mbox{~in~} \Omega, \\
\alpha \nabla u \cdot \vec{n} +\sigma u &= j \mbox{~on~} \partial\Omega
\end{aligned}
\end{equation*}
from an observation $z$ of the state $u$ on a part $\Gamma$ of the boundary $\partial\Omega$, where the functions $\alpha,\beta,\sigma$ and $j$ are given. For the particular interest in reconstructing probably discontinuous sources, we use the standard least squares method with the total variation regularization, i.e. we consider a minimizer of the minimization problem
$$
\min_{f\in F_{ad}} J(f), \quad J(f) := \frac{1}{2}\|u(f)-z\|^2_{\Gamma} + \rho TV(f) \eqno \left(\mathcal{P}\right)
$$
as reconstruction. Here $u(f)$ denotes the unique weak solution of the above elliptic system which depends on the source term $f$, $TV(f)$ is the total variation of $f$, and $\rho>0$ is the regularization parameter. Let $u^h$ be the approximation of $u$
in the finite dimensional space of piecewise linear, continuous finite elements. We then consider the discrete regularized problem corresponding to $\left(\mathcal{P}\right)$, i.e. the following minimization problem
$$
\min_{f\in F^h_{ad}} J^h(f), \quad J^h(f) := \frac{1}{2}\|u^h(f)-z\|^2_{\Gamma} + \rho TV(f). \eqno \big(\mathcal{P}^h\big)
$$
We show the stability and the convergence of solutions to $\big(\mathcal{P}^h\big)$. Furthermore, based on the Bartels' work [SIAM J. Numer. Anal. (2012)], we have proposed an algorithm to stably solve the minimization problem $\big(\mathcal{P}^h\big)$. We prove the iteration sequence $\big(f^h_n\big)_n$ generated by the derived algorithm conversing to a minimizer of $\big(\mathcal{P}^h\big)$, and a convergence measurement of the kind
\begin{align*}
\|f^h_{n+1}-f^h_n\| =\mathcal{O}\left( \frac{1}{\sqrt{n}}\right)
\end{align*}
is also established. Finally, a numerical experiment is presented to illustrate our theoretical findings.
12:20
On the Numerical Study of Obstacle Optimal Control Problem with Source Term
Dr. Yazid Dendani | Université Badji Mokhtar-Annaba | Algeria
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Dr. Yazid Dendani | Université Badji Mokhtar-Annaba | Algeria
In the present work, we study the numerical aspect of the optimality system given by M. Bergounioux and S. Lenhart (SIAMJ. CONTROL OPTIMAL. Vol.43, No. 8, pp. 229-242, 2004). A numerical algorithm is given and its practical feasibility is investigated by several numerical tests in one and two dimensional spaces.