TY - JOUR A1 - Breiten, Tobias A1 - Morandin, Riccardo A1 - Schulze, Philipp T1 - Error bounds for port-Hamiltonian model and controller reduction based on system balancing N2 - Linear quadratic Gaussian (LQG) control design for port-Hamiltonian systems is studied. The recently proposed method from [42] is reviewed and modified such that the resulting controllers have a port-Hamiltonian (pH) realization. Based on this new modification, a reduced-order controller is obtained by truncation of a balanced system. The approach is shown to be closely related to classical LQG balanced truncation and shares a similar a priori error bound with respect to the gap metric. With regard to this error bound, a theoretically optimal pH-representation is derived. Consequences for pH-preserving balanced truncation model reduction are discussed and shown to yield two different classical H∞ -error bounds. Numerical examples illustrate the main theoretical findings. KW - port-Hamiltonian systems KW - model order reduction KW - LQG control design KW - error bound Y1 - 2020 U6 - https://doi.org/10.1016/j.camwa.2021.07.022 CY - Computers & Mathematics with Applications ER - TY - INPR A1 - Breiten, Tobias A1 - Unger, Benjamin T1 - Passivity preserving model reduction via spectral factorization N2 - We present a novel model-order reduction (MOR) method for linear time-invariant systems that preserves passivity and is thus suited for structure-preserving MOR for port-Hamiltonian (pH) systems. Our algorithm exploits the well-known spectral factorization of the Popov function by a solution of the Kalman-Yakubovich-Popov (KYP) inequality. It performs MOR directly on the spectral factor inheriting the original system’s sparsity enabling MOR in a large-scale context. Our analysis reveals that the spectral factorization corresponding to the minimal solution of an associated algebraic Riccati equation is preferable from a model reduction perspective and benefits pH-preserving MOR methods such as a modified version of the iterative rational Krylov algorithm (IRKA). Numerical examples demonstrate that our approach can produce high-fidelity reduced-order models close to (unstructured) H2 -optimal reduced-order models. KW - passivity KW - port-Hamiltonian systems KW - structure-preserving model order reduction KW - spectral factorization KW - H2-optimal Y1 - 2021 ER - TY - INPR A1 - De Santis, Marianna A1 - de Vries, Sven A1 - Schmidt, Martin A1 - Winkel, Lukas T1 - A Penalty Branch-and-Bound Method for Mixed-Binary Linear Complementarity Problems N2 - Linear complementarity problems (LCPs) are an important modeling tool for many practically relevant situations but also have many important applications in mathematics itself. Although the continuous version of the problem is extremely well studied, much less is known about mixed-integer LCPs (MILCPs) in which some variables have to be integer-valued in a solution. In particular, almost no tailored algorithms are known besides reformulations of the problem that allow to apply general-purpose mixed-integer linear programming solvers. In this paper, we present, theoretically analyze, enhance, and test a novel branch-and-bound method for MILCPs. The main property of this method is that we do not ``branch'' on constraints as usual but by adding suitably chosen penalty terms to the objective function. By doing so, we can either provably compute an MILCP solution if one exists or compute an approximate solution that minimizes an infeasibility measure combining integrality and complementarity conditions. We enhance the method by MILCP-tailored valid inequalities, node selection strategies, branching rules, and warmstarting techniques. The resulting algorithm is shown to clearly outperform two benchmark approaches from the literature. KW - Mixed-Integer Programming KW - Linear Complementarity Problems KW - Mixed-Integer Linear Complementarity Problems KW - Branch-and-Bound KW - Penalty Methods Y1 - 2021 ER - TY - THES A1 - Plein, Fränk T1 - When Bilevel Optimization Meets Gas Networks: Feasibility of Bookings in the European Entry-Exit Gas Market. Computational Complexity Results and Bilevel Optimization Approaches N2 - Transport and trade of gas are decoupled after the liberalization of the European gas markets, which are now organized as so-called entry-exit systems. At the core of this market system are bookings and nominations, two special capacity-right contracts that grant traders access to the gas network. The latter is operated by a separate entity, known as the transmission system operator (TSO), who is in charge of the transport of gas from entry to exit nodes. In the mid to long term, traders sign a booking contract with the TSO to obtain injection and withdrawal capacities at entry and exit nodes, respectively. On a day-ahead basis, they then nominate within these booked capacities a balanced load flow of the planned amounts of gas to be injected into and withdrawn from the network the next day. The key property is that by signing a booking contract, the TSO is obliged to guarantee transportability for all balanced load flows in compliance with the booked capacities. To assess the feasibility of a booking, it is therefore necessary to check the feasibility of infinitely many nominations. As a result, deciding if a booking is feasible is a challenging mathematical problem, which we investigate in this dissertation. Our results range from passive networks, consisting of pipes only, to active networks, containing controllable elements to influence gas flows. Since the study of the latter naturally leads to a bilevel framework, we first consider some more general properties of bilevel optimization. For the case of linear bilevel optimization, we consider the hardness of validating the correctness of big-Ms often used in solving these problems via a single-level reformulation. We also derive a family of valid inequalities to be used in a bilevel-tailored branch-and-cut algorithm as a big-M-free alternative. We then turn to the study of feasible bookings. First, we present our results on passive networks, for which bilevel approaches are not required. A characterization of feasible bookings on passive networks is derived in terms of a finite set of nominations. While computing these nominations is a difficult task in general, we present polynomial complexity results for the special cases of tree-shaped or single-cycle passive networks. Finally, we consider networks with linearly modeled active elements. After obtaining a bilevel optimization model that allows us to determine the feasibility of a booking in this case, we derive various single-level reformulations to solve the problem. In addition, we obtain novel characterizations of feasible bookings on active networks, which generalize our characterization in the passive case. The performance of these various approaches is compared in a case study on two networks from the literature, one of which is a simplified version of the Greek gas network. Y1 - 2021 ER - TY - GEN A1 - Beck, Yasmine A1 - Schmidt, Martin T1 - A Gentle and Incomplete Introduction to Bilevel Optimization N2 - These are lecture notes on bilevel optimization. The class of bilevel optimization problems is formally introduced and motivated using examples from different fields. Afterward, the main focus is on how to solve linear and mixed-integer linear bilevel optimization problems. To this end, we first consider various single-level reformulations of bilevel optimization problems with linear or convex follower problems, discuss geometric properties of linear bilevel problems, and study different algorithms for solving linear bilevel problems. Finally, we consider mixed-integer linear bilevel problems, discuss the main obstacles for deriving exact as well as effective solution methods, and derive a branch-and-bound method for solving these problems. KW - Bilevel Optimization KW - Lecture Notes Y1 - 2021 ER - TY - THES A1 - Nowak, Daniel T1 - Nonconvex Nash Games - Solution Concepts and Algorithms N2 - Game theory is a mathematical approach to model competition between several parties, called players. The goal of each player is to choose a strategy, which solves his optimization problem, i.e. minimizes or maximizes his objective function. Due to the competitive setting, this strategy may influence the optimization problems of other players. In the non-cooperative setting each player acts selfish, meaning he does not care about the objective of his opponents. A solution concept for this problem is a Nash equilibrium, which was introduced by John Forbes Nash in his Ph.D. thesis in 1950. Convexity of the optimization problems is a crucial assumption for the existence of Nash equilibria. This work investigates settings, where this convexity assumption fails to hold. The first part of this thesis extends results of Jong-Shi Pang and Gesualdo Scutari from their paper ``Nonconvex Games with Side Constraints'' published in 2011. In this publication, a game with possibly nonconvex objective functions and nonconvex individual and shared inequality constraints was investigated. We extend these results twofold. Firstly, we generalize the individual and shared polyhedral constraints to general convex constraints and, secondly, we introduce convex and nonconvex, individual and shared equality constraints. After a detailed comparison of solution concepts for the generalized Nash game and a related Nash game, we show that so-called quasi-Nash equilibria exist under similar assumptions than in the original work, provided some additional constraint qualification holds. Subsequently, we prove that the existence of Nash equilibria needs additional assumptions on the gradients of the equality constraints. Furthermore, a special case of a multi-leader multi-follower game is investigated. We show the convergence of epsilon-quasi-Nash equilibria to C-stationary points and prove that these are also Clarke-stationary under reasonable assumptions. In the second part of this thesis, an application in computation offloading is investigated. We consider several mobile users that are able to offload parts of a computation task to a connected server. However, the server has limited computation capacities which leads to competition among the mobile users. If a user decides to offload a part of his computation, he needs to wait for the server to finish before he can assemble the results of his computation. This leads to a vanishing constraint in the optimization problem of the mobile users which is a nonconvex and nonsmooth condition. We show the existence of a unique Nash equilibrium for the computation offloading game and provide an efficient algorithm for its computation. Furthermore, we present two extensions to this game, which inherit similar properties and we also show the limitations of these formulations. The third part investigates a hierarchical constrained Cournot game. In the upper level, several firms decide on capacities which act as constraints for the production variables. In the lower level the same firms engage in a Cournot competition, where they choose production variables to maximize profit. The prior chosen capacities are upper bounds on these production variables. This hierarchical setting induces nonconvexity and nonsmoothness in the upper level objective functions. After a detailed sensitivity analysis of the lower level, we give necessary optimality conditions for the upper level, i.e. for the hierarchical Cournot game. Using these conditions, we construct an algorithm which provably finds all Nash equilibria of the game, provided some assumptions are satisfied. This algorithm is numerically tested on several examples which are motivated by the gas market. KW - Game Theory KW - Nash Games KW - Optimization Y1 - 2021 U6 - https://doi.org/10.26083/tuprints-00017637 PB - E-Publishing-Service der TU Darmstadt CY - Darmstadt ER - TY - JOUR A1 - Bárcena-Petisco, J.A. A1 - Cavalcante, M. A1 - Coclite, G.M. A1 - de Nitti, N. A1 - Zuazua, E. T1 - Control of Hyperbolic and Parabolic Equations on Networks and Singular limits N2 - We study the controllability properties of the transport equation and of parabolic equations posed on a tree. Using a control localized on the exterior nodes, we prove that the hyperbolic and the parabolic systems are null-controllable. The hyperbolic proof relies on the method of characteristics, the parabolic one on duality arguments and Carleman inequalities. We also show that the parabolic system may not be controllable if we do not act on all exterior vertices because of symmetries. Moreover, we estimate the cost of the null-controllability of transport-diffusion equations with diffusivity ε > 0ε>0 and study its asymptotic behavior when ε → 0^+ε→0 + . We prove that the cost of the controllability decays for a time sufficiently large and explodes for short times. This is done by duality arguments allowing to reduce the problem to obtain observability estimates which depend on the viscosity parameter. These are derived by using Agmon and Carleman inequalities. Y1 - 2021 ER - TY - INPR A1 - Grübel, Julia A1 - Huber, Olivier A1 - Hümbs, Lukas A1 - Klimm, Max A1 - Schmidt, Martin A1 - Schwartz, Alexandra T1 - Nonconvex Equilibrium Models for Energy Markets: Exploiting Price Information to Determine the Existence of an Equilibrium N2 - Motivated by examples from the energy sector, we consider market equilibrium problems (MEPs) involving players with nonconvex strategy spaces or objective functions, where the latter are assumed to be linear in market prices. We propose an algorithm that determines if an equilibrium of such an MEP exists and that computes an equilibrium in case of existence. Three key prerequisites have to be met. First, appropriate bounds on market prices have to be derived from necessary optimality conditions of some players. Second, a technical assumption is required for those prices that are not uniquely determined by the derived bounds. Third, nonconvex optimization problems have to be solved to global optimality. We test the algorithm on well-known instances from the power and gas literature that meet these three prerequisites. There, nonconvexities arise from considering the transmission system operator as an additional player besides producers and consumers who, e.g., switches lines or faces nonlinear physical laws. Our numerical results indicate that equilibria often exist, especially for the case of continuous nonconvexities in the context of gas market problems. KW - Energy markets KW - Nonconvex games KW - Existence KW - Equilibrium computation KW - Perfect competition Y1 - 2021 ER - TY - INPR A1 - Gugat, Martin A1 - Habermann, Jens A1 - Hintermüller, Michael A1 - Huber, Olivier T1 - Constrained exact boundary controllability of a semilinear model for pipeline gas flow N2 - While the quasilinear isothermal Euler equations are an excellent model for gas pipeline flow, the operation of the pipeline flow with high pressure and small Mach numbers allows us to obtain approximate solutions by a simpler semilinear model. We provide a derivation of the semilinear model that shows that the semilinear model is valid for sufficiently low Mach numbers and sufficiently high pressures. We prove an existence result for continuous solutions of the semilinear model that takes into account lower and upper bounds for the pressure and an upper bound for the magnitude of the Mach number of the gas flow. These state constraints are important both in the operation of gas pipelines and to guarantee that the solution remains in the set where the model is physically valid. We show the constrained exact boundary controllability of the system with the same pressure and Mach number constraints. Y1 - 2021 ER - TY - INPR A1 - Branda, Martin A1 - Henrion, René A1 - Pištěk, Miroslav T1 - Value at risk approach to producer's best response in electricity market with uncertain demand N2 - We deal with several sources of uncertainty in electricity markets. The independent system operator (ISO) maximizes the social welfare using chance constraints to hedge against discrepancies between the estimated and real electricity demand. We find an explicit solution of the ISO problem, and use it to tackle the problem of a producer. In our model, production as well as income of a producer are determined based on the estimated electricity demand predicted by the ISO, that is unknown to producers. Thus, each producer is hedging against the uncertainty of prediction of the demand using the value-at-risk approach. To illustrate our results, a numerical study of a producer's best response given a historical distribution of both estimated and real electricity demand is provided. KW - electricity market KW - multi-leader-common-follower game KW - stochastic demand KW - day-ahead bidding KW - chance constraints Y1 - 2021 ER - TY - JOUR A1 - Berthold, Holger A1 - Heitsch, Holger A1 - Henrion, René A1 - Schwientek, Jan T1 - On the algorithmic solution of optimization problems subject to probabilistic/robust (probust) constraints N2 - We present an adaptive grid refinement algorithm to solve probabilistic optimization problems with infinitely many random constraints. Using a bilevel approach, we iteratively aggregate inequalities that provide most information not in a geometric but in a probabilistic sense. This conceptual idea, for which a convergence proof is provided, is then adapted to an implementable algorithm. The efficiency of our approach when compared to naive methods based on uniform grid refinement is illustrated for a numerical test example as well as for a water reservoir problem with joint probabilistic filling level constraints. KW - probabilistic constraints KW - probust constraints KW - chance constraints KW - bilevel optimization KW - semi-infinite optimization Y1 - 2021 U6 - https://doi.org/10.1007/s00186-021-00764-8 ER - TY - JOUR A1 - Gugat, Martin A1 - Giesselmann, Jan A1 - Kunkel, Teresa T1 - Exponential synchronization of a nodal observer for a semilinear model for the flow in gas networks N2 - 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. Y1 - 2021 U6 - https://doi.org/10.1093/imamci/dnab029 CY - IMA Journal of Mathematical Control and Information ER - TY - THES A1 - Kleinert, Thomas T1 - Algorithms for Mixed-Integer Bilevel Problems with Convex Followers N2 - Bilevel problems are optimization problems for which a subset of variables is constrained to be an optimal solution of another optimization problem. As such, bilevel problems are capable of modeling hierarchical decision processes. This is required by many real-world problems from a broad spectrum of applications such as energy markets, traffic planning, or critical infrastructure defense, to name only a few. However, the hierarchy of decisions makes bilevel optimization problems also very challenging to solve—both in theory and practice. This cumulative PhD thesis is concerned with computational bilevel optimization. In the first part, we summarize several solution approaches that we developed over the last years and highlight the significant computational progress that these methods provide. For linear bilevel problems, we review branch-and-bound methods, critically discuss their practical use, and propose valid inequalities to extend the methods to branch-and-cut approaches. Further, we demonstrate on a large test set that it is no longer necessary to use the well-known but error-prone big-M reformulation to solve linear bilevel problems. We also present a bilevel-specific heuristic that is based on a penalty alternating direction method. This heuristic is applicable to a broad class of bilevel problems, e.g., linear or mixed-integer quadratic bilevel problems. In a computational study, we show that the method computes optimal or close-to-optimal feasible points in a very short time and that it outperforms a state-of-the-art local method from the literature. Finally, we review global approaches for mixed-integer quadratic bilevel problems. In addition to a Benders-like decomposition, we present a multi-tree and a single-tree outer-approximation approach. A computational evaluation demonstrates that both variants outperform known benchmark algorithms. The second part of this thesis consists of reprints of our original articles and preprints. These articles contain all details and are referenced throughout the first part of the thesis. Y1 - 2021 ER - TY - JOUR A1 - Ruiz-Balet, Domenec A1 - Zuazua, Enrique T1 - Neural ODE Control for Classification, Approximation and Transport N2 - We analyze Neural Ordinary Differential Equations (NODEs) from a control theoretical perspective to address some of the main properties and paradigms of Deep Learning (DL), in particular, data classification and universal approximation. These objectives are tackled and achieved from the perspective of the simultaneous control of systems of NODEs. For instance, in the context of classification, each item to be classified corresponds to a different initial datum for the control problem of the NODE, to be classified, all of them by the same common control, to the location (a subdomain of the euclidean space) associated to each label. Our proofs are genuinely nonlinear and constructive, allowing us to estimate the complexity of the control strategies we develop. The nonlinear nature of the activation functions governing the dynamics of NODEs under consideration plays a key role in our proofs, since it allows deforming half of the phase space while the other half remains invariant, a property that classical models in mechanics do not fulfill. This very property allows to build elementary controls inducing specific dynamics and transformations whose concatenation, along with properly chosen hyperplanes, allows achieving our goals in finitely many steps. The nonlinearity of the dynamics is assumed to be Lipschitz. Therefore, our results apply also in the particular case of the ReLU activation function. We also present the counterparts in the context of the control of neural transport equations, establishing a link between optimal transport and deep neural networks. KW - data classification KW - Neural ODEs KW - Optimal Transport KW - simultaneous control KW - deep learning Y1 - 2021 ER - TY - INPR A1 - Egerer, Jonas A1 - Grimm, Veronika A1 - Grübel, Julia A1 - Zöttl, Gregor T1 - Long-run market equilibria in coupled energy sectors: A study of uniqueness N2 - We propose an equilibrium model for coupled markets of multiple energy sectors. The agents in our model are operators of sector-specific production and sector-coupling technologies, as well as price-sensitive consumers with varying demand. We analyze long-run investment in production capacity in each sector and investment in coupling capacity between sectors, as well as production decisions determined at repeated spot markets. We show that in our multi-sector model, multiplicity of equilibria may occur, even if all assumptions hold that would be sufficient for uniqueness in a single-sector model. We then contribute to the literature by deriving sufficient conditions for the uniqueness of short- and long-run market equilibrium in coupled markets of multiple energy sectors. We illustrate via simple examples that these conditions are indeed required to guarantee uniqueness in general. The uniqueness result is an important step to be able to incorporate the proposed market equilibrium problem in more complex computational multilevel equilibrium models, in which uniqueness of lower levels is a prerequisite for obtaining meaningful solutions. Our analysis also paves the way to understand and analyze more complex sector coupling models in the future. KW - Energy Markets KW - Sector Coupling KW - Regional Pricing KW - Uniqueness KW - Short- and Long-Run Market Equilibrium Y1 - 2021 ER - TY - JOUR A1 - Kleinert, Thomas A1 - Manns, Julian A1 - Schmidt, Martin A1 - Weninger, Dieter T1 - Presolving Linear Bilevel Optimization Problems JF - EURO Journal on Computational Optimization N2 - Linear bilevel optimization problems are known to be strongly NP-hard and the computational techniques to solve these problems are often motivated by techniques from single-level mixed-integer optimization. Thus, during the last years and decades many branch-and-bound methods, cutting planes, or heuristics have been proposed. On the other hand, there is almost no literature on presolving linear bilevel problems although presolve is a very important ingredient in state-of-the-art mixed-integer optimization solvers. In this paper, we carry over standard presolve techniques from single-level optimization to bilevel problems and show that this needs to be done with great caution since a naive application of well-known techniques does often not lead to correctly presolved bilevel models. Our numerical study shows that presolve can also be very beneficial for bilevel problems but also highlights that these methods have a more heterogeneous effect on the solution process compared to what is known from single-level optimization. As a side result, our numerical experiments reveal that there is an urgent need for better and more heterogeneous test instance libraries to further propel the field of computational bilevel optimization. KW - Linear Bilevel Optimization KW - Presolve KW - Computational Analysis Y1 - 2021 U6 - https://doi.org/10.1016/j.ejco.2021.100020 IS - 9 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 - 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 - Lang, Jens A1 - Domschke, Pia A1 - Strauch, Elisa T1 - Adaptive Single- and Multilevel Stochastic Collocation Methods for Uncertain Gas Transport in Large-Scale Networks N2 - 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. Y1 - 2021 VL - In: Mesh Generation and Adaptation, Cutting-Edge Techniques. R. Sevilla, S. Perotto, K. Morgan (eds.), SEMA-SIMAI Springer Series IS - Vol. 30 SP - 113 EP - 135 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 - 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 - 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 -