Volume 54, pp. 392-419, 2021.

Optimal Dirichlet control of partial differential equations on networks

Martin Stoll and Max Winkler


Differential equations on metric graphs can describe many phenomena in the physical world but also the spread of information on social media. To efficiently compute the optimal setup of the differential equation for a given desired state is a challenging numerical analysis task. In this work, we focus on the task of solving an optimization problem subject to a linear differential equation on a metric graph with the control defined on a small set of Dirichlet nodes. We discuss the discretization by finite elements and provide rigorous error bounds as well as an efficient preconditioning strategy to deal with the large-scale case. We show in various examples that the method performs very robustly.

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Key words

complex networks, optimal Dirichlet control, preconditioning, saddle point systems, error estimation

AMS subject classifications

65F08, 65F50, 65M60, 65N15,

Links to the cited ETNA articles

[12]Vol. 48 (2018), pp. 97-113 Pia Domschke, Aseem Dua, Jeroen J. Stolwijk, Jens Lang, and Volker Mehrmann: Adaptive refinement strategies for the simulation of gas flow in networks using a model hierarchy
[51]Vol. 40 (2013), pp. 294-310 John W. Pearson and Andrew J. Wathen: Fast iterative solvers for convection-diffusion control problems

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