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MCA
Special Issues
Set Oriented Numerics 2022
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Set Oriented Numerics 2022

A special issue of Mathematical and Computational Applications (ISSN 2297-8747). This special issue belongs to the section "Engineering".

Deadline for manuscript submissions: closed (30 November 2022) | Viewed by 6180

Special Issue Editors

Prof. Dr. Oliver Junge

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Guest Editor
Zentrum Mathematik, Technische Universität München, D-85748 Garching, Germany
Interests: dynamical systems; numerical analysis; systems theory
Prof. Dr. Kathrin Padberg-Gehle

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Guest Editor
Institut für Mathematik und ihre Didaktik, Leuphana Universität Lüneburg, D-21335 Lüneburg, Germany
Interests: dynamical systems; fluid dynamics; set-oriented numerical methods; machine learning
Dr. Sebastian Peitz

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Guest Editor
Data Science for Engineering, Department of Computer Science, Paderborn University, Warburger Str. 100, 33098 Paderborn, Germany
Interests: multiobjective optimization; optimal control; fluid dynamics and flow control; dynamical systems; data-driven modeling; model order reduction
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Oliver Schütze

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Guest Editor
Depto de Computacion, CINVESTAV, Mexico City 07360, Mexico
Interests: multi-objective optimization; evolutionary computation (genetic algorithms and evolution strategies); numerical analysis; engineering applications
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

During the last 25 years, computational techniques for dynamical systems and optimization problems have been developed which allow for the efficient and rigorous approximation of solutions of arbitrary shape and topology. They already have found manifold applications in, e.g., molecular dynamics, engineering, climate research and biology. This issue collects contributions the workshop on Set Oriented Numerics held at the University of Paderborn in 2022 (https://sites.google.com/view/son-2020/home), containing recent advances and exploring connections to other techniques.

Prof. Dr. Oliver Junge
Prof. Dr. Kathrin Padberg-Gehle
Dr. Sebastian Peitz
Prof. Dr. Oliver Schütze
Guest Editors

The article processing charge (APC) is waived for well-prepared manuscripts submitted to this issue.

Published Papers (2 papers)

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Research

28 pages, 23480 KiB  
Article
Learning Motion Primitives Automata for Autonomous Driving Applications
by Matheus V. A. Pedrosa, Tristan Schneider and Kathrin Flaßkamp
Math. Comput. Appl. 2022, 27(4), 54; https://doi.org/10.3390/mca27040054 - 21 Jun 2022
Cited by 5 | Viewed by 2412
Abstract
Motion planning methods often rely on libraries of primitives. The selection of primitives is then crucial for assuring feasible solutions and good performance within the motion planner. In the literature, the library is usually designed by either learning from demonstration, relying entirely on [...] Read more.
Motion planning methods often rely on libraries of primitives. The selection of primitives is then crucial for assuring feasible solutions and good performance within the motion planner. In the literature, the library is usually designed by either learning from demonstration, relying entirely on data, or by model-based approaches, with the advantage of exploiting the dynamical system’s property, e.g., symmetries. In this work, we propose a method combining data with a dynamical model to optimally select primitives. The library is designed based on primitives with highest occurrences within the data set, while Lie group symmetries from a model are analysed in the available data to allow for structure-exploiting primitives. We illustrate our technique in an autonomous driving application. Primitives are identified based on data from human driving, with the freedom to build libraries of different sizes as a parameter of choice. We also compare the extracted library with a custom selection of primitives regarding the performance of obtained solutions for a street layout based on a real-world scenario. Full article
(This article belongs to the Special Issue Set Oriented Numerics 2022)
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23 pages, 746 KiB  
Article
ROM-Based Inexact Subdivision Methods for PDE-Constrained Multiobjective Optimization
by Stefan Banholzer, Bennet Gebken, Lena Reichle and Stefan Volkwein
Math. Comput. Appl. 2021, 26(2), 32; https://doi.org/10.3390/mca26020032 - 15 Apr 2021
Cited by 1 | Viewed by 1988
Abstract
The goal in multiobjective optimization is to determine the so-called Pareto set. Our optimization problem is governed by a parameter-dependent semi-linear elliptic partial differential equation (PDE). To solve it, we use a gradient-based set-oriented numerical method. The numerical solution of the PDE by [...] Read more.
The goal in multiobjective optimization is to determine the so-called Pareto set. Our optimization problem is governed by a parameter-dependent semi-linear elliptic partial differential equation (PDE). To solve it, we use a gradient-based set-oriented numerical method. The numerical solution of the PDE by standard discretization methods usually leads to high computational effort. To overcome this difficulty, reduced-order modeling (ROM) is developed utilizing the reduced basis method. These model simplifications cause inexactness in the gradients. For that reason, an additional descent condition is proposed. Applying a modified subdivision algorithm, numerical experiments illustrate the efficiency of our solution approach. Full article
(This article belongs to the Special Issue Set Oriented Numerics 2022)
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