Publikationen der Gruppe

We propose a new hyper-reduction method for a recently introduced nonlinear model reduction framework based on dynamically transformed basis functions and especially well-suited for transport-dominated systems. Furthermore, we discuss applying this new method to a wildland fire model whose dynamics feature traveling combustion waves and local ignition and is thus challenging for classical model reduction schemes based on linear subspaces. The new hyper-reduction framework allows us to construct parameter-dependent reduced-order models (ROMs) with efficient offline/online decomposition. The numerical experiments demonstrate that the ROMs obtained by the novel method outperform those obtained by a classical approach using the proper orthogonal decomposition and the discrete empirical interpolation method in terms of run time and accuracy.

We study an optimization problem related to the approximation of given data by a linear combination of transformed modes. In the simplest case, the optimization problem reduces to a minimization problem well-studied in the context of proper orthogonal decomposition. Allowing transformed modes in the approximation renders this approach particularly useful to compress data with transported quantities, which are prevalent in many flow applications. We prove the existence of a solution to the infinite-dimensional optimization problem. Towards a numerical implementation, we compute the gradient of the cost functional and derive a suitable discretization in time and space. We demonstrate the theoretical findings with three challenging numerical examples.

Solutions of delay differential‐algebraic equations (DDAEs) may depend on derivatives and future evaluations of some of its equations. Structural analysis can be used to determine how often each equation needs to be differentiated and shifted. Unfortunately, these numbers are not always correct, such that a post‐processing step is required to validate the result. In this contribution, we present the first step towards a success check for DDAEs.

In this contribution, we apply a recently introduced nonlinear model reduction framework based on dynamically transformed modes to the linear wave equation with periodic boundary conditions. We demonstrate that under reasonable assumptions, the reduced‐order model can be evaluated efficiently. Consequently, we obtain that the state variables of the reduced‐order model are constant or linear functions with respect to time.

We prove first-order convergence of the semi-explicit Euler scheme combined with a finite element discretization in space for elliptic-parabolic problems which are weakly coupled. This setting includes poroelasticity, thermoelasticity, as well as multiple-network models used in medical applications. The semi-explicit approach decouples the system such that each time step requires the solution of two small and well-structured linear systems rather than the solution of one large system. The decoupling improves the computational efficiency without decreasing the convergence rates. The presented convergence proof is based on an interpretation of the scheme as an implicit method applied to a constrained partial differential equation with delay term. Here, the delay time equals the used step size. This connection also allows a deeper understanding of the weak coupling condition, which we accomplish to quantify explicitly.

This thesis is dedicated to delay differential-algebraic equations (DDAEs), i.e., constraint dynamical systems where the rate of change depends on the current state and its past. Typical applications include - feedback control, where the delay is a direct consequence of the time required to measure the current state, compute the feedback, and implement the control action, - hybrid numerical-experimental testing environments, used for instance in earthquake engineering, - transmission and propagation delays, encountered for example in chemical reactions connected in series and wide-area power-networks, and - as a mathematical tool to analyze hyperbolic equations and time-integration schemes. The fact that algebraic equations can be included in the implicit system description fosters a rapid model development since complex models can be assembled from a library of existing models with well-defined interaction variables. From a mathematical point of view, DDAEs do not only feature difficulties already known from the theory of differential-algebraic equations (DAEs) and delay differential equations (DDEs) but pose additional challenges. For instance, initial trajectory problems for DDAEs may not be causal. Thus, even in a distributional solution space, they may not have a solution for all initial trajectories. This fact, combined with the infinite-dimensional character of delay equations and the high sensitivity to perturbation known from the theory of DAEs, renders DDAEs a challenging mathematical object. Consequently, the analysis of DDAEs is far from complete. The aim of this thesis is to address some of the many open problems. In the first part of the thesis, initial trajectory problems for linear time-invariant (LTI) and nonlinear DDAEs are discussed. We start our analysis with a distributional solution concept and establish the existence and uniqueness of solutions, whenever the DDAE is delay-regular. Jumps and Dirac-impulses in the solution can be avoided if the coefficient matrices of the LTI DDAE satisfy some algebraic conditions, which are obtained by tracking so-called primary discontinuities. We extend some of the results to the nonlinear setting resulting in existence and uniqueness results for a large class of nonlinear DDAEs. The second part of the thesis is dedicated to constructing a DDAE solely from a prescribed set of data points. Having a time-delay in the realization allows us to build an infinite-dimensional system from finitely many points capable of reproducing the transcendental character of the transfer function of a distributed parameter subsystem that models convection or transport. We construct a realization so that it interpolates the data set in the frequency domain and demonstrate its applicability with several numerical examples.

This paper is devoted to the construction of exponential integrators of first and second order for the time discretization of constrained parabolic systems. For this extend, we combine well-known exponential integrators for unconstrained systems with the solution of certain saddle point problems in order to meet the constraints throughout the integration process. The result is a novel class of semi-explicit time integration schemes. We prove the expected convergence rates and illustrate the performance on two numerical examples including a parabolic equation with nonlinear dynamic boundary conditions.

We consider PDE eigenvalue problems as they occur in two-dimensional photonic crystal modeling. If the permittivity of the material is frequency-dependent, then the eigenvalue problem becomes nonlinear. In the lossless case, linearization techniques allow an equivalent reformulation as an extended but linear and Hermitian eigenvalue problem, which satisfies a Gårding inequality. For this, known iterative schemes for the matrix case such as the inverse power or the Arnoldi method are extended to the infinite-dimensional case. We prove convergence of the inverse power method on operator level and consider its combination with adaptive mesh refinement, leading to substantial computational speed-ups. For more general photonic crystals, which are described by the Drude-Lorentz model, we propose the direct application of a Newton-type iteration. Assuming some a priori knowledge on the eigenpair of interest, we prove local quadratic convergence of the method. Finally, numerical experiments confirm the theoretical findings of the paper.

In this paper formulas are derived for the analytic center of the solution set of linear matrix inequalities (LMIs) defining passive transfer functions. The algebraic Riccati equations that are usually associated with such systems are related to boundary points of the convex set defined by the solution set of the LMI. It is shown that the analytic center is described by closely related matrix equations, and their properties are analyzed for continuous- and discrete-time systems. Numerical methods are derived to solve these equations via steepest descent and Newton methods. It is also shown that the analytic center has nice robustness properties when it is used to represent passive systems. The results are illustrated by numerical examples.

We study linear time-invariant delay differential-algebraic equations (DDAEs). Such equations arise naturally, if a feedback controller is applied to a descriptor system, since the controller requires some time to measure the state and to compute the feedback resulting in the time-delay. We present an existence and uniqueness result for DDAEs within the space of piecewise smooth distributions and an algorithm to determine whether a DDAE is delay-regular.

We present a graph-theoretical approach that can detect which equations of a delay differential-algebraic equation (DDAE) need to be differentiated or shifted to construct a solution of the DDAE. Our approach exploits the observation that differentiation and shifting are very similar from a structural point of view, which allows us to generalize the Pantelides algorithm for differential-algebraic equations to the DDAE setting. The primary tool for the extension is the introduction of equivalence classes in the graph of the DDAE, which also allows us to derive a necessary and sufficient criterion for the termination of the new algorithm.

Kolmogorov $n$-widths and Hankel singular values are two commonly used concepts in model reduction. Here we show that for the special case of linear time-invariant dynamical (LTI) systems, these two concepts are directly connected. More specifically, the greedy search applied to the Hankel operator of an LTI system resembles the minimizing subspace for the Kolmogorov n-width and the Kolmogorov $n$-width of an LTI system equals its $(n+1)st$ Hankel singular value once the subspaces are appropriately defined. We also establish a lower bound for the Kolmorogov $n$-width for parametric LTI systems and illustrate that the method of active subspaces can be viewed as the dual concept to the minimizing subspace for the Kolmogorov $n$-width.

We present a framework for constructing a structured realization of a linear time-invariant dynamical system solely from a discrete sampling of an input and output trajectory of the system. We estimate the transfer function of the original model at selected frequencies using a modification of the empirical transfer function estimation that was recently presented in [Peherstorfer, Gugercin, Willcox, SIAM J. Sci. Comput., 39(5):2152-2178, 2017]. Our realization interpolates the transfer function estimates and can be seen as a generalization of the Loewner framework to structured systems. We demonstrate the presented framework by means of a delay example.

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Python library for computing the analytic center of the passivity LMI in the continuous- and discrete-time setting. See the paper 'Bankmann, D.; Mehrmann, V.; Nesterov, Y.; van Dooren, P., 'Computation of the analytic center of the solution set of the linear matrix inequality arising in continuous- and discrete-time passivity analysis, 2019' for more details see the gitlab page for more information; try mybinder for a live demo;

In this paper we derive a formulation for Model Predictive Control (MPC) of linear time-invariant systems based on H-infinity loop-shaping. The design provides an optimized stability margin for problems that require state estimation. Input and output weights are designed in the frequency domain to satisfy steady-state and transient performance requirements, in lieu of conventional MPC plant model augmentations. The H-infinity loop-shaping synthesis results in an observer-based state feedback structure. Using the linear state feedback law, an inverse optimal control problem is solved to design the MPC cost function, and the H-infinity state estimator is used to initialize the prediction model at each time step. The MPC inherits the closed-loop performance and stability margin of the loop-shaped design when constraints are inactive. We apply the methodology to a multi-zone heat pump system in simulation. The design rejects constant unmeasured disturbances and tracks constant references with zero steady-state error, has good transient performance, provides an excellent stability margin, and enforces input and output constraints.

We consider the problem of finding an optimal data-driven modal decomposition of flows with multiple convection velocities. To this end, we apply the shifted proper orthogonal decomposition (sPOD) which is a recently proposed mode decomposition technique. It overcomes the poor performance of classical methods like the proper orthogonal de- composition (POD) for a class of transport-dominated phenomena with large gradients. This is achieved by identifying the transport directions and velocities and by shifting the modes in space to track the transports. We propose a new algorithm for computing an sPOD which carries out a residual minimization in which the main cost arises from solving a nonlin- ear optimization problem scaling with the snapshot dimension. We apply the algorithm to snapshot data from the simulation of a pulsed detona- tion combuster and observe that very few sPOD modes are sufficient to obtain a good approximation. For the same accuracy, the common POD needs ten times as many modes and, in contrast to the sPOD modes, the POD modes do not reflect the moving front profiles properly.

We introduce a novel model order reduction method for large-scale linear switched systems (LSS) where the coefficient matrices are affected by a low-rank switching. The key idea is to replace the LSS by a non-switched system with extended input and output vectors - called the envelope system - which is able to reproduce the dynamical behavior of the original LSS by applying a certain feedback law. The envelope system can be reduced using standard model order reduction schemes and then transformed back to an LSS. Furthermore, we present an upper bound for the output error of the reduced-order LSS and show how to preserve quadratic Lyapunov stability. The approach is tested by means of various numerical examples demonstrating the efficacy of the presented method.

In view of highly decentralized and diversified power generation concepts, in particular with renewable energies, the analysis and control of the stability and the synchronization of power networks is an important topic that requires different levels of modeling detail for different tasks. A frequently used qualitative approach relies on simplified nonlinear network models like the Kuramoto model with inertia. The usual formulation in the form of a system of coupled ordinary differential equations is not always adequate. We present a new energy-based formulation of the Kuramoto model with inertia as a polynomial port-Hamiltonian system of differential-algebraic equations, with a quadratic Hamiltonian function including a generalized order parameter. This leads to a robust representation of the system with respect to disturbances: it encodes the underlying physics, such as the dissipation inequality or the deviation from synchronicity, directly in the structure of the equations, and it explicitly displays all possible constraints and allows for robust simulation methods. The model is immersed into a system of model hierarchies that will be helpful for applying adaptive simulations in future works. We illustrate the advantages of the modified modeling approach with analytics and numerical results.

We consider linear partial differential equations with an additional delay term, which - under spatial discretization - lead to ordinary differential equations with fixed delay of retarded type. This means that the semi-discrete solution gains smoothness over time. For the concept of classical, mild, and weak solutions we analyse whether this effect also takes place in the original system. We show that some systems behave in a neutral way only. As a result, the smoothness of the exact solution remains unchanged instead of gaining smoothness over time.

In this work we investigate explicit and implicit difference equations and the corresponding infinite time horizon linear-quadratic optimal control problem. We derive conditions for feasibility of the optimal control problem as well as existence and uniqueness of optimal controls under certain weaker assumptions compared to the standard approaches in the literature which are using algebraic Riccati equations. To this end, we introduce and analyse a discrete-time Lur'e equation and a corresponding Kalman– Yakubovich– Popov (KYP) inequality. We show that solvability of the KYP inequality can be characterized via the spectral structure of a certain palindromic matrix pencil. The deflating subspaces of this pencil are finally used to construct solutions of the Lur'e equation. The results of this work are transferred from the continuous-time case. However, many additional technical difficulties arise in this context.

Abstract We present a framework for constructing structured realizations of linear dynamical systems having transfer functions of the form $\widetilde{C}(\sum_{k=1}^K h_k(s)\widetilde{A}_k)^{-1}\widetilde{B}$ where $h_1, h_2, \ldots, h_K$ are prescribed functions that specify the surmised structure of the model. Our construction is data-driven in the sense that an interpolant is derived entirely from measurements of a transfer function. Our approach extends the Loewner realization framework to a more general system structure that includes second-order (and higher) systems as well as systems with internal delays. Numerical examples demonstrate the advantages of this approach.

As a first step towards time-stepping schemes for constrained PDE systems, this paper presents convergence results for the temporal discretization of operator DAEs. We consider linear, semi-explicit systems which includes e.g. the Stokes equations or applications with boundary control. To guarantee unique approximations, we restrict the analysis to algebraically stable Runge-Kutta methods for which the stability functions satisfy $R(\infty)=0$. As expected from the theory of DAEs, the convergence properties of the single variables differ and depend strongly on the assumed smoothness of the data.

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.

We propose a greedy interpolation approach to compute the $\mathcal{L}_\infty$-norm of a possibly irrational $\mathcal{L}_\infty$-function. We approximate this function by a sequence of rational $\mathcal{L}_\infty$-functions. For this we use interpolation employing the Loewner matrix framework. In each iteration, the $\mathcal{L}_\infty$-norm of the rational approximation is computed using established methods. Then a new interpolation point is added where the $\mathcal{L}_\infty$-norm of the function approximation is attained. This way, the $\mathcal{L}_\infty$-norm of the approximation converges superlinearly to the L∞-norm of the original function. We illustrate the efficiency of the resulting algorithm for various numerical examples and compare it to state-of-the-art methods.

The minimization of operation costs for natural gas transport networks is studied. Based on a recently developed model hierarchy ranging from detailed models of instationary partial differential equations with temperature dependence to highly simplified algebraic equations, modeling and discretization error estimates are presented to control the overall error in an optimization method for stationary and isothermal gas flows. The error control is realized by switching to more detailed models or finer discretizations if necessary to guarantee that a prescribed model and discretization error tolerance is satisfied in the end. We prove convergence of the adaptively controlled optimization method and illustrate the new approach with numerical examples.

We consider Lie and Strang splitting for the time integration of constrained partial differential equations with a nonlinear reaction term. Since such systems are known to be sensitive with respect to perturbations, the splitting procedure seems promising as we can treat the nonlinearity separately. This has some computational advantages, since we only have to solve a linear constrained system and a nonlinear ordinary differential equation. However, Strang splitting suffers from order reduction which limits its efficiency. This reduction is caused by the fact that the nonlinear subsystem produces inconsistent initial values for the constrained subsystem. The incorporation of an additional correction term resolves this problem without increasing the computational cost significantly. Numerical examples including a coupled mechanical system illustrate the proved convergence results.

In this article, we present a state estimation method for an approximately constant volume combustion process. This is an Unscented Kalman Filter used to estimate quantities which are of interest for the combustion process inside the considered combustion tube, i.e., the pressure, velocity and temperature field. This algorithm relies only on a small number of discrete pressure measurements along the combustion tube. The proposed methods are applied in numerical simulations to demonstrate their effectiveness. In a one-dimensional simulator, the flow field in the tube is described by the one-dimensional Euler equations with chemical source terms. To keep the computational effort on a manageable level, the Kalman Filter is built based on a reduced model. Thus, the model reduction for the Euler equations in combination with chemical kinetics is another focus of this contribution.

Abstract We consider the problem of finding an energy-based formulation of the Navier–Stokes equations for reactive flows. These equations occur in various applications, e. g., in combustion engines or chemical reactors. After modeling, discretization, and model reduction, important system properties as the energy conservation are usually lost which may lead to unphysical simulation results. In this paper, we introduce a port-Hamiltonian formulation of the one-dimensional Navier–Stokes equations for reactive flows. The port-Hamiltonian structure is directly associated with an energy balance, which ensures that a temporal change of the total energy is only due to energy flows through the boundary. Furthermore, the boundary ports may be used for control purposes.

We study linear semi-explicit stochastic operator differential algebraic equations (DAEs) for which the constraint equation is given in an explicit form. In particular, this includes the Stokes equations arising in fluid dynamics. We combine a white noise polynomial chaos expansion approach to include stochastic perturbations with deterministic regularization techniques. With this, we are able to include Gaussian noise and stochastic convolution terms as perturbations in the differential as well as in the constraint equation. By the application of the polynomial chaos expansion method, we reduce the stochastic operator DAE to an infinite system of deterministic operator DAEs for the stochastic coefficients. Since the obtained system is very sensitive to perturbations in the constraint equation, we analyze a regularized version of the system. This then allows to prove the existence and uniqueness of the solution of the initial stochastic operator DAE in a certain weighted space of stochastic processes.

Abstract We consider the computation of the $\mathcal{L}_\infty$-norm for a general class of $\mathcal{L}_\infty$-functions and focus on the case where the function is represented in terms of large-scale matrix-valued factors. We propose a subspace projection method to obtain reduced approximations of this function by interpolation techniques. The $\mathcal{L}_\infty$-norms are computed for the resulting reduced functions, then the subspaces are refined by means of the optimizer of the $\mathcal{L}_\infty$-norm of the reduced function. In this way we obtain much better performance compared to existing methods. (© 2017 Wiley-VCH Verlag GmbH \& Co. KGaA, Weinheim)

Low rank perturbations of right eigenvalues of quaternion matrices are considered. For real and complex matrices it is well known that under a generic rank-k perturbation the k largest Jordan blocks of a given eigenvalue will disappear while additional smaller Jordan blocks will remain. In this paper, it is shown that the same is true for real eigenvalues of quaternion matrices, but for complex nonreal eigenvalues the situation is different: not only the largest $k$, but the largest 2k Jordan blocks of a given eigenvalue will disappear under generic quaternion perturbations of rank k. Special emphasis is also given to Hermitian and skew-Hermitian quaternion matrices and generic low rank perturbations that are structure-preserving.

We present a data-driven realization for systems with delay, which generalizes the Loewner framework. The realization is obtained with low computational cost directly from measured data of the transfer function. The internal delay is estimated by solving a least-square optimization over some sample data. Our approach is validated by several examples, which indicate the need for preserving the delay structure in the reduced model.

In this report we introduce a Matlab toolbox for the regularization of descriptor systems. We apply it, in particular, for systems resulting from the generalized realization procedure of [16], which generates, via rational interpolation techniques, a linear descriptor system from interpolation data. The resulting system needs to be regularized to make it feasible for the use in simulation, optimization, and control. This process is called regularization.

In this paper we introduce a new matrix nearness problem that is intended to generalize the distance to instability. Due to its applicability in analyzing the robustness of eigenvalues with respect to the arbitrary localization sets (domains) in the complex plane, we call it the distance to delocalization. For the open left half-plane or the unit disk, the distance to the nearest unstable matrix is obtained as a special case. Following the theoretical framework of Hermitian functions and the Lyapunov-type localization approach, we present new Newton-type algorithms for the distance to delocalization: first using an explicit computation of the desired singular values (eD2D), and then using an implicit computation (iD2D). For both algorithms, we introduce a special stabilization technique of the Newton steps and, for a certain class of the localization domains, we provide an additional globality test. Since our investigations are motivated by several practical applications, we illustrate our approach on some of them. Furthermore, in the special case when the distance to delocalization becomes the distance to the continuous time instability, we validate our algorithms against the state-of-the-art computational methods.

Recently in [Gu (2011), Comput. Des. Integr. Circuits Syst., 30(9):1207–1320] a procedure was presented that allows to reformulate nonlinear ordinary differential equations in a way that all the nonlinearities become polynomial on the cost of increasing the dimension of the system. We generalize this procedure (called `polynomialization') to systems of differential-algebraic equations (DAEs). In particular, we show that if the original nonlinear DAE is regular and strangeness-free (i. e., it has differentiation index one) then this property is preserved by the polynomial representation. For systems which are not strangeness-free, i. e., where the solution depends on derivatives of the coefficients and inhomogeneities, we also show that the index is preserved for arbitrary strangeness index. However, to avoid ill-conditioning in the representation one should first perform an index reduction on the nonlinear system and then construct the polynomial representations. Although the analytical properties of the polynomial reformulation are very appealing, care has to be given to the numerical integration of the reformulated system due to additional errors. We illustrate our findings with several examples.

This paper develops a framework to include Dirichlet boundary conditions on a subset of the boundary which depends on time. In this model, the boundary conditions are weakly enforced with the help of a Lagrange multiplier method. In order to avoid that the ansatz space of the Lagrange multiplier depends on time, a bi-Lipschitz transformation, which maps a fixed interval onto the Dirichlet boundary, is introduced. An inf-sup condition as well as existence results are presented for a class of second order initial-boundary value problems. For the semi-discretization in space, a finite element scheme is presented which satisfies a discrete stability condition. Because of the saddle point structure of the underlying PDE, the resulting system is a DAE of index 3.

We present a model describing simultaneous heat and mass transfer of an absorbing or desorbing laminar liquid film flowing over a vertical isothermal plate. We start with a formulation which is comparable to established models by using simplifying assumptions such as homogeneous velocity and constant film thickness. In contrast to those, we allow for effects like change in properties and differential heat of solution within the bulk of the film. Additionally, enthalpy transport due to interdiffusion is accounted for. The impact of the considered effects are discussed and compared. The numerical solution is obtained by utilising a Newton–Raphson scheme to solve the finite difference formulation of the governing equations. Since the temperature gradients adjacent to wall and phase boundary are expected to be large, we discretise the equations on an irregular grid. The results of the model agree very well with established analytical models. It is found that the influence of released differential heat of solution within the bulk is relatively small. However, the impact on the temperature distribution is in the same order of magnitude as the one of a change in properties. Moreover, when comparing desorption with absorption under equivalent conditions, the mass transfer rate during absorption is higher than during desorption.

The $P_1$-nonconforming finite element is introduced for arbitrary triangulations into quadrilaterals and triangles of multiple connected Lipschitz domains. An explicit a priori analysis for the combination of the Park–Sheen and the Crouzeix–Raviart nonconforming finite element methods is given for second-order elliptic PDEs with inhomogeneous Dirichlet boundary conditions.