@article{FrenzelLang2020, author = {Frenzel, David and Lang, Jens}, title = {A Third-Order Weighted Essentially Non-Oscillatory Scheme in Optimal Control Problems Governed by Nonlinear Hyperbolic Conservation Laws}, volume = {Computational Optimization and Applications}, number = {80}, doi = {https://doi.org/10.1007/s10589-021-00295-2}, pages = {301 -- 320}, year = {2020}, abstract = {The weighted essentially non-oscillatory (WENO) methods are popular and effective spatial discretization methods for nonlinear hyperbolic partial differential equations. Although these methods are formally first-order accurate when a shock is present, they still have uniform high-order accuracy right up to the shock location. In this paper, we propose a novel third-order numerical method for solving optimal control problems subject to scalar nonlinear hyperbolic conservation laws. It is based on the first-disretize-then-optimize approach and combines a discrete adjoint WENO scheme of third order with the classical strong stability preserving three-stage third-order Runge-Kutta method SSPRK3. We analyze its approximation properties and apply it to optimal control problems of tracking-type with non-smooth target states. Comparisons to common first-order methods such as the Lax-Friedrichs and Engquist-Osher method show its great potential to achieve a higher accuracy along with good resolution around discontinuities.}, language = {en} } @article{DomschkeKolbLang2021, author = {Domschke, Pia and Kolb, Oliver and Lang, Jens}, title = {Fast and Reliable Transient Simulation and Continuous Optimization of Large-Scale Gas Networks}, publisher = {Mathematical Methods of Operations Research}, doi = {https://doi.org/10.1007/s00186-021-00765-7}, year = {2021}, abstract = {We are concerned with the simulation and optimization of large-scale gas pipeline systems in an error-controlled environment. The gas flow dynamics is locally approximated by sufficiently accurate physical models taken from a hierarchy of decreasing complexity and varying over time. Feasible work regions of compressor stations consisting of several turbo compressors are included by semiconvex approximations of aggregated characteristic fields. A discrete adjoint approach within a first-discretize-then-optimize strategy is proposed and a sequential quadratic programming with an active set strategy is applied to solve the nonlinear constrained optimization problems resulting from a validation of nominations. The method proposed here accelerates the computation of near-term forecasts of sudden changes in the gas management and allows for an economic control of intra-day gas flow schedules in large networks. Case studies for real gas pipeline systems show the remarkable performance of the new method.}, language = {en} } @unpublished{DomschkeHillerLangetal.2021, author = {Domschke, Pia and Hiller, Benjamin and Lang, Jens and Mehrmann, Volker and Morandin, Riccardo and Tischendorf, Caren}, title = {Gas Network Modeling: An Overview}, pages = {53}, year = {2021}, abstract = {With this overview we want to provide a compilation of different models for the description of gas flow in networks in order to facilitate the introduction to the topic. Special attention is paid to the hierarchical structure inherent to the modeling, and the detailed description of individual components such as valves and compressors. Also included are network model classes based on purely algebraic relations, and energy-based port-Hamiltonian models. A short overview of basic numerical methods and concepts for the treatment of hyperbolic balance equations is also given. We do not claim completeness and refer in many places to the existing literature. The idea of a model catalog came to us in the context of the application for the CRC/Transregio 154 ``Mathematical modeling, simulation and optimization using the example of gas networks''. The present English translation is an extension from [P. Domschke, B. Hiller, J. Lang, and C. Tischendorf. Modellierung von Gasnetzwerken: Eine {\"U}bersicht. Preprint, TRR 154, 2017]. At this point we would like to thank the DFG for its support.}, language = {en} } @article{SchusterStrauchGugatetal.2020, author = {Schuster, Michael and Strauch, Elisa and Gugat, Martin and Lang, Jens}, title = {Probabilistic Constrained Optimization on Flow Networks}, volume = {Optimization and Engineering}, doi = {https://doi.org/10.1007/s11081-021-09619-x}, pages = {50}, year = {2020}, abstract = {Uncertainty often plays an important role in dynamic flow problems. In this paper, we consider both, a stationary and a dynamic flow model with uncertain boundary data on networks. We introduce two different ways how to compute the probability for random boundary data to be feasible, discussing their advantages and disadvantages. In this context, feasible means, that the flow corresponding to the random boundary data meets some box constraints at the network junctions. The first method is the spheric radial decomposition and the second method is a kernel density estimation. In both settings, we consider certain optimization problems and we compute derivatives of the probabilistic constraint using the kernel density estimator. Moreover, we derive necessary optimality conditions for the stationary and the dynamic case. Throughout the paper, we use numerical examples to illustrate our results by comparing them with a classical Monte Carlo approach to compute the desired probability.}, language = {en} } @article{LangScheichlSilvester2020, author = {Lang, Jens and Scheichl, Robert and Silvester, David}, title = {A Fully Adaptive Multilevel Stochastic Collocation Strategy for Solving Elliptic PDEs with Random Data}, volume = {Journal of Computational Physics}, number = {419}, doi = {doi:10.1016/j.jcp.2020.109692}, year = {2020}, abstract = {We propose and analyse a fully adaptive strategy for solving elliptic PDEs with random data in this work. A hierarchical sequence of adaptive mesh refinements for the spatial approximation is combined with adaptive anisotropic sparse Smolyak grids in the stochastic space in such a way as to minimize the computational cost. The novel aspect of our strategy is that the hierarchy of spatial approximations is sample dependent so that the computational effort at each collocation point can be optimised individually. We outline a rigorous analysis for the convergence and computational complexity of the adaptive multilevel algorithm and we provide optimal choices for error tolerances at each level. Two numerical examples demonstrate the reliability of the error control and the significant decrease in the complexity that arises when compared to single level algorithms and multilevel algorithms that employ adaptivity solely in the spatial discretisation or in the collocation procedure.}, language = {en} } @article{GraessleHinzeLangetal.2019, author = {Gr{\"a}ßle, Carmen and Hinze, Michael and Lang, Jens and Ullmann, Sebastian}, title = {POD model order reduction with space-adapted snapshots for incompressible flows}, volume = {Advances in Computational Mathematics}, number = {45}, doi = {doi:10.1007/s10444-019-09716-7}, pages = {2401 -- 2428}, year = {2019}, abstract = {We consider model order reduction based on proper orthogonal decomposition (POD) for unsteady incompressible Navier-Stokes problems, assuming that the snapshots are given by spatially adapted finite element solutions. We propose two approaches of deriving stable POD-Galerkin reduced-order models for this context. In the first approach, the pressure term and the continuity equation are eliminated by imposing a weak incompressibility constraint with respect to a pressure reference space. In the second approach, we derive an inf-sup stable velocity-pressure reduced-order model by enriching the velocity reduced space with supremizers computed on a velocity reference space. For problems with inhomogeneous Dirichlet conditions, we show how suitable lifting functions can be obtained from standard adaptive finite element computations. We provide a numerical comparison of the considered methods for a regularized lid-driven cavity problem.}, language = {en} } @unpublished{DomschkeHillerLangetal.2017, author = {Domschke, Pia and Hiller, Benjamin and Lang, Jens and Tischendorf, Caren}, title = {Modellierung von Gasnetzwerken: Eine {\"U}bersicht}, volume = {2717}, pages = {33}, year = {2017}, abstract = {Mit dieser {\"U}bersicht wollen wir eine Zusammenstellung von unterschiedlichen Modellen zur Beschreibung des Gasflusses in Netzwerken bereitstellen, um den Einstieg in das Thema zu erleichtern. Besonderes Augenmerk wird dabei auf die der Modellierung inneliegende hierarchische Struktur und die detaillierte Beschreibung einzelner Bauteile wie Ventile und Kompressoren gelegt. Daneben finden sich ebenfalls Netzmodellklassen, die auf rein algebraische Relationen aufbauen. Am Ende geben wir einen kurzen {\"U}berblick {\"u}ber grundlegende numerische Verfahren und Konzepte zur Behandlung von hyperbolischen Bilanzgleichungen. Wir erheben keinen Anspruch auf Vollst{\"a}ndigkeit und verweisen an vielen Stellen auf die bestehende Literatur. Die Idee eines Modellkataloges ist uns im Rahmen der Antragstellung zum SFB/Transregio 154 „Mathematische Modellierung, Simulation und Optimierung am Beispiel von Gasnetzwerken" gekommen. Wir m{\"o}chten an dieser Stelle die F{\"o}rderung durch die DFG dankend erw{\"a}hnen.}, language = {de} } @article{UllmannRotkvicLang2016, author = {Ullmann, Sebastian and Rotkvic, Marko and Lang, Jens}, title = {POD-Galerkin reduced-order modeling with adaptive finite element snapshots}, doi = {10.1016/j.jcp.2016.08.018}, year = {2016}, abstract = {We consider model order reduction by proper orthogonal decomposition (POD) for parametrized partial differential equations, where the underlying snapshots are computed with adaptive finite elements. We address computational and theoretical issues arising from the fact that the snapshots are members of different finite element spaces. We propose a method to create a POD-Galerkin model without interpolating the snapshots onto their common finite element mesh. The error of the reduced-order solution is not necessarily Galerkin orthogonal to the reduced space created from space-adapted snapshot. We analyze how this influences the error assessment for POD-Galerkin models of linear elliptic boundary value problems. As a numerical example we consider a two-dimensional convection-diffusion equation with a parametrized convective direction. To illustrate the applicability of our techniques to non-linear time-dependent problems, we present a test case of a two-dimensional viscous Burgers equation with parametrized initial data.}, language = {en} } @article{LangHundsdorfer2017, author = {Lang, Jens and Hundsdorfer, Willem}, title = {Extrapolation-based implicit-explicit Peer methods with optimised stability regions}, volume = {J. Comput. Phys.}, number = {337}, doi = {doi:10.1016/j.jcp.2017.02.034}, pages = {203 -- 215}, year = {2017}, abstract = {In this paper we investigate a new class of implicit-explicit (IMEX) two-step methods of Peer type for systems of ordinary differential equations with both non-stiff and stiff parts included in the source term. An extrapolation approach based on already computed stage values is applied to construct IMEX methods with favourable stability properties. Optimised IMEX-Peer methods of order p=2,3,4, are given as result of a search algorithm carefully designed to balance the size of the stability regions and the extrapolation errors. Numerical experiments and a comparison to other implicit-explicit methods are included.}, language = {en} } @article{LangMindt2017, author = {Lang, Jens and Mindt, Pascal}, title = {Entropy-Preserving Coupling Conditions for One-dimensional Euler Systems at Junctions}, volume = {Networks and Heterogeneous Media}, number = {Vol. 13}, doi = {10.3934/nhm.2018008}, pages = {177 -- 190}, year = {2017}, abstract = {This paper is concerned with a set of novel coupling conditions for the 3x3 one-dimensional Euler system with source terms at a junction of pipes with possibly different cross-sectional areas. Beside conservation of mass, we require the equality of the total enthalpy at the junction and that the specific entropy for pipes with outgoing flow equals the convex combination of all entropies that belong to pipes with incoming flow. These conditions ensure energy as well as entropy conservation at the junction. We prove the existence and uniqueness of solutions to the generalised Riemann problem at a junction in the neighbourhood of constant stationary states which belong to the subsonic region. This provides the basis for the well-posedness of the homogeneous and inhomogeneous Cauchy problems for initial data with sufficiently small total variation.}, language = {en} }