Structure-preserving model reduction for dissipative mechanical systems. - In: Calm, smooth and smart, (2024), S. 209-230
Suppressing vibrations in mechanical systems, usually described by second-order dynamical models, is a challenging task in mechanical engineering in terms of computational resources even nowadays. One remedy is structure-preserving model order reduction to construct easy-to-evaluate surrogates for the original dynamical system having the same structure. In our work, we present an overview of recently developed structure-preserving model reduction methods for second-order systems. These methods are based on modal and balanced truncation in different variants, as well as on rational interpolation. Numerical examples are used to illustrate the effectiveness of all described methods.
https://doi.org/10.1007/978-3-031-36143-2_11
Powers of Hamilton cycles in dense graphs perturbed by a random geometric graph. - In: European journal of combinatorics, Bd. 121 (2024), 103848, S. 1-14
Let G be a graph obtained as the union of some n-vertex graph Hn with minimum degree δ (Hn) ≥ αn and a d-dimensional random geometric graph Gd (n,r). We investigate under which conditions for r the graph G will a.a.s. contain the kth power of a Hamilton cycle, for any choice of Hn. We provide asymptotically optimal conditions for r for all values of α, d and k. This has applications in the containment of other spanning structures, such as F-factors.
https://doi.org/10.1016/j.ejc.2023.103848
Long running times for hypergraph bootstrap percolation. - In: European journal of combinatorics, Bd. 115 (2024), 103783, S. 1-18
Consider the hypergraph bootstrap percolation process in which, given a fixed r-uniform hypergraph H and starting with a given hypergraph G0, at each step we add to G0 all edges that create a new copy of H. We are interested in maximising the number of steps that this process takes before it stabilises. For the case where H = Kr+1(r) with r ≥ 3, we provide a new construction for G0 that shows that the number of steps of this process can be of order Θ (nr). This answers a recent question of Noel and Ranganathan. To demonstrate that different running times can occur, we also prove that, if H is K4(3) minus an edge, then the maximum possible running time is 2n − ⌊log2(n−2)⌋ − 6. However, if H is K5(3) minus an edge, then the process can run for Θ (n3) steps.
https://doi.org/10.1016/j.ejc.2023.103783
A hybrid patch decomposition approach to compute an enclosure for multi-objective mixed-integer convex optimization problems. - In: Mathematical methods of operations research, ISSN 1432-5217, Bd. 100 (2024), 1, S. 291-320
In multi-objective mixed-integer convex optimization, multiple convex objective functions need to be optimized simultaneously while some of the variables are restricted to take integer values. In this paper, we present a new algorithm to compute an enclosure of the nondominated set of such optimization problems. More precisely, we decompose the multi-objective mixed-integer convex optimization problem into several multi-objective continuous convex optimization problems, which we refer to as patches. We then dynamically compute and improve coverages of the nondominated sets of those patches to finally combine them to obtain an enclosure of the nondominated set of the multi-objective mixed-integer convex optimization problem. Additionally, we introduce a mechanism to reduce the number of patches that need to be considered in total. Our new algorithm is the first of its kind and guaranteed to return an enclosure of prescribed quality within a finite number of iterations. For selected numerical test instances we compare our new criterion space based approach to other algorithms from the literature and show that much larger instances can be solved with our new algorithm.
https://doi.org/10.1007/s00186-023-00828-x
A test instance generator for multiobjective mixed-integer optimization. - In: Mathematical methods of operations research, ISSN 1432-5217, Bd. 100 (2024), 1, S. 385-410
Application problems can often not be solved adequately by numerical algorithms as several difficulties might arise at the same time. When developing and improving algorithms which hopefully allow to handle those difficulties in the future, good test instances are required. These can then be used to detect the strengths and weaknesses of different algorithmic approaches. In this paper we present a generator for test instances to evaluate solvers for multiobjective mixed-integer linear and nonlinear optimization problems. Based on test instances for purely continuous and purely integer problems with known efficient solutions and known nondominated points, suitable multiobjective mixed-integer test instances can be generated. The special structure allows to construct instances scalable in the number of variables and objective functions. Moreover, it allows to control the resulting efficient and nondominated sets as well as the number of efficient integer assignments.
https://doi.org/10.1007/s00186-023-00826-z
Limit sets in global multiobjective optimization. - In: Optimization, ISSN 1029-4945, Bd. 73 (2024), 1, S. 1-27
Inspired by the recently introduced branch-and-bound method for continuous multiobjective optimization problems from G. Eichfelder, P. Kirst, L. Meng, O. Stein [A general branch-and-bound framework for continuous global multiobjective optimization. J Glob Optim. 2021;80:195-227], we study for a general class of branch-and-bound methods in which sense the generated terminal enclosure and the terminal provisional nondominated set approximate the nondominated set when the termination accuracy is driven to zero. Our convergence analysis of the enclosures relies on constructions from the above paper, but is self-contained and also covers the mixed-integer case. The analysis for the provisional nondominated set is based on general convergence properties of the epsilon-nondominated set, and hence it is also applicable to other algorithms which generate such points. Furthermore, we discuss post-processing steps for the terminal enclosure and provide numerical illustrations for the cases of two and three objective functions.
https://doi.org/10.1080/02331934.2022.2092479
Reprojection methods for Koopman-based modelling and prediction. - In: CDC 2023 Singapore, (2023), S. 315-321
Extended Dynamic Mode Decomposition (eDMD) is a powerful tool to generate data-driven surrogate models for the prediction and control of nonlinear dynamical systems in the Koopman framework. In eDMD a compression of the lifted system dynamics on the space spanned by finitely many observables is computed, in which the original space is embedded as a low-dimensional manifold. While this manifold is invariant for the infinite-dimensional Koopman operator, this invariance is typically not preserved for its eDMD-based approximation. Hence, an additional (re-)projection step is often tacitly incorporated to improve the prediction capability. We propose a novel framework for consistent reprojectors respecting the underlying manifold structure. Further, we present a new geometric reprojector based on maximum-likelihood arguments, which significantly enhances the approximation accuracy and preserves known finite-data error bounds.
https://doi.org/10.1109/CDC49753.2023.10383796
Linear versus quadratic detrending in analyzing simultaneous changes in DC-EEG and transcutaneous pCO2. - In: 2023 45th Annual International Conference of the IEEE Engineering in Medicine & Biology Conference (EMBC), (2023), insges. 4 S.
Physiological direct current (DC) potential shifts in electroencephalography (EEG) can be masked by artifacts such as slow electrode drifts. To reduce the influence of these artifacts, linear detrending has been proposed as a pre-processing step. We considered quadratic detrending, which has hardly been addressed for ultralow frequency components in EEG. We compared the performance of linear and quadratic detrending in simultaneously acquired DC-EEG and transcutaneous partial pressure of carbon dioxide during two activation methods: hyperventilation (HV) and apnea (AP). Quadratic detrending performed significantly better than linear detrending in HV, while for AP, our analysis was inconclusive with no statistical significance. We conclude that quadratic detrending should be considered for DC-EEG preprocessing.
https://doi.org/10.1109/EMBC40787.2023.10340855
Funnel control of linear systems with arbitrary relative degree under output measurement losses. - In: IMA journal of mathematical control and information, ISSN 1471-6887, Bd. 40 (2023), 4, S. 691-713
We consider tracking control of linear minimum phase systems with known arbitrary relative degree which are subject to possible output measurement losses. We provide a control law which guarantees the evolution of the tracking error within a (shifted) prescribed performance funnel whenever the output signal is available. The result requires a maximal duration of measurement losses and a minimal time of measurement availability, which both strongly depend on the internal dynamics of the system, and are derived explicitly. The controller is illustrated by a simulation of a mass-on-car system.
https://doi.org/10.1093/imamci/dnad029
Safe data-driven reference tracking with prescribed performance. - In: 2023 27th International Conference on System Theory, Control and Computing (ICSTCC), (2023), S. 454-460
ISBN 979-8-3503-3798-3
We study output reference tracking for unknown continuous-time systems with arbitrary relative degree. The control objective is to keep the tracking error within predefined time-varying bounds while measurement data is only available at discrete sampling times. To achieve the control objective, we propose a two-component controller. One part is a recently developed sampled-data zero-order hold controller, which achieves reference tracking within prescribed error bounds. To further improve the control signal, we explore the system dynamics via input-output data, and include as the second component a data-driven MPC scheme based on Willems et al.’s fundamental lemma. This combination yields significantly improved input signals as illustrated by a numerical example.
https://doi.org/10.1109/ICSTCC59206.2023.10308521