Publications at the Institute of Mathematics

Results: 1877
Created on: Wed, 04 Aug 2021 23:09:41 +0200 in 0.0922 sec


Grundel, Sara M.; Heyder, Stefan; Hotz, Thomas; Ritschel, Tobias K. S.; Sauerteig, Philipp; Worthmann, Karl;
How to coordinate vaccination and social distancing to mitigate SARS-CoV-2 outbreaks. - In: SIAM journal on applied dynamical systems. - Philadelphia, Pa. : SIAM, ISSN 1536-0040, Bd. 20 (2021), 2, S. 1135-1157

Most countries have started vaccinating people against COVID-19. However, due to limited production capacities and logistical challenges it will take months/years until herd immunity is achieved. Therefore, vaccination and social distancing have to be coordinated. In this paper, we provide some insight on this topic using optimization-based control on an age-differentiated compartmental model. For real-life decision-making, we investigate the impact of the planning horizon on the optimal vaccination/social distancing strategy. We find that in order to reduce social distancing in the long run, without overburdening the health care system, it is essential to vaccinate the people with the highest contact rates first. That is also the case if the objective is to minimize fatalities provided that the social distancing measures are sufficiently strict. However, for short-term planning it is optimal to focus on the high-risk group.



https://doi.org/10.1137/20M1387687
Kriesell, Matthias;
A note on uniquely 10-colorable graphs. - In: Journal of graph theory. - New York, NY [u.a.] : Wiley, ISSN 1097-0118, Bd. 98 (2021), 1, S. 24-26

Hadwiger conjectured that every graph of chromatic number k admits a clique minor of order k. Here we prove for k ≤ 10, that every graph of chromatic number k with a unique k-coloring (up to the color names) admits a clique minor of order k. The proof does not rely on the Four Color Theorem.



https://doi.org/10.1002/jgt.22679
Grüne, Lars; Schaller, Manuel; Schiela, Anton;
Abstract nonlinear sensitivity and turnpike analysis and an application to semilinear parabolic PDEs. - In: Control, optimisation and calculus of variations : COCV.. - Les Ulis : EDP Sciences, ISSN 1262-3377, Bd. 27 (2021), S. 1-28

We analyze the sensitivity of the extremal equations that arise from the first order necessary optimality conditions of nonlinear optimal control problems with respect to perturbations of the dynamics and of the initial data. To this end, we present an abstract implicit function approach with scaled spaces. We will apply this abstract approach to problems governed by semilinear PDEs. In that context, we prove an exponential turnpike result and show that perturbations of the extremal equation's dynamics, e.g., discretization errors decay exponentially in time. The latter can be used for very efficient discretization schemes in a Model Predictive Controller, where only a part of the solution needs to be computed accurately. We showcase the theoretical results by means of two examples with a nonlinear heat equation on a two-dimensional domain.



https://doi.org/10.1051/cocv/2021030
Gernandt, Hannes; Trunk, Carsten;
The spectrum and the Weyr characteristics of operator pencils and linear relations. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2021. - 1 Online-Ressource (18 Seiten). . - (Preprint. - M21,05)

The relation between the spectra of operator pencils with unbounded coeficients and of associated linear relations is investigated. It turns out that various types of spectrum coincide and the same is true for the Weyr characteristics. This characteristic describes how many independent Jordan chains up to a certain length exist. Furthermore, the change of this characteristic subject to one-dimensional perturbations is investigated.



https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2021200091
Gernandt, Hannes; Trunk, Carsten;
Locally finite extensions and Gesztesy-Šeba realizations for the Dirac operator on a metric graph. - In: Operator theory. - Berlin : De Gruyter, (2021), S. 25-54

Behrndt, Jussi; Schmitz, Philipp; Teschl, Gerald; Trunk, Carsten;
Perturbations of periodic Sturm-Liouville operators. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2021. - 1 Online-Ressource (17 Seiten). . - (Preprint. - M21,04)

We study perturbations of self-adjoint periodic Sturm-Liouville operators and conclude under L1-assumptions on the differences of the coeffcients that the essential spectrum and absolutely continuous spectrum remain the same. If a finite first moment condition holds for the differences of the coeffcients, then at most finitely many eigenvalues appear in the spectral gaps. This observation extends a seminal result by Rofe-Beketov from the 1960s. Finally, imposing a second moment condition we show that the band edges are no eigenvalues of the perturbed operator.



https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2021200075
De Santis, Marianna; Eichfelder, Gabriele;
A decision space algorithm for multiobjective convex quadratic integer optimization. - In: Computers & operations research : an international journal.. - Amsterdam [u.a.] : Elsevier, ISSN 0305-0548, Bd. 134 (2021)

We present a branch-and-bound algorithm for minimizing multiple convex quadratic objective functions over integer variables. Our method looks for efficient points by fixing subsets of variables to integer values and by using lower bounds in the form of hyperplanes in the image space derived from the continuous relaxations of the restricted objective functions. We show that the algorithm stops after finitely many fixings of variables with detecting both the full efficient and the nondominated set of multiobjective strictly convex quadratic integer problems. A major advantage of the approach is that the expensive calculations are done in a preprocessing phase so that the nodes in the branch-and-bound tree can be enumerated fast. We show numerical experiments on biobjective instances and on instances with three and four objectives.



https://doi.org/10.1016/j.cor.2021.105396
Chen, Liangzhi; Hotz, Thomas; Zhang, Haizhang;
Admissible kernels for RKHS embedding of probability distributions. - In: Statistical papers. - Berlin : Springer, ISSN 1613-9798, Bd. 62 (2021), 3, S. 1499-1518

Similarity measurement of two probability distributions is important in many applications of statistics. Embedding such distributions into a reproducing kernel Hilbert space (RKHS) has many favorable properties. The choice of the reproducing kernel is crucial in the approach. We study this question by considering the similarity of two distributions of the same class. In particular, we investigate when the RKHS embedding is "admissible" in the sense that the distance between the embeddings should become smaller when the expectations are getting closer or when the variance is increasing to infinity. We give conditions on the widely-used translation-invariant reproducing kernels to be admissible. We also extend the study to multivariate non-symmetric Gaussian distributions.



https://doi.org/10.1007/s00362-019-01144-5
Eichfelder, Gabriele; Kirst, Peter; Meng, Laura; Stein, Oliver;
A general branch-and-bound framework for continuous global multiobjective optimization. - In: Journal of global optimization : an international journal dealing with theoretical and computational aspects of seeking global optima and their applications in science, management and engineering.. - Dordrecht [u.a.] : Springer Science + Business Media B.V, ISSN 1573-2916, Bd. 80 (2021), 1, S. 195-227

Current generalizations of the central ideas of single-objective branch-and-bound to the multiobjective setting do not seem to follow their train of thought all the way. The present paper complements the various suggestions for generalizations of partial lower bounds and of overall upper bounds by general constructions for overall lower bounds from partial lower bounds, and by the corresponding termination criteria and node selection steps. In particular, our branch-and-bound concept employs a new enclosure of the set of nondominated points by a union of boxes. On this occasion we also suggest a new discarding test based on a linearization technique. We provide a convergence proof for our general branch-and-bound framework and illustrate the results with numerical examples.



https://doi.org/10.1007/s10898-020-00984-y
Derkach, Volodymyr; Hassi, Seppo; Malamud, Mark;
Generalized boundary triples, II : some applications of generalized boundary triples and form domain invariant Nevanlinna functions. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2021. - 1 Online-Ressource (54 Seiten). . - (Preprint. - M21,03)
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2021200058