TY - JOUR A1 - Frenzel, David A1 - Lang, Jens T1 - A Third-Order Weighted Essentially Non-Oscillatory Scheme in Optimal Control Problems Governed by Nonlinear Hyperbolic Conservation Laws N2 - 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. Y1 - 2021 U6 - https://doi.org/https://doi.org/10.1007/s10589-021-00295-2 VL - Computational Optimization and Applications IS - 80 SP - 301 EP - 320 ER - TY - JOUR A1 - Domschke, Pia A1 - Kolb, Oliver A1 - Lang, Jens T1 - Fast and Reliable Transient Simulation and Continuous Optimization of Large-Scale Gas Networks N2 - 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. Y1 - 2021 U6 - https://doi.org/https://doi.org/10.1007/s00186-021-00765-7 PB - Mathematical Methods of Operations Research ER - TY - INPR A1 - Domschke, Pia A1 - Hiller, Benjamin A1 - Lang, Jens A1 - Mehrmann, Volker A1 - Morandin, Riccardo A1 - Tischendorf, Caren T1 - Gas Network Modeling: An Overview N2 - 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 Übersicht. Preprint, TRR 154, 2017]. At this point we would like to thank the DFG for its support. Y1 - 2021 ER - TY - JOUR A1 - Schuster, Michael A1 - Strauch, Elisa A1 - Gugat, Martin A1 - Lang, Jens T1 - Probabilistic Constrained Optimization on Flow Networks N2 - 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. KW - Probabilistic Constraints KW - Flow Networks KW - Gas Networks KW - Spheric Radial Decomposition KW - Kernel Density Estimator Y1 - 2020 U6 - https://doi.org/https://doi.org/10.1007/s11081-021-09619-x VL - Optimization and Engineering ER - TY - JOUR A1 - Lang, Jens A1 - Scheichl, Robert A1 - Silvester, David T1 - A Fully Adaptive Multilevel Stochastic Collocation Strategy for Solving Elliptic PDEs with Random Data N2 - 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. Y1 - 2021 U6 - https://doi.org/doi:10.1016/j.jcp.2020.109692 VL - Journal of Computational Physics IS - 419 ER - TY - JOUR A1 - Gräßle, Carmen A1 - Hinze, Michael A1 - Lang, Jens A1 - Ullmann, Sebastian T1 - POD model order reduction with space-adapted snapshots for incompressible flows N2 - 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. Y1 - 2019 U6 - https://doi.org/doi:10.1007/s10444-019-09716-7 VL - Advances in Computational Mathematics IS - 45 SP - 2401 EP - 2428 ER - TY - INPR A1 - Domschke, Pia A1 - Hiller, Benjamin A1 - Lang, Jens A1 - Tischendorf, Caren T1 - Modellierung von Gasnetzwerken: Eine Übersicht N2 - Mit dieser Übersicht 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 Überblick über grundlegende numerische Verfahren und Konzepte zur Behandlung von hyperbolischen Bilanzgleichungen. Wir erheben keinen Anspruch auf Vollständigkeit 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 möchten an dieser Stelle die Förderung durch die DFG dankend erwähnen. KW - Euler-Gleichungen, isotherme Euler-Gleichungen, Modellhierarchie, Netzelemente Y1 - 2017 VL - 2717 ER - TY - JOUR A1 - Ullmann, Sebastian A1 - Rotkvic, Marko A1 - Lang, Jens T1 - POD-Galerkin reduced-order modeling with adaptive finite element snapshots N2 - 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. Y1 - 2016 U6 - https://doi.org/10.1016/j.jcp.2016.08.018 ER - TY - JOUR A1 - Lang, Jens A1 - Hundsdorfer, Willem T1 - Extrapolation-based implicit–explicit Peer methods with optimised stability regions N2 - 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. Y1 - 2017 U6 - https://doi.org/doi:10.1016/j.jcp.2017.02.034 VL - J. Comput. Phys. IS - 337 SP - 203 EP - 215 ER - TY - JOUR A1 - Lang, Jens A1 - Mindt, Pascal T1 - Entropy-Preserving Coupling Conditions for One-dimensional Euler Systems at Junctions N2 - 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. Y1 - 2017 U6 - https://doi.org/10.3934/nhm.2018008 VL - Networks and Heterogeneous Media IS - Vol. 13 SP - 177 EP - 190 ER - TY - GEN A1 - Lang, Jens A1 - Leugering, Günter A1 - Martin, Alexander A1 - Tischendorf, Caren T1 - Gasnetzwerke: Mathematische Modellierung, Simulation und Optimierung N2 - Im Mai 2014 wurde seitens der DFG der Transregio 154 Mathematische Modellierung, Simulation und Optimierung am Beispiel von Gasnetzwerken bewilligt. Die Forschungsarbeiten an den beteiligten Standorten, der Friedrich-Alexander-Universität Erlangen-Nürnberg (Sprecheruniversität; Sprecher: Alexander Martin), der Technischen Universität Darmstadt (stellvertretender Sprecher: Jens Lang), der Technischen Universität Berlin, der Humboldt Universität (stellvertretende Sprecherin: Caren Tischendorf) sowie den Partnerinstitutionen Weierstraß-Institut (Berlin), Konrad-Zuse-Zentrum (Berlin) und Universität Duisburg-Essen haben im Oktober 2014 begonnen. Y1 - 2015 U6 - https://doi.org/10.1515/dmvm-2015-0013 ER - TY - JOUR A1 - Domschke, Pia A1 - Dua, Aseem A1 - Stolwijk, Jeroen J. A1 - Lang, Jens A1 - Mehrmann, Volker T1 - Adaptive Refinement Strategies for the Simulation of Gas Flow in Networks using a Model Hierarchy N2 - A model hierarchy that is based on the one-dimensional isothermal Euler equations of fluid dynamics is used for the simulation and optimisation of gas flow through a pipeline network. Adaptive refinement strategies have the aim of bringing the simulation error below a prescribed tolerance while keeping the computational costs low. While spatial and temporal stepsize adaptivity is well studied in the literature, model adaptivity is a new field of research. The problem of finding an optimal refinement strategy that combines these three types of adaptivity is a generalisation of the unbounded knapsack problem. A refinement strategy that is currently used in gas flow simulation software is compared to two novel greedy-like strategies. Both a theoretical experiment and a realistic gas flow simulation show that the novel strategies significantly outperform the current refinement strategy with respect to the computational cost incurred. KW - gas supply networks KW - model hierarchy KW - error estimators KW - model adaptivity KW - refinement strategies Y1 - 2017 U6 - https://doi.org/10.1553/etna_vol48s97 VL - Electronic Transactions on Numerical Analysis IS - Vol. 48 SP - 97 EP - 113 ER - TY - JOUR A1 - Schneider, Moritz A1 - Lang, Jens A1 - Hundsdorfer, Willem T1 - Extrapolation-Based Super-Convergent Implicit-Explicit Peer Methods with A-stable Implicit Part N2 - In this paper we extend the implicit-explicit (IMEX) methods of Peer type recently developed in [Lang, Hundsdorfer, J. Comp. Phys., 337:203–215, 2017] to a broader class of two-step methods that allow the construction of super- convergent IMEX-Peer methods with A-stable implicit part. IMEX schemes combine the necessary stability of implicit and low computational costs of ex- plicit methods to efficiently solve systems of ordinary differential equations with both stiff and non-stiff parts included in the source term. To construct super- convergent IMEX-Peer methods with favourable stability properties, we derive necessary and sufficient conditions on the coefficient matrices and apply an extrapolation approach based on already computed stage values. Optimised super-convergent IMEX-Peer methods of order s + 1 for s = 2, 3, 4 stages 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 IMEX-Peer methods are included. Y1 - 2017 U6 - https://doi.org/10.1016/j.jcp.2018.04.006 VL - J. Comput. Physics IS - Vol. 367 SP - 121 EP - 133 ER - TY - JOUR A1 - Schneider, Moritz A1 - Lang, Jens A1 - Weiner, Rüdiger T1 - Super-Convergent Implicit-Explicit Peer Methods with Variable Step Sizes N2 - Dynamical systems with sub-processes evolving on many different time scales are ubiquitous in applications. Their efficient solution is greatly enhanced by automatic time step variation. This paper is concerned with the theory, construction and application of IMEX-Peer methods that are super-convergent for variable step sizes and A-stable in the implicit part. IMEX schemes combine the necessary stability of implicit and low computational costs of explicit methods to efficiently solve systems of ordinary differential equations with both stiff and non-stiff parts included in the source term. To construct super-convergent IMEX-Peer methods which keep their higher order for variable step sizes and exhibit favourable linear stability properties, we derive necessary and sufficient conditions on the nodes and coefficient matrices and apply an extrapolation approach based on already computed stage values. New super-convergent IMEX-Peer methods of order s + 1 for s = 2, 3, 4 stages are given as result of additional order conditions which maintain the super-convergence property independent of step size changes. Numerical experiments and a comparison to other super-convergent IMEX-Peer methods show the potential of the new methods when applied with local error control. Y1 - 2019 U6 - https://doi.org/doi:10.1016/j.cam.2019.112501 VL - J. Comput. Appl. Math. IS - 387 SP - 112501 ER - TY - JOUR A1 - Ullmann, Sebastian A1 - Müller, Christopher A1 - Lang, Jens T1 - Stochastic Galerkin Reduced Basis Methods for Parametrized Linear Convection-Diffusion-Reaction Equations N2 - We consider the estimation of parameter-dependent statistics of functional outputs of steady-state convection–diffusion–reaction equations with parametrized random and deterministic inputs in the framework of linear elliptic partial differential equations. For a given value of the deterministic parameter, a stochastic Galerkin finite element (SGFE) method can estimate the statistical moments of interest of a linear output at the cost of solving a single, large, block-structured linear system of equations. We propose a stochastic Galerkin reduced basis (SGRB) method as a means to lower the computational burden when statistical outputs are required for a large number of deterministic parameter queries. Our working assumption is that we have access to the computational resources necessary to set up such a reduced-order model for a spatial-stochastic weak formulation of the parameter-dependent model equations. In this scenario, the complexity of evaluating the SGRB model for a new value of the deterministic parameter only depends on the reduced dimension. To derive an SGRB model, we project the spatial-stochastic weak solution of a parameter-dependent SGFE model onto a reduced basis generated by a proper orthogonal decomposition (POD) of snapshots of SGFE solutions at representative values of the parameter. We propose residual-corrected estimates of the parameter-dependent expectation and variance of linear functional outputs and provide respective computable error bounds.We test the SGRB method numerically for a convection–diffusion–reaction problem, choosing the convective velocity as a deterministic parameter and the parametrized reactivity or diffusivity field as a random input. Compared to a standard reduced basis model embedded in a Monte Carlo sampling procedure, the SGRB model requires a similar number of reduced basis functions to meet a given tolerance requirement. However, only a single run of the SGRB model suffices to estimate a statistical output for a new deterministic parameter value, while the standard reduced basis model must be solved for each Monte Carlo sample. Y1 - 2018 U6 - https://doi.org/doi:10.3390/fluids6080263 VL - Fluids IS - 6 SP - 263 ER - TY - JOUR A1 - Mindt, Pascal A1 - Lang, Jens A1 - Domschke, Pia T1 - Entropy-Preserving Coupling of Hierarchical Gas Models N2 - This paper is concerned with coupling conditions at junctions for transport models which differ in their fidelity to describe transient flow in gas pipelines. It also includes the integration of compressors between two pipes with possibly different models. A hierarchy of three one-dimensional gas transport models is built through the 3 × 3 polytropic Euler equations, the 2 × 2 isentropic Euler equations and a simplified version of it for small velocities. To ensure entropy preservation, we make use of the novel entropy-preserving coupling conditions recently proposed by Lang and Mindt [Netw. Heterog. Media, 13:177-190, 2018] and 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. We prove the existence and uniqueness of solutions to generalised Riemann problems at a junction in the neighbourhood of constant coupling functions and stationary states which belong to the subsonic region. This provides the basis for the well-posedness of certain Cauchy problems for initial data with sufficiently small total variation. Y1 - 2018 U6 - https://doi.org/doi:10.1137/19M1240034 VL - SIAM Journal on Mathematical Analysis IS - 51 SP - 4754 EP - 4775 ER - TY - INPR A1 - Lang, Jens A1 - Schmitt, Bernhard A. T1 - Implicit Peer Triplets in Gradient-Based Solution Algorithms for ODE Constrained Optimal Control N2 - It is common practice to apply gradient-based optimization algorithms to numerically solve large-scale ODE constrained optimal control problems. Gradients of the objective function are most efficiently computed by approximate adjoint variables. High accuracy with moderate computing time can be achieved by such time integration methods that satisfy a sufficiently large number of adjoint order conditions and supply gradients with higher orders of consistency. In this paper, we upgrade our former implicit two-step Peer triplets constructed in [Algorithms, 15:310, 2022] to meet those new requirements. Since Peer methods use several stages of the same high stage order, a decisive advantage is their lack of order reduction as for semi-discretized PDE problems with boundary control. Additional order conditions for the control and certain positivity requirements now intensify the demands on the Peer triplet. We discuss the construction of 4-stage methods with order pairs (4,3) and (3,3) in detail and provide three Peer triplets of practical interest. We prove convergence for s-stage methods, for instance, order s for the state variables even if the adjoint method and the control satisfy the conditions for order s-1, only. Numerical tests show the expected order of convergence for the new Peer triplets. Y1 - 2023 VL - http://arxiv.org/abs/2303.18180 ER - TY - INPR A1 - Domschke, Pia A1 - Giesselmann, Jan A1 - Lang, Jens A1 - Breiten, Tobias A1 - Mehrmann, Volker A1 - Morandin, Riccardo A1 - Hiller, Benjamin A1 - Tischendorf, Caren T1 - Gas Network Modeling: An Overview (Extended English Version) N2 - 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. Y1 - 2023 ER - TY - JOUR A1 - Lang, Jens A1 - Schmitt, Bernhard A. T1 - Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems N2 - This paper is concerned with the construction and convergence analysis of novel implicit Peer triplets of two-step nature with four stages for nonlinear ODE constrained optimal control problems. We combine the property of superconvergence of some standard Peer method for inner grid points with carefully designed starting and end methods to achieve order four for the state variables and order three for the adjoint variables in a first-discretize-then-optimize approach together with A-stability. The notion triplets emphasizes that these three different Peer methods have to satisfy additional matching conditions. Four such Peer triplets of practical interest are constructed. Also as a benchmark method, the well-known backward differentiation formula BDF4, which is only A(73.35)-stable, is extended to a special Peer triplet to supply an adjoint consistent method of higher order and BDF type with equidistant nodes. Within the class of Peer triplets, we found a diagonally implicit A(84)-stable method with nodes symmetric in [0,1] to a common center that performs equally well. Numerical tests with three well established optimal control problems confirm the theoretical findings also concerning A-stability. Y1 - U6 - https://doi.org/https://doi.org/10.3390/a15090310 VL - Algorithms IS - Vol. 15 ER - TY - JOUR A1 - Lang, Jens A1 - Schmitt, Bernhard A. T1 - Discrete Adjoint Implicit Peer Methods in Optimal Control N2 - It is well known that in the first-discretize-then-optimize approach in the control of ordinary differential equations the adjoint method may converge under additional order conditions only. For Peer two-step methods we derive such adjoint order conditions and pay special attention to the boundary steps. For $s$-stage methods, we prove convergence of order s for the state variables if the adjoint method satisfies the conditions for order s-1, at least. We remove some bottlenecks at the boundaries encountered in an earlier paper of the first author et al. [J. Comput. Appl. Math., 262:73--86, 2014] and discuss the construction of 3-stage methods for the order pair (3,2) in detail including some matrix background for the combined forward and adjoint order conditions. The impact of nodes having equal differences is highlighted. It turns out that the most attractive methods are related to BDF. Three 3-stage methods are constructed which show the expected orders in numerical tests. Y1 - 2020 U6 - https://doi.org/https://doi.org/10.1016/j.cam.2022.114596 VL - Journal of Computational and Applied Mathematics IS - 416:114596 ER -