FG Numerische Mathematik

17 Items

Recent Submissions
Approximation of stability radii for large-scale dissipative Hamiltonian systems

Aliyev, Nicat ; Mehrmann, Volker ; Mengi, Emre (2020-02-06)

A linear time-invariant dissipative Hamiltonian (DH) system x ̇ = ( J − R ) Q x, with a skew-Hermitian J , a Hermitian positive semidefinite R , and a Hermitian positive definite Q , is always Lyapunov stable and under further weak conditions even asymptotically stable. By exploiting the characterizations from Mehl et al. (SIAM J. Matrix Anal. Appl. 37 (4), 1625–1654, 2016 ), we focus on the es...

Computation of the analytic center of the solution set of the linear matrix inequality arising in continuous- and discrete-time passivity analysis

Bankmann, Daniel ; Mehrmann, Volker ; Nesterov, Yurii ; Van Dooren, Paul (2020-07-23)

In this paper formulas are derived for the analytic center of the solution set of linear matrix inequalities (LMIs) defining passive transfer functions. The algebraic Riccati equations that are usually associated with such systems are related to boundary points of the convex set defined by the solution set of the LMI. It is shown that the analytic center is described by closely related matrix e...

Multilevel optimization problems with linear differential-algebraic equations

Bankmann, Daniel Steffen (2021)

This thesis is about various multilevel optimization problems in the context of general parameter-dependent linear differential-algebraic equations. In the first part we analyze problems where the lowest level consists of an optimal control problem for linear differential-algebraic equations. The optimal control problem depends on higher level variables that take the role of a parameter. A so...

Stability Assessment of Stochastic Differential-Algebraic Systems via Lyapunov Exponents with an Application to Power Systems

González-Zumba, Andrés ; Fernández-de-Córdoba, Pedro ; Cortés, Juan-Carlos ; Mehrmann, Volker (2020-08-20)

In this paper, we discuss stochastic differential-algebraic equations (SDAEs) and the asymptotic stability assessment for such systems via Lyapunov exponents (LEs). We focus on index-1 SDAEs and their reformulation as ordinary stochastic differential equations (SDEs). Via ergodic theory, it is then feasible to analyze the LEs via the random dynamical system generated by the underlying SDEs. Onc...

Model Reduction for Kinetic Models of Biological Systems

Ali Eshtewy, Neveen ; Scholz, Lena (2020-05-25)

High dimensionality continues to be a challenge in computational systems biology. The kinetic models of many phenomena of interest are high-dimensional and complex, resulting in large computational effort in the simulation. Model order reduction (MOR) is a mathematical technique that is used to reduce the computational complexity of high-dimensional systems by approximation with lower dimension...

Semi‐active ℋ∞ damping optimization by adaptive interpolation

Tomljanović, Zoran ; Voigt, Matthias (2020-04-06)

In this work we consider the problem of semi‐active damping optimization of mechanical systems with fixed damper positions. Our goal is to compute a damping that is locally optimal with respect to the ℋ∞‐norm of the transfer function from the exogenous inputs to the performance outputs. We make use of a new greedy method for computing the ℋ∞‐norm of a transfer function based on rational interpo...

Well-posedness and realization theory for delay differential-algebraic equations

Unger, Benjamin (2020)

This thesis is dedicated to delay differential-algebraic equations (DDAEs), i.e., constraint dynamical systems where the rate of change depends on the current state and its past. Typical applications include - feedback control, where the delay is a direct consequence of the time required to measure the current state, compute the feedback, and implement the control action, - hybrid numerical...

Error analysis and model adaptivity for flows in gas networks

Stolwijk, Jeroen Johannes ; Mehrmann, Volker (2018-11-22)

In the simulation and optimization of natural gas flow in a pipeline network, a hierarchy of models is used that employs different formulations of the Euler equations. While the optimization is performed on piecewise linear models, the flow simulation is based on the one to three dimensional Euler equations including the temperature distributions. To decide which model class in the hierarchy is...

On the sign characteristics of Hermitian matrix polynomials

Mehrmann, Volker ; Noferini, Vanni ; Tisseur, Françoise ; Xu, Hongguo (2016-09-14)

The sign characteristics of Hermitian matrix polynomials are discussed, and in particular an appropriate definition of the sign characteristics associated with the eigenvalue infinity. The concept of sign characteristic arises in different forms in many scientific fields, and is essential for the stability analysis in Hamiltonian systems or the perturbation behavior of eigenvalues under structu...

Inexact methods for the solution of large scale Hermitian eigenvalue problems

Kandler, Ute (2019)

This thesis focuses on the solution of high dimensional Hermitian eigenproblems in situations where vector operations cannot be carried out exactly. To this end an inexact Arnoldi method with the aim to approximate extreme eigenvalues and eigenvectors is developed. This method is particularly wellsuited for large scale problems as it efficiently reduces the storage and computational requiremen...

Coupled system of Maxwell equations and circuit equations in electro-magnetism

Niroomand Rad, Helia (2019)

This thesis is devoted to modeling electro-magnetic coupling, the so-called crosstalk phenomenon, in circuit simulation, where a new modeling approach via bilateral coupling of the Maxwell equations with circuit equations is considered. Using the bilaterally coupled model allows full simulation of the crosstalk phenomenon in the evolution of time, when the disturbances generated by the excitati...

Operator differential-algebraic equations with noise arising in fluid dynamics

Altmann, Robert ; Levajković, Tijana ; Mena, Hermann (2016)

We study linear semi-explicit stochastic operator differential algebraic equations (DAEs) for which the constraint equation is given in an explicit form. In particular, this includes the Stokes equations arising in fluid dynamics. We combine a white noise polynomial chaos expansion approach to include stochastic perturbations with deterministic regularization techniques. With this, we are able ...

Backward error analysis of the shift-and-invert Arnoldi algorithm

Schröder, Christian ; Taslaman, Leo (2015)

We perform a backward error analysis of the inexact shift-and-invert Arnoldi algorithm. We consider inexactness in the solution of the arising linear systems, as well as in the orthonormalization steps, and take the non-orthonormality of the computed Krylov basis into account. We show that the computed basis and Hessenberg matrix satisfy an exact shift-and-invert Krylov relation for a perturbed...

Simulation of multibody systems with servo constraints through optimal control

Altmann, Robert ; Heiland, Jan (2016)

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Wavelet methods for a weighted sparsity penalty for region of interest tomography

Klann, Esther ; Quinto, Eric Todd ; Ramlau, Ronny (2015)

We consider region of interest (ROI) tomography of piecewise constant functions. Additionally, an algorithm is developed for ROI tomography of piecewise constant functions using a Haar wavelet basis. A weighted ℓp–penalty is used with weights that depend on the relative location of wavelets to the region of interest. We prove that the proposed method is a regularization method, i.e., that the r...

Regularization properties of Mumford–Shah-type functionals with perimeter and norm constraints for linear ill-posed problems

Klann, Esther ; Ramlau, Ronny (2013)

In this paper we consider the simultaneous reconstruction and segmentation of a function f from measurements g = Kf, where K is a linear operator. Assuming that the inversion of K is illposed, regularization methods have to be used for the inversion process in case of inexact data. We propose using a Mumford–Shah-type functional for the stabilization of the inversion. Restricting our analysis t...

A Mumford–Shah-type approach to simultaneous reconstruction and segmentation for emission tomography problems with Poisson statistics

Klann, Esther ; Ramlau, Ronny ; Sun, Peng (2017)

We propose a variational model to simultaneous reconstruction and segmentation in emission tomography. As in the original Mumford–Shah model [27] we use the contour length as penalty term to preserve edge information whereas a different data fidelity term is used to measure the information discrimination between the computed tomography data of the reconstructed object and the observed (or simul...