Rückblick

• Absolventen Seminar WS 16/17 [14]
• Absolventen Seminar SS 16 [15]
• Absolventen Seminar WS 15/16 [16]
• Absolventen Seminar SS 15 [17]
• Absolventen Seminar WS 14/15 [18]
• Absolventen Seminar SS 14 [19]
• Absolventen Seminar WS 13/14 [20]
• Absolventen Seminar SS 13 [21]
• Absolventen Seminar WS 12/13 [22]
• Absolventen Seminar SS 12 [23]
• Absolventen Seminar WS 11/12 [24]

Structure-preserving and time-integration of pHDAE

Port-Hamiltonian theory is intended to provide a unified framework for analysis, modeling, control and simulation of multi-physics systems. An essential element of this theory is the Dirac structure that interconnects systems with its environment. This structure can be represented in different ways like Image representation, input-output to mention some of them. A special class of pHSys is the constrained input-output representation of the Dirac structure. This formulation is a DAE where the constraints rise in a structured way. The task of my master thesis is to analyze and simulate this kind of pHSys with special attention on conservative systems.It is known that conservative systems require symplectic integrators, and the DAE can only be simulated via BDF or implicit methods. These facts pose some interesting questions like, how suitable pH theory is for simulations, and which numerical schemes are really available for this kind of systems. In this talk, I will present some results of my thesis and how to adapt the general regularization method for descriptor systems in order to apply any symplectic integrator.

Space-time Galerkin POD for Optimal Control of Burgers' Equation

In the context of Galerkin discretizations of a PDE, the modes of the classical method of Proper Orthogonal Decomposition (POD) can be interpreted as the ansatz and trial functions of a low-dimensional Galerkin scheme. If one also considers a Galerkin method for the time integration, one can similarly define a POD reduction of the temporal component. This has been described earlier but not expanded upon - probably because the reduced time discretization globalizes time which is computationally inefficient. However, in finite-time optimal control systems, time is a global variable and there is no disadvantage from using a POD reduced Galerkin scheme in time.

In this talk, we provide a newly developed generalized theory for space-time Galerkin POD, show its application for the control of the nonlinear Burgers' equation, and discuss the competitiveness by comparing to standard approaches like classical POD combined with gradient-based methods.

On random perturbations of matrix polynomials

We will consider matrix polynomials of degree <3, where one of the coefficients is a large NxN random matrix while the other coefficients are nonrandom and constant. In particular we will be interested in the rate of convergence (with N to infinity) of some special eigenvalues appearing in this construction. Although the main tools come from probability theory, the work is motivated by numerical analysis problems.

Block Kronecker Linearizations of Matrix Polynomials and their Backward Errors

We introduce a new family of strong linearizations of matrix polynomials--which we call "block Kronecker pencils"-- and perform a backward stability analysis of complete polynomial eigenproblems. These problems are solved by applying any backward stable algorithm to a block Kronecker pencil, such as the staircase algorithm for singular pencils or the QZ algorithm for regular pencils. This analysis allows us to identify those block Kronecker pencils that yield a computed complete eigenstructure which is exactly that of a slightly perturbed matrix polynomial. These favorable pencils include the famous Fiedler linearizations, which are just a very particular case of block Kronecker pencils. Thus, our analysis offers the first proof available in the literature of global backward stability for Fiedler pencils. In addition, the theory developed for block Kronecker pencils is much simpler than the theory available for Fiedler pencils, especially in the case of rectangular matrix polynomials. Joint work with Froilán Dopico, Piers Lawrence and Javier Pérez

Computing the distance to the nearest unstable pencil

In this paper, we study the nearest stable matrix pair problem: given a square matrix pair $(E,A)$,  minimize the Frobenius norm of $(Delta_E,Delta_A)$ such that $(E+Delta_E,A+Delta_A)$ is a stable matrix pair.
We propose a reformulation of the problem with a simpler feasible set by introducing dissipative Hamiltonian (DH) matrix pairs: A matrix pair $(E,A)$ is DH if $A=(J-R)Q$ with skew-symmetric $J$, positive semidefinite $R$, and an invertible $Q$ such that $Q^TE$ is positive semidefinite. This reformulation has a convex feasible domain onto which it is easy to project. This allows us to employ a fast gradient method to obtain a nearby stable approximation of a given matrix pair.

Random walks in the quarter-plane: a numerical approach

Random walks in the quarter-plane are frequently used to model queueing problems. In particular, we are interested in computing their steady-state performance measures. Performance measures can be readily computed once the invariant probability distribution of the random walk is known. Various approaches for finding the invariant distribution exist. In this talk, we consider a numerical method first presented by Chen et al., that computes exactly the invariant probability distribution for a certain class of random walks. This method has the advantage that it gives a closed form for the invariant probability distribution. For other random walks, a general approximation scheme is developed, by perturbing the transition probabilities along the boundaries of the state space. We further enlarge the class of random walks of which the performance measures can be computed exactly with this method, by relaxing some significant hypotheses. We also find necessary and sufficient conditions for convergence when the the closed form provided by the algorithm should be infinite. These enhancements permit us to apply the algorithm to the join the shortest queue model, obtaining arbitrarily precise results.

Parameter-Dependent Rank-One Perturbations of Singular Hermitian Pencils

We investigate the effect of perturbations of singular Hermitian pencils that are (1) of rank one, (2) structure-preserving, (3) generic, (4) parameter-dependent, and (5) regularizing, i.e., the perturbed pencil will be regular. Surprisingly, the perturbed pencil will show a behaviour that differs significantly from the one that can be observed for perturbations satisfying (1)-(4) when we start with a pencil already being regular.

Boundary Value Perturbation Analysis for a Y-Shaped Gas Network

Natural gas plays a crucial role in the energy supply of the world. After oil, it is the second most used energy supplier in Germany. The high demand for natural gas calls for an accurate, efficient and robust mathematical modeling, simulation and optimisation of the gas transport through the existing pipeline network. The most accurate model to describe the gas flow through a pipe is given by the three-dimensional Euler equations of fluid dynamics. However, often it is not necessary to consider this highly detailed model and simplifying assumptions can be made in order to save computational cost. In this talk we analyse the one-dimensional isothermal Euler equations, which are discretised using the implicit box scheme. The friction coefficient is given by the Prandtl-Colebrook law. As an example, we consider the gas flow through a Y-shaped network with pipes of different lengths. After the discretisation of the Euler equations, a nonlinear system of equations is obtained for every time integration step. A boundary value perturbation analysis for this nonlinear system is performed using componentwise relative amplification factors and componentwise relative condition numbers. The advantage over their normwise counterparts was the topic of a previous presentation and is shortly recalled. We conclude with a discussion of the partly surprising results. This is joint work with V. Mehrmann. It is supported by the German Research Foundation DFG in the Collaborative Research Centre TRR 154, subproject B03.

Minimal realization of MIMO linear systems using Hermite form and coprime fractions

Many mathematical representations exist for control systems. Minimal realization allows to accurately represent a system with a minimal order in state space form. The discussed method is used to compute the minimal realization of linear time-invariant (LTI) systems. It is based on Hermite form of the matrix transfer function and coprime fractions. This technique is implemented in MATLAB and compared to other traditional techniques in terms of three different aspects: The configuration of the realization, memory space and time complexity of the algorithm.

State Space Formulations of the Navier-Stokes Equations for Reactive Flows

In this talk I present results of a joint work with Christoph Zimmer (TU-Berlin).

The first part of my talk is closely related to the work of our colleagues Philipp Schulze (TU-Berlin) and Robert Altmann (TU-Berlin). In their work which has been published under the title "A port-Hamiltonian Formulation of the Navier–Stokes Equations for Reactive Flows" they introduce a state space formulation of the one-dimensional reactive Navier-Stokes with simplified constitutive relations. A weak formulation that encodes the corresponding governing partial-differential equation given as a system of ordinary-differential equations defined on an abstract state space that due to a state space dependent skew-adjoint operator guarantees passivity.

We have been able to generalize the results of our colleagues to the three-dimensional case and general linear constitutive equations. I will show that the structure induced by the skew-adjoint operator not only guarantees passivity (conservation of energy in case of a closed system) but also guarantees that the infinite-dimensional state space formulation satisfies the second law of thermodynamics.

In the second part of my talk I will present some of the results related to our attempt to formulate the reactive three-dimensional Navier-Stokes as an open metriplectic system. Metriplectic systems have become famous under the name GENERIC (General Equations for Non-Equilibrium Reversible Irreversible Coupling).

Iterative Methods for the Laplace Eigenvalue Problem

Within the ECMath project OT10 on photonic crystals, which started last week, we consider a nonlinear PDE eigenvalue problem. Although this model is based on the Maxwell equations including the double curl operator, we first try to tackle the Laplace eigenvalue problem. Instead of discretizing the problem and then applying iterative eigenvalue solvers, we apply the inverse power method as well as a Krylov iteration method directly to the PDE problem.

Symmetry-reduced Reduced-order Models

In previous talks, I have introduced symmetry reduction as a potential enhancement of reduced basis methods for reduced-order modelling. This was achieved in converting a PDE into a PDAE; the reduced-order model (ROM) would then be constructed from the PDAE, leading to a DAE. The primary benefit is a smaller Kolmogorov n-width, determined by an exponential decay in the singular value spectrum of the PDAE snapshot data, which means that a smaller basis can be used for the reduced-order model without losing accuracy. However, the DAE comes at a cost of increased complexity, which can offset the gains of a smaller basis; this is evident in linear systems where the resulting symmetry-reduced DAEs are nonlinear. Increases in the complexity could be tolerated if the error of the symmetry-reduced model were better than the standard ROM.

In this talk, I will focus on the ROMs. I show results from symmetry-reduced ROMs and demonstrate that the reduction in error between the PDAE and its ROM is parameter-dependent: in cases with strong advection and (damped) shocks, symmetry reduction performs better than standard reduced-basis methods, but in cases with smoother solution profiles, the extra terms generated by the symmetry reduction lead to a larger asymptotic error (with respect to the number of basis modes) when compared to the standard ROMs.

Half-explicit Runge-Kutta methods for overdetermined semi-implicit differential-algebraic equations

The numerical solution of differential-algebraic equations can be expensive, if the underlying model has high dimensions. When using Runge-Kutta methods, large nonlinear systems of equations have to be solved during the iteration. The main focus for this thesis is to study half-explicit Runge-Kutta methods for a special class of differential-algebraic equations to make these computations more efficient. We exploit the structure and properties of the given differential-algebraic equations to construct an algorithm that allows to solve reduced nonlinear systems of equations. The algorithm is tested with multiple examples from the fields of mechanical, electrical and chemical sciences to validate the approach.

Structured discretization of linear port-Hamiltonian differential-algebraic equations

Port-Hamiltonian systems (pH-systems) provide a systematic framework for energy-based modelling and simulation of multi-physics systems. As a generalization of classical Hamiltonian systems, the pH-structure implies preferable system properties such as passivity and stability.

A special case of pH-systems arise when there are further state constraints obtained. Those systems are called port-Hamiltonian differential-algebraic equations (pHDAEs).

In my master’s thesis, I consider linear pHDAEs with constant coefficients with respect to two major tasks: firstly, it is preferable to describe the system in such a way that both the pH-structure and the DAE-structure are reflected and that is well-suited for the numerical treatment. Secondly, I will consider different discretization methods that allow to preserve the system properties.

Structure Preserving Model Order Reduction Methods for port-Hamiltonian Systems

Port-Hamiltonian systems are often used to model large physical systems, which might become even larger after discretization. Therefore, model reduction methods are needed. Furthermore, these methods have to be structure preserving to keep all the nice properties of pH-systems.

In my talk I am going to summarize some structure preserving model reduction methods for ordinary port-Hamiltonian systems, like Moment Matching and power conservation based methods.

Furthermore, I will give an outlook on the topic of my master thesis.

Reduced Order Modeling of Linear Transport Phenomena

Model order reduction (MOR) techniques have experienced a rapid development in the past decades and are applied in various fields of science and industry. However, standard MOR methods reveal major deficiencies when dealing with problems which are dominated by transport, e. g., moving shocks.

In this talk we present the shifted proper orthogonal decomposition (sPOD) as a mode decomposition explicitly accounting for transport. This is achieved by shifting the modes in space by a time-dependent shift. In the offline phase the modes, their amplitudes, and their shifts are computed based on snapshots of the solution. Further, we show how the sPOD modes can be used to construct a reduced order model (ROM) via projection. Afterwards, in the online phase the ROM can be used to simulate various parameter configurations. We show that for linear systems an efficient offline/online decomposition can be achieved, i. e., the ROM can be evaluated with a computational complexity which is independent from the order of the original model. The method is demonstrated by numerical simulations of the linear acoustic wave equation.

This is joint work with Volker Mehrmann and Julius Reiss.

Behavior-like approach to delay differential-algebraic equations

One of the main challenges in the analysis of delay differential-algebraic equations (DDAEs) is the so called type of the system. Roughly speaking, the type classifies whether the solution becomes smoother or less smooth over time. In this talk we propose a new framework to tackle this problem, which is based on a combination of the method of steps and the behavior-approach from control theory. This allows us to establish existence and uniqueness theory based solely on DAE theory.

Some ideas on H_∞ optimization of large-scale systems

In my last seminar talk I have shown how to compute the H_∞-norm of a large-scale dynamical system using interpolation techniques. In this talk I will discuss some further ideas on this problem that we are pursuing at the moment. I will briefly show how we want to apply these techniques in H_∞ optimization.

Implementation of a Stripboard Layout Algorithm Using Orthogonal Graph Drawing

These days almost every design and layout process of electronic circuits is supported by software programs. Hardware companies make use of highly specialized layout design software for manufacturing integrated circuits while the board sizes become smaller and smaller. Nevertheless, simple stripboards are widely used, especially for developing and testing smaller circuits. Withal software applications, which supports the stripboard layout process, are mostly restricted and an automatic processing of the schematics into a finished board layout is not supported. The intention of my bachelor thesis is to elaborate and expand on one approach for the automation of creating stripboard layouts using orthogonal graph drawing.

In this talk, I will briefly present my work and its most important results. This includes the specific characteristics of the stripboard layout process, the basic approach of the thesis, the so- called visibility approach, which is based on an algorithm belonging to area of orthogonal graph drawing, and the problems and possibilities of practical implementation. Furthermore, I will discuss two modifications of the algorithm presented within the visibility approach, and provide some points for further research.

Checksum-Based Fault Tolerance for solving large linear systems on multicore architectures

High-Performance Computing (HPC) systems were initially utilized with executing code on parallel and distributed platforms. The advent of exascale computational architectures has shifted the focus from achieving massive parallelism to using it in a more efficient manner; modern computational models now aim to support accurate and reliable scientific operations.

As computational demands in various scientific fields have increased, it is more than essential to provide methods that deliver acceptable level of user-visible service and enable the system's continuous operation, even in the presence of soft errors. Locating and correcting faults due to numerical computations, while minimizing performance loss, is a primary goal that needs to be achieved rapidly.

In this talk I will present a technique for tolerating faults when solving large linear systems on HPC platforms with the Generalized Minimum Residual method of Saad and Schultz. The way of achieving fault resilience is by using checkpointing and rollback recovery protocols, in combination with Algorithm-Based Fault Tolerance using Checksums. The algorithm provides detection and correction of software faults due to incorrect parallel numerical computations on a large-scale platform.

Runge-Kutta Methods for Operator Differential-Algebraic Equations

Operator differential-algebraic equations (DAEs) are a helpful tool to investigate constrained partial differential equations which arise for example in the description of fluids, elastic multibody systems or circuit networks. One of the simplest operator DAEs is given by the unsteady Stokes equation. The normal approach to solve this constrained PDE is to discretize it first in space, possibly reduce the index, and then use a standard time-integration scheme. In this talk we will use another approach; we will first use a regularization ansatz and then apply a Runge-Kutta method directly to the unsteady Stokes equation. 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.

This is joint work with Robert Altmann.

Error balancing for eigenvalue problems arising from photonic crystal simulation

Photonic crystals are particular materials that can affect the propagation and reflection of light thanks to their periodic structure. They have been known for a few decades already, but their conception and analysis are still very challenging tasks. Yet the applications of such crystals cover a wide range of domains and new promising technologies would greatly benefit from efficient numerical methods that are able to simulate accurately the interaction between photons and matter in periodic structures.

Adaptive finite element methods (AFEM) offer a great framework for discretizing the PDE Maxwell eigenvalue problem (EVP) based on error bounds derived from the PDE. Standard AFEM usually base the grid refinement procedure on the assumption that the algebraic EVP is solved exactly. Yet, this assumption is generally not true because of limited machine precision and, more importantly, because of the error tolerance introduced by iterative algebraic methods.

In this talk, we show how balancing the “algebraic error” made by the iterative solver and the discretization error can increase the performance of AFEM. We will look in particular at the flux reconstruction framework to combine both types of errors.

This is a joint work with Robert Altmann.

Computing the Passivity Radius with the Help of the Analytic Center

Passivity is a desired property of a dynamical system, since physical systems without any external input do not produce energy, i.e., are passive. Sometimes, it might happen though, that the linear time-invariant system obtained from linearization and discretatization approaches applied to nonlinear PDEs are not passive anymore. Hence, one is interested in analyzing the distance to passivity of a non-passive system and the radius of stability, i.e., robustness, of a passive system.

In this talk we will consider a new approach for the computation of the passivity radius with the help of the so-called analytic center. The analytic center is the minimizer of a certain convex functional with certain nice properties, and, thus, the hope is that being at this point will provide a 'good' method for computing the passivity radius.

We will show how one can compute the analytic center and consider a simple example for the computation of the passivity radius.

This is joint work with Volker Mehrmann and Paul van Dooren.