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## Absolventen-Seminar • Numerische Mathematik

Verantwortliche Dozenten: | Prof. Dr. Christian Mehl, Prof. Dr. Volker Mehrmann |
---|---|

Koordination: | Federico Poloni |

Termine: | Do 10:00-12:00 in MA 376 |

Inhalt: | Vorträge von Diplomanden, Doktoranden, Postdocs und manchmal auch Gästen zu aktuellen Forschungsthemen |

## Rückblick

- Absolventen Seminar WS 2011/2012
- Diplomanden- und Doktorandenseminar SS 2011
- Diplomanden- und Doktorandenseminar WS 2010/11
- Diplomanden- und Doktorandenseminar SS 2010
- Diplomanden- und Doktorandenseminar WS 2009/10
- Diplomanden- und Doktorandenseminar SS 2009
- Diplomanden- und Doktorandenseminar WS 2008/09
- Diplomanden- und Doktorandenseminar SS 2008
- Diplomanden- und Doktorandenseminar WS 2007/08
- Diplomanden- und Doktorandenseminar SS 2007
- Diplomanden- und Doktorandenseminar WS 2006/07
- Diplomanden- und Doktorandenseminar SS 2006
- Diplomanden- und Doktorandenseminar WS 2005/06
- Diplomanden- und Doktorandenseminar SS 2005

### Sarosh Quraishi (TU Berlin)

Donnerstag, 19. April 2012

**Compressed sensing: A new paradigm for finite elements using sparse representations**

Partial differential equations on complex domains and with discontinuities and high gradients solutions are very difficult to approximate with standard finite element method (FEM). We survey existing methods and their shortcomings and propose a new method which is basically a unification of modern day methods. Our method is exploits the sparsity of solution, and solves the problem using compressed sensing algorithms. Compared with FEM, the proposed method requires computation of a minimal number of degrees on unknowns, it is unaffected by system conditioning, and it is scalable to very large systems. We demonstrate our method by solving simple elliptic equations.

### Tobias Brüll (TU Berlin)

Donnerstag, 19. April 2012

**The behavioral nature of infinite-time linear quadratic optimal control**

I will first give an introduction to optimal control in the behavioral setting and then show a result which characterizes the solvability of the linear quadratic optimal control problem. The optimal solution can be derived from an optimality system which involves the adjoint equation. In a second result I will show that the Lagrange multiplier depends linearly on the state.

Both results are, of course, standard for linear ODEs with regular cost functionals. However, since we exploit the behavioral nature of the problem, the presented results can be applied to differential-algebraic systems with singular cost functionals. In particular, the second result formulates a generalization of the algebraic Riccati equation.

### Manideepa Saha (IIT Guwahati)

Donnerstag, 26. April 2012

**The Frobenius-Jordan form of nonnegative matrices**

The invariant subspace associated with the spectral radius of a nonnegative matrix possesses some nonnegative bases which have some nice combinatorial structure. Using some of those bases we introduced a variant of Jordan canonical form named as Frobenius Jordan Form. In this talk I will discuss about the existence of such canonical form and some graphical properties of a nonnegative matrices with the help of this form. Furthermore, I will discuss about the special graph-representation of nonnegative bases for nonnegative matrices and give a necessary condition for the existence of such graph bases.

### Phi Ha (TU Berlin)

Donnerstag, 03. Mai 2012

**Differential-algebraic equations with delay --- A 25 minute-journey**

This talk aims at linear time-invariant delay differential-algebraic equations (DDAEs) of the form

\begin{equation}\label{eq1} A_k x^{(k)}(t) + \dots + A_0 x(t) + A_{-1} x(t-\tau) + \dots + A_{-\kappa} x^{(\kappa)}(t-\tau) = f(t), \tag{1} \end{equation}

and its special case

\begin{equation}\label{eq2} A_1\dot{x}(t) + A_0 x(t) + A_{-1}x(t-\tau) = f(t).\tag{2} \end{equation}

Surprisingly, already in order to deal with \eqref{eq2}, it is necessary to study linear high-order differential-algebraic equations of the form

\begin{equation}\label{eq3} A_m x^{(m)}(t) + \dots + A_0 x(t) = f(t), \tag{3} \end{equation}

for different orders $m$. Therefore, in the first part of the talk, we study system \eqref{eq3}. The second part of the talk is about the solvability analysis of systems \eqref{eq1} and \eqref{eq2}. The key tool of our analysis here is the combination of algebraic and behavior approaches \cite{KunM06,PolW98,Ste11b}.

### Jan Heiland (TU Berlin)

Donnerstag, 03. Mai 2012

**Numerical Realization of Differential-algebraic Riccati Decoupling in Optimal Control of Flows **

Linear-time varying differential-algebraic equations (DAE) of index 2 are the key to the numerical treatment of Navier-Stokes equations, since sooner or later most solution procedures carry out a spatial discretization and a linearization. I will introduce this particular class of DAEs. In view of optimal control a general approach is to find an equivalent formulation in terms of an ordinary differential equation (ODE) and use the well known linear-quadratic control theory to write down the optimality conditions. When it comes to the numerical solution, however, this ODE-approach may be infeasible in terms of computational costs and numerical stability.

In my talk I will formulate the optimality system in terms of the original DAE and its solution set. I will present a decoupling of the solution that leads to a differential-algebraic Riccati equation for the gain-matrix that establishes the optimal control via a feedback law.

When approaching the Riccati-DAE numerically one has to face the high-dimensionality of the system. Thus I will adress a promising numerical solution procedure to the Riccati-DAE that bases on the solutions of large scale differential-algebraic Lyapunov equations by ADI-methods, as it is under investigation in the group by Peter Benner at the MPI Magdeburg. It will turn out that this particular approach to the Riccati-DAE is equivalent to the saddle-point formulation that was proposed by Heinkenschloss, Sörensen and Sun in 2008 to overcome some of the difficulties with the mentioned above ODE interpretation.

### Robert Altmann (TU Berlin)

Donnerstag, 24. Mai 2012

**Gelfand Triples, Second Order Operator DAEs and Unique Solutions**

We consider the equations of elastodynamics with weakly enforced Dirichlet boundary conditions. A straight forward semi-discretization in space by finite elements then leads to a DAE of index 3. In this talk, I present an index reduction on operator level. This procedure acts directly on the partial differential equation, i.e., on the continuous model. As a result, we obtain a reformulation of the original problem of which a semi-discretization leads directly to a DAE of index 1. In order to work with operator DAEs, I will introduce some tools from functional analysis such as Gelfand triples.

### Giacomo Sbrana (Rouen Business School)

Donnerstag, 24. Mai 2012

**On the use and importance of the palindromic matrix equation in econometrics**

The exponential smoothing method represents one of the most popular approach in modeling and forecasting economic and financial time series. An interesting feature of this method is that its parameters can be derived as the solution of a quadratic equation. In a multivariate context, the matrices of parameters can be derived as the solution of a quadratic matrix equation. In the presentation I show the importance of the palindromic matrix solution when employing some models frequently used in financial econometrics. This approach represents a very easy and fast alternative to standard (but generally very difficult) estimation methods such as the Maximum Likelihood approach.

### Kristin Steinberg (TU Berlin)

Donnerstag, 07. Juni 2012

** WaveTUB: A Numerical Wave Tank**

**Mathematical Foundation and Analysis of the Programme Structure**

For the analysis of loads and motions of marine structures in extreme seaways, knowledge of the hydrodynamics of the exciting wave field is required. While the surface motion of waves can easily be obtained from tests in physical wave tanks, other characteristics of the seaway such as the particle velocity and acceleration as well as the pressure field are difficult to determine through physical measurements. Therefore a wave simulation code called WaveTUB was developed at Technical University Berlin which can numerically simulate waves in a two-dimensional wave tank.

The simulation is based on potential flow theory with the flow field described by a velocity potential which satisfies the Laplace equation. At each time step the velocity potential is calculated in the entire fluid domain using a finite element method. From this solution the velocities at the free surface are determined by second-order differences. To develop the solution in the time domain the classical fourth-order Runge-Kutta formula is applied.

As WaveTUB has repeatedly been complemented and changed since its inception, but a proper documentation is not available, the use of the programme in its current state is impeded by a number of issues concerning its mathematical foundations, the implementation of numerical algorithms in Matlab and Fortran, its programme structure, and its overall usability.

The present Bachelor thesis confronts these issues by functioning as a fundamental numerics-based documentation of WaveTUB, providing both the mathematical foundations of the programme and an improvement-oriented analysis of the programme structure and the programme code.

### Hermann Mena (Escuela Politécnica Nacional - Quito)

Donnerstag, 07. Juni 2012

**Simulation of the Glyphosate Aerial Spray Drift at the Ecuador-Colombia border **

Glyphosate is one of the herbicides used by the Colombian government to spray coca fields. Sprays took place for a number of years and were more frequent between 2000 and 2006. The spray drifts at the Ecuador-Colombia border had become a big issue for people living close to the border. Hence, in 2005 Ecuador and Colombia signed an agreement to stop the sprays along a 10 km stripe at the border. However, measurements on Ecuadorian territory indicated that significant amounts of Glyphosate spray had still drifted into Ecuador. The latter end up in a trail that the two countries are still running in The International Court of Justice located in The Hague. In this talk, in addition to the political context, we present a simple model that takes into account the particular guidelines that aircrafts follow to perform the sprays. Numerical simulations in 2D and 3D are performed at sensitive zones along the Ecuador-Colombia border.

### Carola Schütt (TU Berlin)

Donnerstag, 14. Juni 2012

**Generic rank one perturbation theory**

In the first part of this talk, I will summarize well known results about the perturbation theory of matrices. From which general statements of the structure of the perturbed Jordan canonical form can be deduced. Furthermore, I will speak about the differences between the unstructured and structured perturbation, the Hamiltonian case. I survey existing results and then one special example is regarded.

In the second part, the perturbation theory of matrix pencil is considered. Also in this part, I will first survey on general results of rank one perturbation before the symmetric/skew-symmetric case is considered. At the end I am going to analyse one special example. In this case the result of the Jordan structure differs from the general perturbation results.

### Ann-Kristin Baum (TU Berlin)

Donnerstag, 14. Juni 2012

**Positivity preserving simulation of Differential-Algebraic Equations**

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, Zürich, CH, 2006.

### André Gaul (TU Berlin)

Donnerstag, 28. Juni 2012

**Experiences with deflated MINRES and the Ginzburg-Landau equations **

In the context of Krylov subspace methods for solving linear algebraic systems a "deflated" method tries to modify the operator's spectrum in order to improve convergence. In this talk I want to share some experiences with the deflated MINRES method which is used to speed up the computation of solutions of the (extreme type-II) Ginzburg-Landau equations. The used deflation data consists of auxiliary information (that can be derived from theoretical properties of the operators) and of information that is generated within previous runs of MINRES ("recycling"). After a brief introduction I will concentrate on results obtained from experiments with authentic 2d and 3d data.

This is joint work with Nico Schlömer (U Antwerpen).

### Federico Poloni (TU Berlin)

Donnerstag, 28. Juni 2012

**A duality relation for matrix pencils (with applications to linearizations)**

Let A-xE be a matrix pencil, and C,S such that [C S] spans the left nullspace of [E;-A]. Then there is an explicit relation between the Kronecker structure of C-xS and that of the original pencil; we call this new pencil a dual of the original one. This has some applications in the study of linearizations of matrix polynomials: we show that most known linearizations can be viewed in terms of duals, and this interpretation allows one to deduce easily their Kronecker structure and related properties.

### Lennart Jansen (U Köln)

Donnerstag, 05. Juli 2012

**Semi-explicit methods for coupled circuit/field problems **

The need of combining circuit simulation directly with complex device models to refine critical circuit parts becomes more and more urgent, since the classical circuit simulation can no longer

supply sufficiently accurate results. The simulation of such coupled problems leads to large systems and therefore to high computing times.

We consider a set of differential-algebraic equations, which arise from an electric circuit modeled by the modified nodal analysis coupled with electromagnetic devices. While the normal circuit elements are 0d-elements, the electromagnetic devices are given by a three dimensional model. Therefore the number of variables can easily go beyond millions, if we refine the spatial discretization.

Since we are confronted by a system of DAEs we can not make use of explicit methods in general. So we are forced to solve very large implicit systems.

We analyze the structure of the discretized coupled system and present a way to transform it into a semi-explicit system of differential-algebraic equations. In the process we make use of a new decoupling method for DAEs which results from a mix of the strangeness index and the tractability index. After this remodelling the electromagnetic part of the equation will be a system of ordinary differential equations with sparse matrices only.

### Christian Schröder (TU Berlin)

Donnerstag, 05. Juli 2012

**A real Jacobi-Davidson Algorithm for the 2-real-Parameter Eigenvalue Problem**

We consider the nonlinear eigenvalue problem

( i*omega*M + A + exp(-i*omega*tau*)*B )u=0

for real n-times-n matrices M,A,B with invertible M. Sought are triples (omega, tau, u) consisting of a complex eigenvector u and two _real_ eigenvalues omega and tau.

Problems of this type appear e.g. in the search for critical delays of linear time invariant delay differential equations (LTI-DDEs).

In [Meerbergen, Schroeder, Voss, 2010] this problem is discussed for complex M,A,B. Like there we are considering a Jacobi-Davidson-like projection method. The main differences in the real case are that a) the search space is kept real and b) a specialized method for the small projected problem is used.

Numerical experiments show the effectiveness of the method.

### Helia Niroomand Rad (TU Berlin)

Donnerstag, 12. Juli 2012

**An Introduction to Modeling the Crosstalk Phenomenon in Electromagnetic Systems**

Electromagnetic interferences play a crucial role in designing electronic chips, i.e. integrated circuits (ICs). Specifically, the interference can lead to the parasitic effects, a.k.a. crosstalk phenomenon, which is defined as the the energy coupling between two neighboring systems. The crosstalk can deteriorate the operation of electrical circuits integrated in a chip, which in turn limits the achievable number of electronics elements and circuits integrated into a limited space of a chip. In this talk, I will give an introduction to the concept of crosstalk and the problem of modeling this phenomenon. Our focus is mainly on the coupling of current in electromagnetic systems that are typically integrated in ICs. In modeling the crosstalk we need to start with Maxwell's set of equations as the fundamentals of electricity and magnetism, which govern almost every phenomenon in electromagnetic systems.