@article{MartinGiesselmannKunkel2021, author = {Martin, Gugat and Giesselmann, Jan and Kunkel, Teresa}, title = {Exponential synchronization of a nodal observer for a semilinear model for the flow in gas networks}, address = {IMA Journal of Mathematical Control and Information}, doi = {10.1093/imamci/dnab029}, year = {2021}, abstract = {The flow of gas through networks of pipes can be modeled by coupling hyperbolic systems of partial differential equations that describe the flow through the pipes that form the edges of the graph of the network by algebraic node conditions that model the flow through the vertices of the graph. In the network, measurements of the state are available at certain points in space.Based upon these nodal observations, the complete system state can be approximated using an observer system. In this paper we present a nodal observer, and prove that the state of the observer system converges to the original state exponentially fast. Numerical experiments confirm the theoretical findings.}, language = {en} } @unpublished{GiesselmannEgger2020, author = {Giesselmann, Jan and Egger, Herbert}, title = {Stability and asymptotic analysis for instationary gas transport via relative energy estimates}, year = {2020}, abstract = {We consider the transport of gas in long pipes and pipeline networks for which the dynamics are dominated by friction at the pipe walls. The governing equations can be formulated as an abstract dissipative Hamiltonian system which allows us to derive perturbation bounds by means of relative energy estimates. As particular consequences, we obtain stability with respect to initial conditions and model parameters and quantitative estimates in the high friction limit. Our results are established in detail for the flow in a single pipe and through the energy-based modelling they naturally generalize also to pipe networks.}, language = {en} } @unpublished{SarnaGiesselmannBenner2020, author = {Sarna, Neeraj and Giesselmann, Jan and Benner, Peter}, title = {Data-Driven Snapshot Calibration via Monotonic Feature Matching}, year = {2020}, abstract = {Snapshot matrices of hyperbolic equations have a slow singular value decay, resulting in inefficient reduced-order models. We develop on the idea of inducing a faster singular value decay by computing snapshots on a transformed spatial domain, or the so-called snapshot calibration/transformation. We are particularly interested in problems involving shock collision, shock rarefaction-fan collision, shock formation, etc. For such problems, we propose a realizable algorithm to compute the spatial transform using monotonic feature matching. We consider discontinuities and kinks as features, and by carefully partitioning the parameter domain, we ensure that the spatial transform has properties that are desirable both from a theoretical and an implementation standpoint. We use these properties to prove that our method results in a fast \$m\$-width decay of a so-called calibrated manifold. A crucial observation we make is that due to calibration, the \$m\$-width does not only depend on \$m\$ but also on the accuracy of the full order model, which is in contrast to elliptic and parabolic problems that do not need calibration. The method we propose only requires the solution snapshots and not the underlying partial differential equation (PDE) and is therefore, data-driven. We perform several numerical experiments to demonstrate the effectiveness of our method.}, language = {en} } @article{GugatGiesselmann2020, author = {Gugat, Martin and Giesselmann, Jan}, title = {Boundary feedback stabilization of a semilinear model for the flow in star-shaped gas networks}, address = {ESAIM:COCV}, doi = {10.1051/cocv/2021061}, year = {2020}, abstract = {The flow of gas through a pipeline network can be modelled by a coupled system of 1-d quasilinear hyperbolic equations. In this system, the influence of certain source terms that model friction effects is essential. Often for the solution of control problems it is convenient to replace the quasilinear model by a simpler semilinear model. In this paper, we analyze the behavior of such a semilinear model on a star-shaped network. The model is derived from the diagonal form of the quasilinear model by replacing the eigenvalues by the sound speed multiplied by 1 or -1 respectively. Thus in the corresponding eigenvalues the influence of the gas velocity is neglected, which is justified in the applications since it is much smaller than the sound speed in the gas. For a star-shaped network of horizontal pipes for suitable coupling conditions we present boundary feedback laws that stabilize the system state exponentially fast to a position of rest for sufficiently small initial data. We show the exponential decay of the \$H^1\$-norm for arbitrarily long pipes. This is remarkable since in general even for linear systems, for certain source terms the system can become exponentially unstable if the space interval is too long. Our proofs are based upon observability inequalities for the \$L^2\$ and the \$H^1\$-norm.}, language = {en} } @unpublished{EggerGiesselmannPhilippietal.2021, author = {Egger, Herbert and Giesselmann, Jan and Philippi, Nora and Kunkel, Teresa}, title = {An asymptotic-preserving discretization scheme for gas transport in pipe networks}, year = {2021}, abstract = {We consider the simulation of barotropic flow of gas in long pipes and pipe networks. Based on a Hamiltonian reformulation of the governing system, a fully discrete approximation scheme is proposed using mixed finite elements in space and an implicit Euler method in time. Assuming the existence of a smooth subsonic solution bounded away from vacuum, a full convergence analysis is presented based on relative energy estimates. Particular attention is paid to establishing error bounds that are uniform in the friction parameter. As a consequence, the method and results also cover the parabolic problem arising in the asymptotic large friction limit. The error estimates are derived in detail for a single pipe, but using appropriate coupling conditions and the particular structure of the problem and its discretization, the main results directly generalize to pipe networks. Numerical tests are presented for illustration.}, language = {en} } @unpublished{GiesselmannGugatKunkel2023, author = {Giesselmann, Jan and Gugat, Martin and Kunkel, Teresa}, title = {Observer-based data assimilation for barotropic gas transport using distributed measurements}, year = {2023}, abstract = {We consider a state estimation problem for gas pipeline flow modeled by the one-dimensional barotropic Euler equations. In order to reconstruct the system state, we construct an observer system of Luenberger type based on distributed measurements of one state variable. First, we show the existence of Lipschitz-continuous semi-global solutions of the observer system and of the original system for initial and boundary data satisfying smallness and compatibility conditions for a single pipe and for general networks. Second, based on an extension of the relative energy method we prove that the state of the observer system converges exponentially in the long time limit towards the original system state. We show this for a single pipe and for star-shaped networks.}, language = {en} } @unpublished{GiesselmannKrupa2023, author = {Giesselmann, Jan and Krupa, Sam}, title = {Theory of shifts, shocks, and the intimate connections to L2-type a posteriori error analysis of numerical schemes for hyperbolic problems}, year = {2023}, abstract = {In this paper, we develop reliable a posteriori error estimates for numerical approximations of scalar hyperbolic conservation laws in one space dimension. Our methods have no inherent small-data limitations and are a step towards error control of numerical schemes for systems. We are careful not to appeal to the Kruzhkov theory for scalar conservation laws. Instead, we derive novel quantitative stability estimates that extend the theory of shifts, and in particular, the framework for proving stability first developed by the second author and Vasseur. This is the first time this methodology has been used for quantitative estimates. We work entirely within the context of the theory of shifts and a-contraction, techniques which adapt well to systems. In fact, the stability framework by the second author and Vasseur has itself recently been pushed to systems [Chen-Krupa-Vasseur. Uniqueness and weak-BV stability for 2×2 conservation laws. Arch. Ration. Mech. Anal., 246(1):299--332, 2022]. Our theoretical findings are complemented by a numerical implementation in MATLAB and numerical experiments.}, language = {en} } @unpublished{GiesselmannKolbe2023, author = {Giesselmann, Jan and Kolbe, Niklas}, title = {A posteriori error analysis of a positivity preserving scheme for the power-law diffusion Keller-Segel model}, year = {2023}, abstract = {We study a finite volume scheme approximating a parabolic-elliptic Keller-Segel system with power law diffusion with exponent γ∈[1,3] and periodic boundary conditions. We derive conditional a posteriori bounds for the error measured in the L∞(0,T;H1(Ω)) norm for the chemoattractant and by a quasi-norm-like quantity for the density. These results are based on stability estimates and suitable conforming reconstructions of the numerical solution. We perform numerical experiments showing that our error bounds are linear in mesh width and elucidating the behaviour of the error estimator under changes of γ.}, language = {en} } @unpublished{AlldredgeFrankGiesselmann2023, author = {Alldredge, Graham and Frank, Martin and Giesselmann, Jan}, title = {On the convergence of the regularized entropy-based moment method for kinetic equations}, volume = {9}, doi = {https://doi.org/10.5802/smai-jcm.93}, pages = {1-29}, year = {2023}, abstract = {The entropy-based moment method is a well-known discretization for the velocity variable in kinetic equations which has many desirable theoretical properties but is difficult to implement with high-order numerical methods. The regularized entropy-based moment method was recently introduced to remove one of the main challenges in the implementation of the entropy-based moment method, namely the requirement of the realizability of the numerical solution. In this work we use the method of relative entropy to prove the convergence of the regularized method to the original method as the regularization parameter goes to zero and give convergence rates. Our main assumptions are the boundedness of the velocity domain and that the original moment solution is Lipschitz continuous in space and bounded away from the boundary of realizability. We provide results from numerical simulations showing that the convergence rates we prove are optimal.}, language = {en} } @unpublished{GiesselmannKwon2023, author = {Giesselmann, Jan and Kwon, Kiwoong}, title = {A posteriori error control for a Discontinuous Galerkin approximation of a Keller-Segel model}, year = {2023}, abstract = {We provide a posteriori error estimates for a discontinuous Galerkin scheme for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The estimates are conditional, in the sense that an a posteriori computable quantity needs to be small enough - which can be ensured by mesh refinement - and optimal in the sense that the error estimator decays with the same order as the error under mesh refinement. A specific feature of our error estimator is that it can be used to prove existence of a weak solution up to a certain time based on numerical results.}, language = {en} }