Publications at the Institute of Mathematics

Results: 1962
Created on: Thu, 02 Feb 2023 23:08:09 +0100 in 0.0682 sec

Derkach, Volodymyr; Trunk, Carsten
PT-symmetric couplings of dual pairs. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (24 Seiten). - (Preprint ; M23,03)
Eichfelder, Gabriele; Gerlach, Tobias; Rocktäschel, Stefan
Convexity and continuity of specific set-valued maps and their extremal value functions. - In: Journal of applied and numerical optimization, ISSN 2562-5535, Bd. 5 (2023), 1, S. 71-92

In this paper, we study several classes of set-valued maps, which can be used in set-valued optimization and its applications, and their respective maximum and minimum value functions. The definitions of these maps are based on scalar-valued, vector-valued, and cone-valued maps. Moreover, we consider those extremal value functions which are obtained when optimizing linear functionals over the image sets of the set-valued maps. Such extremal value functions play an important role for instance for derivative concepts for set-valued maps or for algorithmic approaches in set-valued optimization. We formulate conditions under which the set-valued maps and their extremal value functions inherit properties like (Lipschitz-)continuity and convexity.
Qi, Yaru; Qiu, Wenwen; Trunk, Carsten; Wilson, Mitsuru
Spectral inclusion property for a class of block operator matrices. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (12 Seiten). - (Preprint ; M23,02)

The numerical range and the quadratic numerical range is used to study the spectrum of a class of block operator matrices. We show that the approximate point spectrum is contained in the closure of the quadratic numerical range. In particular, the spectral enclosures yield a spectral gap. It is shown that these spectral bounds are tighter than classical numerical range bounds.
Behrndt, Jussi; Gesztesy, Fritz; Schmitz, Philipp; Trunk, Carsten
Lower bounds for self-adjoint Sturm-Liouville operators. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2023. - 1 Online-Ressource (11 Seiten). - (Preprint ; M23,01)
Hörsch, Florian;
Globally balancing spanning trees. - In: European journal of combinatorics, Bd. 109 (2023), 103644
Philipp, Friedrich;
Relatively bounded perturbations of J-non-negative operators. - In: Complex analysis and operator theory, ISSN 1661-8262, Bd. 17 (2023), 1, 14, insges. 30 S.

We improve known perturbation results for self-adjoint operators in Hilbert spaces and prove spectral enclosures for diagonally dominant J-self-adjoint operator matrices. These are used in the proof of the central result, a perturbation theorem for J-non-negative operators. The results are applied to singular indefinite Sturm-Liouville operators with Lp-potentials. Known bounds on the non-real eigenvalues of such operators are improved.
Hörsch, Florian; Szigeti, Zoltán
On the complexity of finding well-balanced orientations with upper bounds on the out-degrees. - In: Journal of combinatorial optimization, ISSN 1573-2886, Bd. 45 (2023), 1, 30, S. 1-14
Lee, Dae Gwan; Philipp, Friedrich; Voigtlaender, Felix
A note on the invertibility of the Gabor frame operator on certain modulation spaces. - In: The journal of Fourier analysis and applications, ISSN 1531-5851, Bd. 29 (2023), 1, 3, S. 1-20

We consider Gabor frames generated by a general lattice and a window function that belongs to one of the following spaces: the Sobolev space $$V_1 = H^1(\mathbb {R}^d)$$, the weighted $$L^2$$-space $$V_2 = L_{1 + |x|}^2(\mathbb {R}^d)$$, and the space $$V_3 = \mathbb {H}^1(\mathbb {R}^d) = V_1 \cap V_2$$consisting of all functions with finite uncertainty product; all these spaces can be described as modulation spaces with respect to suitable weighted $$L^2$$spaces. In all cases, we prove that the space of Bessel vectors in $$V_j$$is mapped bijectively onto itself by the Gabor frame operator. As a consequence, if the window function belongs to one of the three spaces, then the canonical dual window also belongs to the same space. In fact, the result not only applies to frames, but also to frame sequences.
Viehweg, Johannes; Worthmann, Karl; Mäder, Patrick
Parameterizing echo state networks for multi-step time series prediction. - In: Neurocomputing, ISSN 1872-8286, Bd. 522 (2023), S. 214-228

Prediction of multi-dimensional time-series data, which may represent such diverse phenomena as climate changes or financial markets, remains a challenging task in view of inherent nonlinearities and non-periodic behavior In contrast to other recurrent neural networks, echo state networks (ESNs) are attractive for (online) learning due to lower requirements data and computational power. However, the randomly-generated reservoir renders the choice of suitable hyper-parameters as an open research topic. We systematically derive and exemplarily demonstrate design guidelines for the hyper-parameter optimization of ESNs. For the evaluation, we focus on the prediction of chaotic time series, an especially challenging problem in machine learning. Our findings demonstrate the power of a hyper-parameter-tuned ESN when auto-regressively predicting time series over several hundred steps. We found that ESNs’ performance improved by 85.1%-99.8% over an already wisely chosen default parameter initialization. In addition, the fluctuation range is considerably reduced such that significantly worse performance becomes very unlikely across random reservoir seeds. Moreover, we report individual findings per hyper-parameter partly contradicting common knowledge to further, help researchers when training new models.
Nüske, Feliks; Peitz, Sebastian; Philipp, Friedrich; Schaller, Manuel; Worthmann, Karl
Finite-data error bounds for Koopman-based prediction and control. - In: Journal of nonlinear science, ISSN 1432-1467, Bd. 33 (2023), 1, 14, S. 1-34

The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points, for both ordinary and stochastic differential equations while using either ergodic trajectories or i.i.d. samples. We illustrate these bounds by means of an example with the Ornstein-Uhlenbeck process. Moreover, we extend our analysis to (stochastic) nonlinear control-affine systems. We prove error estimates for a previously proposed approach that exploits the linearity of the Koopman generator to obtain a bilinear surrogate control system and, thus, circumvents the curse of dimensionality since the system is not autonomized by augmenting the state by the control inputs. To the best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the bilinear approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled.