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Certified Reduced Basis Methods for Parametrized Elliptic Optimal Control Problems with Distributed Controls

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Abstract

In this paper, we consider the efficient and reliable solution of distributed optimal control problems governed by parametrized elliptic partial differential equations. The reduced basis method is used as a low-dimensional surrogate model to solve the optimal control problem. To this end, we introduce reduced basis spaces not only for the state and adjoint variable but also for the distributed control variable. We also propose two different error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. The reduced basis optimal control problem and associated a posteriori error bounds can be efficiently evaluated in an offline–online computational procedure, thus making our approach relevant in the many-query or real-time context. We compare our bounds with a previously proposed bound based on the Banach–Nečas–Babuška theory and present numerical results for two model problems: a Graetz flow problem and a heat transfer problem. Finally, we also apply and test the performance of our newly proposed bound on a hyperthermia treatment planning problem.

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Notes

  1. The subscripts “\(\mathrm {e}\)” denote “exact”.

  2. Our framework covers spatially distributed controls \(U_\mathrm {e}= L^2({\varOmega }_U)\), \({\varOmega }_U\subset {\varOmega }\), and Neumann boundary controls \(U_\mathrm {e}= L^2({\varGamma }_U)\), \({\varGamma }_U\subset {\varGamma }\). It also applies to finite-dimensional control spaces \(U_\mathrm {e}= \mathbb {R}^m\).

  3. Here and in the following we often omit the dependence on \(\mu \) to simplify notation.

  4. We again note that we omit the dependence on \(\mu \) to simplify notation, i.e., we write \(y = y(\mu )\), \(p = p(\mu )\), and \(u = u(\mu )\).

  5. The bilinear form \(b(\cdot ,\cdot ;\mu ): U\times Y \rightarrow \mathbb {R}\) defines an associated mapping \(\mathscr {B}(\mu ): U \rightarrow Y'\) given by \(\langle \mathscr {B}(\mu ) \psi , \phi \rangle _{Y',Y} = b(\psi ,\phi ;\mu )\) for all \(\psi \in U\), \(\phi \in Y\), \(\mu \in \mathscr {D}\).

  6. The error for a POD basis of size \(N_\text {POD}\) is given by \((\sum _{i=N_\text {POD}+1}^{n_\mathrm {train}} \sigma _i^2)^{1/2}\), where \(\sigma _i\), \(1 \le i \le n_\mathrm {train}\), are the singular values (in decreasing order) of \(\frac{1}{\sqrt{n_\mathrm {train}}} \mathbb {U}^{1/2} S\). Here, \(\mathbb {U}\) is the finite element matrix associated with the reference inner product \((\cdot ,\cdot )_{U} = (\cdot ,\cdot )_{U(\mu ^\mathrm {ref})}\), and \(S \in \mathbb {R}^{\mathscr {N}_U\times n_\mathrm {train}}\) is the snapshot matrix of optimal controls \(u^*(\mu )\) for all \(\mu \in {\varXi }_\mathrm {train}\). See also [36].

  7. The superscript “o” indicates quantities related to the original parameter-dependent domain \({\varOmega }^o(\mu )\), whereas no superscript refers to the parameter-independent reference domain \({\varOmega }= {\varOmega }^o\big (\mu ^\mathrm {ref}\big )\).

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Authors and Affiliations

  1. Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, Schinkelstraße 2, 52062, Aachen, Germany

    Mark Kärcher, Zoi Tokoutsi & Karen Veroy

  2. Philips Research Eindhoven, Eindhoven, The Netherlands

    Zoi Tokoutsi

  3. Numerical Mathematics, RWTH Aachen University, Templergraben 55, 52056, Aachen, Germany

    Martin A. Grepl

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  1. Mark Kärcher
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  2. Zoi Tokoutsi
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  3. Martin A. Grepl
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  4. Karen Veroy
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Correspondence to Martin A. Grepl.

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This work was supported by the Excellence Initiative of the German federal and state governments and the German Research Foundation through Grant GSC 111 and by the European Commission through the Marie Skłodowska-Curie Actions (Innovative Training Program—European Industrial Doctorate, Project Number 642445).

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Kärcher, M., Tokoutsi, Z., Grepl, M.A. et al. Certified Reduced Basis Methods for Parametrized Elliptic Optimal Control Problems with Distributed Controls. J Sci Comput 75, 276–307 (2018). https://doi.org/10.1007/s10915-017-0539-z

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