@phdthesis{Bernreuther2023Nonsm-69009,
  year={2023},
  title={Nonsmooth PDEs: Efficient Algorithms, Model Order Reduction, Multiobjective PDE-Constrained Optimization},
  author={Bernreuther, Marco},
  address={Konstanz},
  school={Universität Konstanz}
}

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    <dcterms:title>Nonsmooth PDEs: Efficient Algorithms, Model Order Reduction, Multiobjective PDE-Constrained Optimization</dcterms:title>
    <dc:date rdf:datatype="http://www.w3.org/2001/XMLSchema#dateTime">2024-01-11T11:47:26Z</dc:date>
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    <dcterms:abstract>This thesis considers the efficient numerical treatment of nonsmooth parameter dependent partial differential equations (PDEs). The PDE is either time-independent (elliptic) or time-dependent (parabolic). A PDE with max-type nonsmoothness serves as model problem. Scalar and multiobjective PDE-constrained optimization problems are considered for the case of an elliptic max-type PDE.

To efficiently treat parameter dependent, nonsmooth elliptic and parabolic PDEs, a model order reduction (MOR) approach by the reduced basis (RB) method is applied in order to obtain a computationally efficient solution approach with a certified a-posteriori error estimator. In case of the parabolic problem, this is based on space-time formulations. To efficiently evaluate the nonsmoothness on the reduced level, the discrete empirical interpolation method (DEIM) is incorporated and the error estimator is adopted accordingly. The separability of the error estimator into an RB and a DEIM part allows the developement of an adaptive RB-DEIM algorithm in the parabolic case, which combines the offline phases of both methods. Numerical examples for both elliptic and parabolic max-type PDEs show
the capabilities of those approaches.

For scalar PDE-constrained optimization problems governed by elliptic max-type PDEs, strong stationarity conditions are known in case of ample controls. It is shown that those results can be extended to the case of finite-dimensional controls, though numerically only an adjoint-based system sufficient for stationarity is solved. Afterwards, a classical offline / online RB method and an adaptive algorithm that combines MOR and optimization are developed and tested successfully for several numerical examples. However, it turns out that DEIM is not suitable for the adjoint equation and the adaptive approach.

For multiobjective PDE-constrained optimization problems governed by elliptic max-type PDEs, the strong stationarity conditions, which are known in case of ample controls, do not extend to the case of finite-dimensional controls. Nonetheless, a sufficient adjoint-based system can be derived and is considered numerically to characterize the Pareto stationary fronts. This is further justified, since these stationarity conditions correspond to strong stationarity systems for problems obtained by scalarization approaches by means of the weighted-sum and the reference point method. The performance of both methods is compared for several numerical examples. Instead of a MOR approach, a matrix-free, preconditioned, iterative approach is incorporated to deal with memory and complexity requirements.</dcterms:abstract>
    <dc:contributor>Bernreuther, Marco</dc:contributor>
    <dc:language>eng</dc:language>
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