Absolventen-Seminar • Numerische Mathematik

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

Abstracts zu den Vorträgen:

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

Abstract in PDF format (PDF, 127,7 KB)

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.