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Optimal Input Design for Large-Scale Inverse Problems using PDE-Constrained Optimization

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2024

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In this thesis we consider the optimal input design of inverse problems using partial differential equation (PDE)-constrained optimization. While the analysis and numerical solution of inverse problems has made extensive progress in the last decades, some of these problems might depend on the model input or control. The field of optimal input/experiment design is an established framework for generating experiments or input functions in order to satisfy certain theoretical conditions. In our case we aim to optimize the observability of the parameters, i.e. their impact the model output. We analyze two classes of inverse problems, one driven by a general linear parabolic system and the other by a nonlinearly coupled elliptic-parabolic system. Then, we consider an optimal input design algorithm, whose purpose is to improve the parameter estimation process. Numerical experiments of the design algorithm conclude this work, and deliver satisfying results to both the mathematical problem and the industrial application. We use, in particular, the reduced basis (RB) approximation in the framework of model order reduction (MOR) to define fast-computable approximated PDE-solutions. To solve the finite-dimensions inverse problem, we consider a trust-region optimization method, which requires RB error estimates in order to be properly defined and to guarantee its convergence. While the study and optimization of general linear parabolic systems is already well-studied, the nonlinear model lacks efficient and certified error estimates. Hence, one of the novelties of this dissertation is the approximation of hierarchical error estimates and especially their use in the optimization algorithm. For the optimal input design we describe a standard algorithm and propose a variation for an industrial application. The standard formulation relies on an adaptive argument, where one finds a sequence of input functions that maximize the overall observability of the parameters, and therefore makes the convergence to the true parameters easier. Even though the input design problem lacks a theoretical framework, we observe a general success in its purpose, namely gaining trust in the estimated parameter. Furthermore, the algorithm is independent of the model and can be applied to general inverse problems. Finally, we apply the input design algorithm to the industrial problem of estimating parameters for Lithium-ion battery cell systems. Since these systems are very complex and depend on many parameters, the estimation process is usually long and resource-expensive. Hence, the input design finds a shorter optimal input function, in this case the battery current profile. We apply two variations of the optimal input design algorithm previously defined in order to solve this problem.

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ISO 690PETROCCHI, Andrea, 2024. Optimal Input Design for Large-Scale Inverse Problems using PDE-Constrained Optimization [Dissertation]. Konstanz: Universität Konstanz
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@phdthesis{Petrocchi2024Optim-70280,
  year={2024},
  title={Optimal Input Design for Large-Scale Inverse Problems using PDE-Constrained Optimization},
  author={Petrocchi, Andrea},
  address={Konstanz},
  school={Universität Konstanz}
}
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For the optimal input design we describe a standard algorithm and propose a variation for an industrial application. The standard formulation relies on an adaptive argument, where one finds a sequence of input functions that maximize the overall observability of the parameters, and therefore makes the convergence to the true parameters easier. Even though the input design problem lacks a theoretical framework, we observe a general success in its purpose, namely gaining trust in the estimated parameter. Furthermore, the algorithm is independent of the model and can be applied to general inverse problems.
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June 18, 2024
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Konstanz, Univ., Diss., 2024
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