Projector chains for semi-explicit index-2 DAEs and ADAEs

To start with, we consider a linear differential-algebraic Equation (DAE) with a finite-dimensional state. In a DAE, differential and algebraic equations are interlocked, what makes their analysis and numerical treatment involved. This interlocking is quantified by various index concepts. Simply put, the lower the index, the better the algebraic and differential parts are seperated. Via the projector chain approach by Griepentrog and März*, one can recursively define a projector that is used to decouple the algebraic and differential part of the linear DAE. This is achieved by a scaling of the equations from the right and variable substitutions, but without variable transformations. Having discussed, how this operator chain applies to semi-explicit partially nonlinear index-2 DAEs, I will turn towards DAEs with states in Banach spaces, often referred to as abstract DAEs (ADAEs). I will introduce the functional analytical setting, the particular type of ADAEs considered here, and the difficulties that come with the infinite dimensional setting. Then I will show that, under additional regularity conditions, the ADAE has the same solutions as a decoupling of the ADAE via a projector. To assure you that I haven't lost touch with reality, I will illustrate why the chosen approach is very natural if considering the weak formulation of the Navier-Stokes equations. *Griepentrog, März: DAEs and their numerical treatment, 1986

Adaptive simulation of an elastic pendulum

Mathematical models combining rigid and elastic bodies are developed and analysed in recent years. An easy example of a non-rigid body is the elastic pendulum. In this talk, I will present methods for modelling and simulation of an oscillating, flexible bar. I will give a short glimpse of a discretization of an elastic pendulum. After this I will show first numerical results and explain three ways for an adaptive >simulation. Finally I will compare these three methods.

A dictionary based adaptive finite element approach

In this talk we present a dictionary based approach for finite element discretization. We construct a basic dictionary using tensor product of simple functions like multilevel B-spline functions. The special features of solution (like singularities and shape of domain which is available from the problem) are subsequently incorporated into basic dictionary. In this approach the geometric model of the domain and inherent singularities does not influence the mesh generation or mesh refinement. Given the high accuracy of dictionary based discretization, the system size is relatively small and can be solved by a variety of techniques. Finally, we present simple examples to illustrate our approach.

Projections, angles and spectra in Hilbert spaces

In this talk I will present some fundamental yet very interesting results (old and new ones) from the wonderful world of projections, angles between subspaces and spectral perturbation theory in Hilbert spaces.

Anti-Triangularizing T-alternating Matrix Polynomials

T-alternating matrix polynomials can occur in different applications. While it has recently been shown that any square matrix polynomial is triangularizable, the structure-preserving triangularization of alternating matrix polynomials has not yet been investigated. In this talk, we present an algorithm to create a triangular form for alternating matrix polynomials of the same structure and degree. Unfortunately, this procedure is restricted to the case that all elementary divisors/ eigenvalues are finite, the consequences of allowing infinite elementary divisors and a possible solution to the arising problems are presented as well.

Spectral Error Bounds for an Inexact Arnoldi Method

We investigate the behavior of an inexact Arnoldi method to approximate eigenvalues, eigenvectors and/or invariant subspaces  corresponding to k<<n eigenvalues of a Hermitian, complex n by n matrix A.
The dominant operations of Arnoldi's method are matrix-vector products and weighted vector sums (for the sake of orthogonalization).
Also scalar products and vector scalings are necessary.

In practice neither matrix-vector multiplication nor vector sums nor vector scaling can be done exactly. Instead only approximations of the intended quantities are available. However it is asumed that scalar products can be evaluated exactly.

Our main goal is to give bounds on the approximation quality of the approximate eigenvalues, -vectors, and invariant subspaces obtained using an inexact Arnoldi method. The presented error bounds generalize the known results for the unperturbed Arnoldi method.   Furthermore, as a prerequisite we also analyze the distance to orthonormality of the now-not-anymore orthogonal basis vectors.

Joint work with Christian Schroeder

Stability and robust stability of linear differential-algebraic equations with delay

In this talk, we deal with the exponential stability and the stability radius of linear differential-algebraic equations with delay when the equation is subjected to the structured perturbations. Necessary and sufficient conditions for the exponential stability are studied. Computable formulas for the stability radius are given. Examples are derived to illustrate results.

This is joint work with Volker Mehrmann, Vu Hoang Linh and Nguyen Huu Du

Regularization via Overdetermined Formulations and Numerical
Simulation of Multibody Systems Modeled with Modelica (I)

In this talks we will discuss the efficient and robust
numerical simulation of dynamical systems modeled with Modelica.

We will present an approach which mainly consists in two steps.

The first step consists in an automatic regularization of the model
equations provided in Modelica source code. The obtained overdetermined
formulation satisfies the requirements for a regularization and, in
particular, is equivalent to the original DAE in the sense that both
have the same solution set.

The second step in the approach is the subsequent robust and efficient
numerical integration of the regularized overdetermined formulation with
adapted discretization methods.
Within the Modelica framework, there currently exist no numerical
integrator which is suited for overdetermined systems.

This is joint work with Volker Mehrmann and Andreas Steinbrecher.

Regularization via Overdetermined Formulations and Numerical
Simulation of Multibody Systems Modeled with Modelica (II)

In this talks we will discuss the efficient and robust
numerical simulation of dynamical systems modeled with Modelica.

We will present an approach which mainly consists in two steps.

The first step consists in an automatic regularization of the model
equations provided in Modelica source code. The obtained overdetermined
formulation satisfies the requirements for a regularization and, in
particular, is equivalent to the original DAE in the sense that both
have the same solution set.

The second step in the approach is the subsequent robust and efficient
numerical integration of the regularized overdetermined formulation with
adapted discretization methods.
Within the Modelica framework, there currently exist no numerical
integrator which is suited for overdetermined systems.

This is joint work with Volker Mehrmann and Andreas Steinbrecher.

Analysis and reformulation of linear delay differential-algebraic equations

Delay differential equations (DDEs) arise in a variety of applications, including physical systems, biological systems and electronic networks. If the states of the physical system are constrained, e.g., by conservation laws or interface conditions, or some economical interest are involved in the biological model, then algebraic equations have to be included and one has to analyze delay differential-algebraic equations (Delay-DAEs).

In this talk, we study the analysis of linear Delay-DAEs from both theoretical and numerical viewpoints. First, we generalize some well-known results in DAEs theory to Delay-DAEs.
Then, we propose algorithms to reformulate a Delay-DAE into its underlying delay system. Moreover, the constructed forms are also used to address structural properties of the system like solvability, regularity, consistency and smoothness requirements.

This is joint work with Volker Mehrmann and Andreas Steinbrecher.

On three problems on H-selfadjoint matrices with one eigenvalue of nonpositive type.

Let H be a hermitian--symmetric, invertible  matrix with one negative eigenvalue and let A be H-symmetric, that is HA=A^*H.
It is well known, that the spectrum of A is real, except the eigenvalue of nonpositive type, which may be real or complex (in the latter case its cojugate is an eigenvalue as well).  The main object of the presented research is the study the behavior of this eigenvalue under a certain change of the matrix A. We consider three instances

- A is a one dimensional perturbation of a given matrix (or operator)
- A is a finite dimensional truncation of a given  infinite Jacobi matrix
- A is a large random matrix.

Apparently, these three problems  can be treated with the same method of Weyl functions and Nevanlinna functions with one negative square.
The results are obtained in cooperation with H.S.V. de Snoo (RUG Groningen), H. Winkler (TU Ilmenau), M. Derevyagin (TU Berlin) and P. Pagacz (Jagiellonian University).

Structural-Algebraic Remodeling of Coupled Dynamical Systems

The automated modeling of multi-physical dynamical systems is usually realized by coupling different subsystems together via certain interface or coupling conditions. This approach results in large-scale high-index differential-algebraic equations (DAEs). Since the direct numerical simulation of these kind of systems leads to instabilities and possibly non-convergence of the numerical methods a regularization or remodeling of the system is required. In many simulation environments a kind of structural analysis based on the sparsity pattern of the system is used to determine the index and a reduced system model. However, this approach is not reliable for certain problem classes, in particular not for coupled systems of DAEs. We will present a new approach for the remodeling of coupled dynamical systems that combines the structural analysis, in particular the Signature Method, with classical algebraic regularization techniques and thus allows to handle so-called structurally singular systems and also enables  a proper treatment of redundancies or inconsistencies in the system.

Structured backward errors for eigenvalues of Hermitian pencils

In this talk, we consider the structured backward errors for eigenvalues of Hermitian pencils or, in other words, the problem of finding the
smallest Hermitian perturbation so that a given value is an eigenvalue
of the perturbed Hermitian pencil.

The answer is well known for the case that the eigenvalue real, but
in the case of nonreal eigenvalues, only the structured backward error for eigenpairs has been considered so far, i.e., the problem of finding the smallest
Hermitian perturbation so that a given pair is an eigenpair of the perturbed Hermitian pencil.

In this talk, we give a complete answer to the question by reducing the problem to an eigenvalue minimization problem of Hermitian matrices depending on two real parameters. We will see that
the structured backward error of complex nonreal eigenvalues may be significatly different from the corresponding unstructured backward error - which is in conrast to the case of real eigenvalues where
the structured and unstructured backward errors coincide.

Riccati-Based Boundary Feedback Stabilization of Flow Problems

In order to explore boundary feedback stabilization of flow problems, we consider the(Navier-) Stokes equations that describe instationary, incompressible flows for moderate Reynolds (in case of Navier-Stokes) or viscosity (in case of Stokes) numbers. Following the analytic approach by Raymond [Raymond ’06], we have to find a numerical treat-ment of the Leray projector. After a finite element discretization we get a differential-algebraic system of differential index two. We show how to reduce this index with a projection method based on [Heinkenschloss/Sorensen/Sun ’08] and point out the connection to the Leray projector. This leads to generalized state space systems where a linear quadratic control approach can be applied. Avoiding the explicit projection,we end up with large-scale saddle point systems which have to be solved. We will show numerical results regarding the arising nested iteration and how the different parameters influence its convergence. Furthermore, we show some examples where we want to apply these ideas in multi-field flow problems. Finally, we point out recent problems and tasks related to our approach.The work is developed within the project Optimal Control-Based Feedback Stabilization in Multi-Field Flow Problems with Eberhard B ̈nsch and Peter Benner as principal investigators. This project is part of the DFG priority program 1253: Optimization With Partial Differential Equations. Furthermore, this is a joined work with Jens Saak(general state space systems), Martin Stoll (preconditioning of iterative solvers) and Friedhelm Schieweck, Piotr Skrzypacz (finite element techniques).

Numerical Computation of the Polar Decomposition

The factorization A=UH of a complex matrix A into an isometric matrix U and a Hermitian positive semidefinite matrix H is called polar decomposition.. I am going to summarize some basic properties of this factorization like existence and uniqueness and afterwards I will discuss some numerical methods for computing the polar factor U, e.g. the so-called scaled Newton iteration.

Matrix functions that commute with their derivative

Joint work with Olga Holtz and Hans Schneider

We examine when a matrix whose elements are differentiable functions in one variable commutes with its derivative. This problem was discussed in a letter from Issai Schur to Helmut Wielandt written in 1934, which we found in Wielandt's Nachlass. The topic was rediscovered later and partial results were proved. However, there are many subtle observations in Schur's letter which were not obtained in later years. Using an algebraic setting, we put these into perspective and extend them in several directions. We present in detail the relationship between several conditions mentioned in Schur's letter. We also present several examples that demonstrate Schur's observations.

Equivalence Transformations for Quadratic Eigenvalue Problems

Since there does not exist a simple equivalence transformation which can triangularize or diagonalize a quadratic matrix polynomial, we study more general equivalence transformations for quadratic eigenvalue problems in this paper. In order to well understand the equivalence of quadratic matrix polynomials, we first analyze properties of their decomposable pairs, which are closely related to the equivalence. Furthermore, based on the theory about the equivalence of linear matrix polynomials, we construct equivalence transformations for quadratic matrix polynomials. Specifically, first, linearize a regular quadratic matrix polynomial as a linear matrix polynomial; second, apply a structure preserving transformation to the linear matrix polynomial to get another one; finally, recover a new regular quadratic matrix polynomial from the transformed linear matrix polynomial. In particular, we propose our definitions of structure preserving transformations and analyze their properties. We find that those transformations satisfying some constraints can transform a symmetric linearization for a regular quadratic $Q(lambda)$ into another symmetric linearization for a new regular quadratic $wQ(lambda)$. Meanwhile, we discover that they can preserve the structures of the original structured quadratic matrix polynomial and its structured linearizations.

Positivity preserving simulation of  DAEs with variable coefficients

Positive dynamical systems arise in every application in which the
considered variables represent a material quantity that does not take
negative values, like e.g. the concentration of chemical and biological
species or the amount of goods and individuals in economic and social
sciences.
Beside positivity, the dynamics are often subject to constraints resulting
from limitation of resources, conservation or balance laws, which extend
the differential system by additional algebraic equations.
In order to obtain a physically meaningful simulation of such processes,
both properties, the positivity and the constraints, should be reflected
in the numerical solution.
In this talk, we discuss these issues for linear time-varying systems, as
they arise for example in the linearization of non-linear systems in
chemical reaction kinetics or process engineering.

As for linear time-invariant systems [1], we pursue a projection approach
based on generalized inverses that admits to separate the differential and
algebraic components without changing coordinates.

We first consider index-1 problems, in which the differential and
algebraic equations are explicitly given and explain under which
conditions we can expect a positive numerical approximation that meets the
algebraic constraints.

We then extend these results to higher index problems, i.e., problems in
which some of the algebraic equations are hidden in the system, using
derivative arrays and the index reduction developed by Kunkel and Mehrmann
[2].

[1] Numerical Integration of Positive Linear Differential-Algebraic
Systems. A.K. Baum and V. Mehrmann, Preprint TU Berlin, 2012.
www3.math.tu-berlin.de/multiphysics/Publications/Articles/BauM12_ppt.pdf

[2]  Differential-Algebraic Equations. Analysis and Numerical Solution,
P. Kunkel and V. Mehrmann, EMS Publishing House, Zuerich, CH, 2006.

Moving Dirichlet Boundary Conditions

In my last talk we analysed the dynamics of elastic media involving the Dirichlet boundary conditions as a weak constraint. Therefore, we introduced suitable spaces such that the resulting operator DAE has a unique solution. In this talk we derive a suitable model for moving Dirichlet boundary conditions. This means that we demand Dirichlet conditions on a part of the boundary which changes with time. In order to avoid ansatz spaces which depend on time, we need an additional coordinate transformation. In the second part of the talk, a stable discretization scheme is presented.

Eigenvalue problems for Hamiltonian matrices - The curse of Van Loan

I will present Van Loan's approach to computing all the eigenvalues of a Hamiltonian matrix. For this purpose I will give a short summary of the properties of Hamiltonian and skew-Hamiltonian matrices and matrices wich preserve their structure under similarity transformations. Afterwards I will explain the steps of the algorithm developed by Van Loan. I will compare the computational cost of Van Loan's algorithm to the QR algorithm and present the results of the error analysis of Van Loan's algorithm. 

• Absolventen Seminar WS 2013
• Absolventen Seminar SS 2012 [3]
• Absolventen Seminar WS 2011/2012 [4]
• Diplomanden- und Doktorandenseminar SS 2011 [5]
• Diplomanden- und Doktorandenseminar WS 2010/11 [6]
• Diplomanden- und Doktorandenseminar SS 2010 [7]
• Diplomanden- und Doktorandenseminar WS 2009/10 [8]
• Diplomanden- und Doktorandenseminar SS 2009 [9]
• Diplomanden- und Doktorandenseminar WS 2008/09 [10]
• Diplomanden- und Doktorandenseminar SS 2008 [11]
• Diplomanden- und Doktorandenseminar WS 2007/08 [12]
• Diplomanden- und Doktorandenseminar SS 2007 [13]
• Diplomanden- und Doktorandenseminar WS 2006/07 [14]
• Diplomanden- und Doktorandenseminar SS 2006 [15]
• Diplomanden- und Doktorandenseminar WS 2005/06 [16]
• Diplomanden- und Doktorandenseminar SS 2005 [17]