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Nonoverlapping Domain Decomposition in Space and Time for Optimal Control Problems on Metric Graphs by the Example of Gas Flow in Pipe Networks

  • We consider non-overlapping domain decomposition methods for ordinary and partial differential equations and corresponding optimal control problems on metric graphs. As an exemplary context, we chose a semilinear approximation of the Euler system and a doubly nonlinear parabolic model that has come to be known as friction dominated flow in gas pipe networks. By this choice, we encounter hyperbolic, parabolic and elliptic linear and nonlinear problems in a single highly motivating application. We depart from the classical domain decomposition methods described by P.L. Lions and J.L. Lions and O. Pironneau and extend those to problems on metric graphs. The choice of methods is determined by the desire to use a control concept that has come to be known as virtual controls which, in turn, possibly lead to a fully parallel decomposition of the corresponding optimality systems. In a second step, we extend the methods to p-Laplace problems on networks. The analysis, due to space limitations, will appear in a forthcoming publication. See however J.E. Lagnese and G. Leugering. Furthermore, we then describe methods for space and time domain decomposition or optimal control problems in the spirit of J.E. Lagnese and G. Leugering. We finally provide some comments on PINN-based approximations of the methods described before. We provide numerical evidence for all algorithms discussed.

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Metadaten
Author:Günter Leugering
Document Type:Part of a Book
Language:English
Date of Publication (online):2022/04/05
Release Date:2022/04/05
Tag:Optimal control; PDEs on graphs; instantaneous control; p-Laplace problem on a graph; p-parabolic problems
Institutes:Friedrich-Alexander-Universität Erlangen-Nürnberg
Subprojects:A05
Licence (German):License LogoCreative Commons - CC BY - Namensnennung 4.0 International