On Snapshot-Based Model Reduction Under Compatibility Conditions for a Nonlinear Flow Problem on Networks

This paper is on the construction of structure-preserving, online-efficient reduced models for the barotropic Euler equations with a friction term on networks. The nonlinear flow problem finds broad application in the context of gas distribution networks. We propose a snapshot-based reduction approach that consists of a mixed variational Galerkin approximation combined with quadrature-type complexity reduction. Its main feature is that certain compatibility conditions are assured during the training phase, which make our approach structure-preserving. The resulting reduced models are locally mass conservative and inherit an energy bound and port-Hamiltonian structure. We also derive a wellposedness result for them. In the training phase, the compatibility conditions pose challenges, we face constrained data approximation problems as opposed to the unconstrained training problems in the conventional reduction methods. The training of our model order reduction consists of a principal component analysis under a compatibility constraint and, notably, yields reduced models that fulfill an optimality condition for the snapshot data. The training of our quadrature-type complexity reduction involves a semi-definite program with combinatorial aspects, which we approach by a greedy procedure. Efficient algorithmic implementations are presented. The robustness and good performance of our structure-preserving reduced models are showcased at the example of gas network simulations.

Automated summary

This paper focuses on the construction of structure-preserving, online-efficient reduced models for the barotropic Euler equations with a friction term on networks. The proposed snapshot-based reduction approach ensures compatibility conditions during the training phase and produces locally mass conservative reduced models with energy bounds and port-Hamiltonian structure. The training involves constrained data approximation and combinatorial semi-definite programming, but efficient algorithmic implementations are presented to address these challenges. Gas network simulations demonstrate the robustness and good performance of the structure-preserving reduced models.