arXiv subjects

Dominik Garmatter

Publications and source records attributed to Dominik Garmatter.

4 recordsLinked to original sources

arXiv2021

An Improved Penalty Algorithm using Model Order Reduction for MIPDECO problems with partial observations

This work addresses optimal control problems governed by a linear time-dependent partial differential equation (PDE) as well as integer constraints on the control. Moreover, partial observations are assumed in the objective function. The resulting problem poses several numerical challenges due to the mixture of combinatorial aspects, induced by integer variables, and large scale linear algebra issues, arising from the PDE discretization. Since classical solution approaches such as the branch-and-bound framework are typically overwhelmed by such large-scale problems, this work extends an improved penalty algorithm proposed by the authors, to the time-dependent setting. The main contribution is a novel combination of an interior point method, preconditioning, and model order reduction yielding a tailored local optimization solver at the heart of the overall solution procedure. A thorough numerical investigation is carried out both for a Poisson problem as well as a convection-diffusion problem demonstrating the versatility of the approach.

math.OC↗

arXiv2021

Improved penalty algorithm for Mixed Integer PDE Constrained Optimization Problems

Optimal control problems including partial differential equation (PDE) as well as integer constraints merge the combinatorial difficulties of integer programming and the challenges related to large-scale systems resulting from discretized PDEs. So far, the Branch-and-Bound framework has been the most common solution strategy for such problems. In order to provide an alternative solution approach, especially in a large-scale context, this article investigates penalization techniques. Taking inspiration from a well-known family of existing exact penalty algorithms, a novel improved penalty algorithm is derived, whose key ingredients are a basin hopping strategy and an interior point method, both of which are specialized for the problem class. A thorough numerical investigation is carried out for a standard stationary test problem. Extensions to a convection-diffusion as well as a nonlinear test problem finally demonstrate the versatility of the approach.

math.NA↗

arXiv2017

Magnet Resonance Electrical Impedance Tomography (MREIT): Convergence and Reduced Basis Approach

This article considers the inverse problem of Magnet resonance electrical impedance tomography (MREIT) in two dimensions. A rigorous mathematical framework for this inverse problem as well as the existing Harmonic $B_z$ Algorithm as a solution algorithm are presented. The convergence theory of this algorithm is extended, such that the usage an approximative forward solution of the underlying partial differential equation (PDE) in the algorithm is sufficient for convergence. Motivated by this result, a novel algorithm is developed where it is the aim to speed-up the existing Harmonic $B_z$ Algorithm. This is achieved by combining it with an adaptive variant of the reduced basis method, a model order reduction technique. In a numerical experiment a high-resolution image of the shepp-logan phantom is reconstructed and both algorithms are compared.

math.NA↗

arXiv2015

A Reduced Basis Landweber method for nonlinear inverse problems

We consider parameter identification problems in parametrized partial differential equations (PDE). This leads to nonlinear ill-posed inverse problems. One way to solve them are iterative regularization methods, which typically require numerous amounts of forward solutions during the solution process. In this article we consider the nonlinear Landweber method and want to couple it with the reduced basis method as a model order reduction technique in order to reduce the overall computational time. In particular, we consider PDEs with a high-dimensional parameter space, which are known to pose difficulties in the context of reduced basis methods. We present a new method that is able to handle such high-dimensional parameter spaces by combining the nonlinear Landweber method with adaptive online reduced basis updates. It is then applied to the inverse problem of reconstructing the conductivity in the stationary heat equation.

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