|Dozenten:||Jörg Liesen, Christian Mehl, Volker Mehrmann, Reinhard Nabben, Tatjana Stykel|
|LV-Termine:||Do 10-12 in MA 376|
|Inhalt:||Vorträge von Diplomanden, Doktoranden, Postdocs und manchmal auch Gästen zu aktuellen Forschungsthemen|
|Do 14.04.2011||10:15||MA 376
||Philipp Wolters||Circuit Synthesis (Abstract)|
|im Anschluss||MA 376
|Do 05.05.2011||10:15||MA 376
||Ina Thies||Numerical Solution of Linear Systems Arising from Fokker-Planck Equations (Abstract)|
|Do 12.05.2011||10:15||MA 376
||Prof. Dr. Friedhelm Schieweck||Higher order variational time discretizations for nonlinear systems of ordinary differential equations (Abstract)|
|Do 19.05.2011||10:15||MA 376
||Pia L. Kempker||Coordinated Linear Systems (Abstract)|
|Do 26.05.2011||10:15||MA 376
||Saskia Zurth||Solving differential-algebraic equations by half-explicit Runge-Kutta methods (Abstract)|
|im Anschluss||MA 376
||Linh Vu Hoang||Some complementary remarks on half-explicit methods (Abstract)|
|Do 09.06.2011||10:15||MA 376
||Robert Altmann||An adaptive hp Approach for the Non-Intrusive Projection of Stochastic Model Output (Abstract)|
|Do 23.06.2011||10:15||MA 376
||Jan Heiland||Boundary Control of Turbulent Flows - I Try a Transformation to an ODE Constraint Optimal Control Problem (Abstract)|
|Do 30.06.2011||10:15||MA 376
||Robert Luce||On fill, FLOPs and treewidth in sparse Cholesky factorization (Abstract)|
|im Anschluss||MA 376
||Federico Poloni||Two numerical algorithms for the solution of Lur'e equations (Abstract)|
|Do 07.07.2011||10:15||MA 376
||David Hafemann||Solving the linear-quadratic optimal control problem s.t. semi explicit DAE constraint, with half explicit methods (Abstract)|
|im Anschluss||MA 376
||Phi Ha||Remodeling Delay Differential Algebraic Equations (Abstract)|
|Do 14.07.2011||10:15||MA 376
||Ann-Kristin Baum||Numerical simulation of constrained positive systems - How do we stay positive? (Abstract)|
|im Anschluss||MA 376
||Johanna Ridder||Symplectic Methods for Hamiltonian Systems (Abstract)|
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The presentation deals with a modified nodal approach (MNA), a part of mathematical control theory. First, a short derivation of equations to gain the MNA form is shown. The main focus of the presentation is a numerical algorithm to synthesize linear electric circuits. That means a transformation of a given descriptor system to a MNA form. Then it is possible to reinterpret a descriptor system as an electric circuit.
Excitations of technical systems, such as wind gusts or road roughness, can be described by random variables. This leads to stochastic differential equations. Their probability density function solves (under certain conditions) the Fokker-Planck equation (FPE). This talk is about the numerical solution of linear homogeneous systems of equations arising from solving the FPE.
We discuss different time discretizations of variational type applied to a nonlinear system of ordinary differential equations which is typically generated by a semi-discretization in space of a given nonlinear parabolic partial differential equation like, for instance, the non-stationary Burgers equation.
Among these methods we compare the known continuous Galerkin-Petrov and the discontinuous Galerkin method with time polynomial ansatz functions of order k (cGP(k)- and dG(k)-method) with respect to accuracy, stability and computational costs. Moreover, we propose two new extended methods (cGP-C1(k+1)- and dG-C0(k+1)-method) which have on the one hand a one higher degree of ansatz functions and accuracy, the same stability properties and on the other hand the same computational costs as the original methods. We prove A-stability for the cGP-methods and strong A-stability for the dG-methods.
We present optimal error estimates and the close relationship between the original and extended methods which prove as a byproduct the super-convergence of the original methods cGP(2) and dG(1) in the endpoints of all discrete time intervals. Finally, we present first numerical results for the non-stationary Burgers equation in one space dimension.
The concept of a coordinated linear system is introduced, and the construction and control of coordinated linear systems is discussed.
Differential Algebraic Equations (DAEs) are efficient tools for modelling and simulation in various areas such as electrical circuit design, multibody mechanics, fluid dynamics, etc. For solving DAEs, well-known implicit schemes like Backward Differentiation Formula (BDF) or collocation Runge-Kutta methods are most popular. For semi-explicit DAEs, a class of so-called half-explicit methods provides an alternative efficient solution. The main idea is to apply an explicit discretization scheme to the differential subsystem and an implicit one to the algebraic subsystem. Thus the computational cost due to solving nonlinear system at each time step can be be reduced significantly. In this talk I will give an overview on Half-Explicit Runge-Kutta methods (HERKs) for index 1 or index 2 DAEs of semi-explicit form. The construction as well as the convergence analysis of these methods are discussed.
I will give some remarks on the use of half-explicit methods in solving a special class of matrix-valued DAEs that arise in the computation of spectral intervals of linear DAEs by QR or SVD methods.
We consider a model involving uncertain data and are interested in propagating uncertainty of the models parameters into the solution. To obtain the stochastic modes of the solution, we integrate over d-dimensional domains with the help of Smolyak sparse grids. Higher accuracy of the approximation is obtained by either splitting the expansion domains (h-refinement) or increasing the local projection order (p-refinement).
Consider a stirred vessel filled with two immiscible fluids. To control the turbulent mixing via the stirrerspeed one has to face three major difficulties.
Firstly, in the mathematical model for the control problem the control variable appears in the boundary conditions. Secondly, the model equations contain algebraic constraints. Thirdly, the equations are nonlinear.
In my talk I will present two transformations, that reformulate the model as a nonlinear ODE. For this form the design of the optimal controller is straight forward.
Ordering heuristics for the Cholesky factorization of a sparse symmetric positive definite matrix aim at minimizing the number of elements in the Cholesky factor. Intuitively this should also minimize the number of FLOPs necessary to compute it. The class of graphs that we present shows that this is not the case. The minima for these two objectives may even be attained on disjoint sets of minimal triangulations. The same class of graph shows that the minimum treewidth problem is different from minimum fill and minimum FLOPs.
Continuous-time control problems with a singular matrix R arise in several applications, both in optimal control and model reduction, but are often neglected in literature. When R is singular, one cannot reduce the problem to a Hamiltonian subspace problem nor form an algebraic Riccati equation. Instead, a system of equations called Lur'e equations can be constructed from the invariant subspace problem in the same fashion.
We present two approaches for the solution of Lur'e equations with singular R. The first approach consists in a deflation technique different from the usual one, that reduces the problem to a pencil in standard symplectic form while deflating some of the eigenvalues at infinity. This method transforms the problem to one that can easily be solved using methods such as the structure-preserving doubling algorithm (SDA) of Chu, Fan and Lin. The second approach, which is suitable to sparse large-scale equations, consists in computing the infinite and singular structure of the pencil and projecting it to zero. The problem can then be reduced to an algebraic Riccati equation, whose coefficients are represented implicitly as the solutions of singular linear systems. Using some care in representing these coefficients, we can solve the ARE with the Newton-ADI method using the Lyapack toolbox, without resorting to dense matrix arithmetic.
This talk is based on joint work with Timo Reis.
In this talk I will present the results of V. Mehrmann, P. Kunkel and Kurina and März from the study of the lnear quadratic optimal control problem s.t. DAE discryptorsystem. Formulate them for the semi explicit case and try to solve the Riccati-equations with a half explicit method.
This talk aims at Delay Differential Algebraic Equations of Neutral and Retarded type with linear time invariant matrix coefficients (LTI DDAEs). We propose a reformulation of a given system that can help us partially answer the question "How can we characterize structural properties of the system like regularity, solvability, model consistency, etc. via the structure of their matrix coefficients?"
Positive differential equations arise in every application, in which the considered quantity is real and cannot take negative values; for example in biological or chemical processes, where one deals with densities or concentrations of a given species. Besides positivity, such processes often satisfy additional constraints that result from limitation of resources, balance equations or equilibrium conditions and which lead to a differential-algebraic system structure. To obtain a physical meaningful numerical solution of these Differential-Algebraic-Equations (DAEs), the applied discretization methods need to meet the algebraic constraints and preserve the positivity of the solution. In this talk, we analyze this topic for Runge-Kutta and Multistep methods and present conditions under which these methods provide a consistent a positive discretization of positive DAEs. The crucial tool in this analysis is the decomposition of the problem into its dynamic and constrained components, which admits a separate discussion of each part. For the differential part, we extend the already existing result for positive ODEs, while for the algebraic part we exploit the consistency of the applied method. We check the derived conditions for some common one- and multistep methods and show possibilities to obtain better results for smaller classes of positive DAEs.
In this talk I will give an introduction to symplectic methods for Hamlitonian systems. Hamiltonian systems are very useful for modeling conservative mechanical systems. A characteristic property of the flow of Hamiltonian systems is symplecticity which in the two dimensional case can be interpreted as area preservation of the flow. Numerical methods that preserve symplecticity when applied to a Hamiltonian system give better long-time approximations than non-symplectic methods. In the talk I will give some examples of symplectic methods and analyze their behavior. The talk is based on my bachelor's thesis.
|Impressum||Agnieszka Międlar 26.04.2011|