@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} } @misc{LangLeugeringMartinetal., author = {Lang, Jens and Leugering, G{\"u}nter and Martin, Alexander and Tischendorf, Caren}, title = {Gasnetzwerke: Mathematische Modellierung, Simulation und Optimierung}, doi = {10.1515/dmvm-2015-0013}, abstract = {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-Universit{\"a}t Erlangen-N{\"u}rnberg (Sprecheruniversit{\"a}t; Sprecher: Alexander Martin), der Technischen Universit{\"a}t Darmstadt (stellvertretender Sprecher: Jens Lang), der Technischen Universit{\"a}t Berlin, der Humboldt Universit{\"a}t (stellvertretende Sprecherin: Caren Tischendorf) sowie den Partnerinstitutionen Weierstraß-Institut (Berlin), Konrad-Zuse-Zentrum (Berlin) und Universit{\"a}t Duisburg-Essen haben im Oktober 2014 begonnen.}, language = {de} } @article{DomschkeDuaStolwijketal.2017, author = {Domschke, Pia and Dua, Aseem and Stolwijk, Jeroen J. and Lang, Jens and Mehrmann, Volker}, title = {Adaptive Refinement Strategies for the Simulation of Gas Flow in Networks using a Model Hierarchy}, volume = {Electronic Transactions on Numerical Analysis}, number = {Vol. 48}, doi = {10.1553/etna_vol48s97}, pages = {97 -- 113}, year = {2017}, abstract = {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.}, language = {en} } @article{SchneiderLangHundsdorfer, author = {Schneider, Moritz and Lang, Jens and Hundsdorfer, Willem}, title = {Extrapolation-Based Super-Convergent Implicit-Explicit Peer Methods with A-stable Implicit Part}, volume = {J. Comput. Physics}, number = {Vol. 367}, doi = {10.1016/j.jcp.2018.04.006}, pages = {121 -- 133}, abstract = {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.}, language = {en} } @article{SchneiderLangWeiner2019, author = {Schneider, Moritz and Lang, Jens and Weiner, R{\"u}diger}, title = {Super-Convergent Implicit-Explicit Peer Methods with Variable Step Sizes}, volume = {J. Comput. Appl. Math.}, number = {387}, doi = {doi:10.1016/j.cam.2019.112501}, pages = {112501}, year = {2019}, abstract = {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.}, language = {en} } @article{UllmannMuellerLang2018, author = {Ullmann, Sebastian and M{\"u}ller, Christopher and Lang, Jens}, title = {Stochastic Galerkin Reduced Basis Methods for Parametrized Linear Convection-Diffusion-Reaction Equations}, volume = {Fluids}, number = {6}, doi = {doi:10.3390/fluids6080263}, pages = {263}, year = {2018}, abstract = {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.}, language = {en} } @article{MindtLangDomschke2018, author = {Mindt, Pascal and Lang, Jens and Domschke, Pia}, title = {Entropy-Preserving Coupling of Hierarchical Gas Models}, volume = {SIAM Journal on Mathematical Analysis}, number = {51}, doi = {doi:10.1137/19M1240034}, pages = {4754 -- 4775}, year = {2018}, abstract = {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.}, language = {en} } @unpublished{LangSchmitt2023, author = {Lang, Jens and Schmitt, Bernhard A.}, title = {Implicit Peer Triplets in Gradient-Based Solution Algorithms for ODE Constrained Optimal Control}, volume = {http://arxiv.org/abs/2303.18180}, year = {2023}, abstract = {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.}, language = {en} } @unpublished{DomschkeGiesselmannLangetal.2023, author = {Domschke, Pia and Giesselmann, Jan and Lang, Jens and Breiten, Tobias and Mehrmann, Volker and Morandin, Riccardo and Hiller, Benjamin and Tischendorf, Caren}, title = {Gas Network Modeling: An Overview (Extended English Version)}, pages = {54}, year = {2023}, 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.}, language = {en} } @article{LangSchmitt, author = {Lang, Jens and Schmitt, Bernhard A.}, title = {Implicit A-Stable Peer Triplets for ODE Constrained Optimal Control Problems}, volume = {Algorithms}, number = {Vol. 15}, doi = {https://doi.org/10.3390/a15090310}, abstract = {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.}, language = {en} } @article{LangSchmitt2020, author = {Lang, Jens and Schmitt, Bernhard A.}, title = {Discrete Adjoint Implicit Peer Methods in Optimal Control}, volume = {Journal of Computational and Applied Mathematics}, number = {416:114596}, doi = {https://doi.org/10.1016/j.cam.2022.114596}, year = {2020}, abstract = {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.}, language = {en} } @article{StrelowGerischLangetal., author = {Strelow, Erik Laurin and Gerisch, Alf and Lang, Jens and Pfetsch, Marc E.}, title = {Physics-Informed Neural Networks: A Case Study for Gas Transport Problems}, volume = {Journal of Computational Physics}, number = {Vol. 481}, pages = {112041}, abstract = {Physics informed neural networks have been recently proposed and offer a new promising method to solve differential equations. They have been adapted to many more scenarios and different variations of the original method have been proposed. In this case study we review many of these variations. We focus on variants that can compensate for imbalances in the loss function and perform a comprehensive numerical comparison of these variants with application to gas transport problems. Our case study includes different formulations of the loss function, different algorithmic loss balancing methods, different optimization schemes and different numbers of parameters and sampling points. We conclude that the original PINN approach with specifically chosen constant weights in the loss function gives the best results in our tests. These weights have been obtained by a computationally expensive random-search scheme. We further conclude for our test case that loss balancing methods which were developed for other differential equations have no benefit for gas transport problems, that the control volume physics informed formulation has no benefit against the initial formulation and that the best optimization strategy is the L-BFGS method.}, language = {en} } @article{LangDomschkeStrauch2020, author = {Lang, Jens and Domschke, Pia and Strauch, Elisa}, title = {Adaptive Single- and Multilevel Stochastic Collocation Methods for Uncertain Gas Transport in Large-Scale Networks}, volume = {In: Mesh Generation and Adaptation, Cutting-Edge Techniques. R. Sevilla, S. Perotto, K. Morgan (eds.), SEMA-SIMAI Springer Series}, number = {Vol. 30}, pages = {113 -- 135}, year = {2020}, abstract = {In this paper, we are concerned with the quantification of uncertainties that arise from intra-day oscillations in the demand for natural gas transported through large-scale networks. The short-term transient dynamics of the gas flow is modelled by a hierarchy of hyperbolic systems of balance laws based on the isentropic Euler equations. We extend a novel adaptive strategy for solving elliptic PDEs with random data, recently proposed and analysed by Lang, Scheichl, and Silvester [J. Comput. Phys., 419:109692, 2020], to uncertain gas transport problems. Sample-dependent adaptive meshes and a model refinement in the physical space is combined with adaptive anisotropic sparse Smolyak grids in the stochastic space. A single-level approach which balances the discretization errors of the physical and stochastic approximations and a multilevel approach which additionally minimizes the computational costs are considered. Two examples taken from a public gas library demonstrate the reliability of the error control of expectations calculated from random quantities of interest, and the further use of stochastic interpolants to, e.g., approximate probability density functions of minimum and maximum pressure values at the exits of the network.}, language = {en} } @unpublished{WilkaLang, author = {Wilka, Hendrik and Lang, Jens}, title = {Adaptive hp-Polynomial Based Sparse Grid Collocation Algorithms for Piecewise Smooth Functions with Kinks}, abstract = {High-dimensional interpolation problems appear in various applications of uncertainty quantification, stochastic optimization and machine learning. Such problems are computationally expensive and request the use of adaptive grid generation strategies like anisotropic sparse grids to mitigate the curse of dimensionality. However, it is well known that the standard dimension-adaptive sparse grid method converges very slowly or even fails in the case of non-smooth functions. For piecewise smooth functions with kinks, we construct two novel hp-adaptive sparse grid collocation algorithms that combine low-order basis functions with local support in parts of the domain with less regularity and variable-order basis functions elsewhere. Spatial refinement is realized by means of a hierarchical multivariate knot tree which allows the construction of localised hierarchical basis functions with varying order. Hierarchical surplus is used as an error indicator to automatically detect the non-smooth region and adaptively refine the collocation points there. The local polynomial degrees are optionally selected by a greedy approach or a kink detection procedure. Three numerical benchmark examples with different dimensions are discussed and comparison with locally linear and highest degree basis functions are given to show the efficiency and accuracy of the proposed methods.}, language = {en} } @article{LangSchmitt, author = {Lang, Jens and Schmitt, Bernhard A.}, title = {A Stiff MOL Boundary Control Problem for the 1D Heat Equation with Exact Discrete Solution}, volume = {Journal of Optimization Theory and Applications}, number = {Vol. 196}, doi = {https://doi.org/10.1007/s10957-022-02154-4}, pages = {1106 -- 1118}, abstract = {Method-of-lines discretizations are demanding test problems for stiff inte- gration methods. However, for PDE problems with known analytic solution the presence of space discretization errors or the need to use codes to compute reference solutions may limit the validity of numerical test results. To over- come these drawbacks we present in this short note a simple test problem with boundary control, a situation where one-step methods may suffer from order reduction. We derive exact formulas for the solution of an optimal boundary control problem governed by a one-dimensional discrete heat equation and an objective function that measures the distance of the final state from the target and the control costs. This analytical setting is used to compare the numeri- cally observed convergence orders for selected implicit Runge-Kutta and Peer two-step methods of classical order four which are suitable for optimal control problems.}, language = {en} }