[{"publication_identifier":{"isbn":["978-3-030-79392-0"]},"page":"43-76","citation":{"ieee":"S. Banholzer, B. Gebken, M. Dellnitz, S. Peitz, and S. Volkwein, “ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical Continuation,” in <i>Non-Smooth and Complementarity-Based Distributed Parameter Systems</i>, H. Michael, H. Roland, K. Christian, U. Michael, and U. Stefan, Eds. Cham: Springer, 2022, pp. 43–76.","chicago":"Banholzer, Stefan, Bennet Gebken, Michael Dellnitz, Sebastian Peitz, and Stefan Volkwein. “ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical Continuation.” In <i>Non-Smooth and Complementarity-Based Distributed Parameter Systems</i>, edited by Hintermüller Michael, Herzog Roland, Kanzow Christian, Ulbrich Michael, and Ulbrich Stefan, 43–76. Cham: Springer, 2022. <a href=\"https://doi.org/10.1007/978-3-030-79393-7_3\">https://doi.org/10.1007/978-3-030-79393-7_3</a>.","ama":"Banholzer S, Gebken B, Dellnitz M, Peitz S, Volkwein S. ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical Continuation. In: Michael H, Roland H, Christian K, Michael U, Stefan U, eds. <i>Non-Smooth and Complementarity-Based Distributed Parameter Systems</i>. Springer; 2022:43-76. doi:<a href=\"https://doi.org/10.1007/978-3-030-79393-7_3\">10.1007/978-3-030-79393-7_3</a>","mla":"Banholzer, Stefan, et al. “ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical Continuation.” <i>Non-Smooth and Complementarity-Based Distributed Parameter Systems</i>, edited by Hintermüller Michael et al., Springer, 2022, pp. 43–76, doi:<a href=\"https://doi.org/10.1007/978-3-030-79393-7_3\">10.1007/978-3-030-79393-7_3</a>.","short":"S. Banholzer, B. Gebken, M. Dellnitz, S. Peitz, S. Volkwein, in: H. Michael, H. Roland, K. Christian, U. Michael, U. Stefan (Eds.), Non-Smooth and Complementarity-Based Distributed Parameter Systems, Springer, Cham, 2022, pp. 43–76.","bibtex":"@inbook{Banholzer_Gebken_Dellnitz_Peitz_Volkwein_2022, place={Cham}, title={ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical Continuation}, DOI={<a href=\"https://doi.org/10.1007/978-3-030-79393-7_3\">10.1007/978-3-030-79393-7_3</a>}, booktitle={Non-Smooth and Complementarity-Based Distributed Parameter Systems}, publisher={Springer}, author={Banholzer, Stefan and Gebken, Bennet and Dellnitz, Michael and Peitz, Sebastian and Volkwein, Stefan}, editor={Michael, Hintermüller and Roland, Herzog and Christian, Kanzow and Michael, Ulbrich and Stefan, Ulbrich}, year={2022}, pages={43–76} }","apa":"Banholzer, S., Gebken, B., Dellnitz, M., Peitz, S., &#38; Volkwein, S. (2022). ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical Continuation. In H. Michael, H. Roland, K. Christian, U. Michael, &#38; U. Stefan (Eds.), <i>Non-Smooth and Complementarity-Based Distributed Parameter Systems</i> (pp. 43–76). Springer. <a href=\"https://doi.org/10.1007/978-3-030-79393-7_3\">https://doi.org/10.1007/978-3-030-79393-7_3</a>"},"place":"Cham","year":"2022","author":[{"last_name":"Banholzer","full_name":"Banholzer, Stefan","first_name":"Stefan"},{"id":"32643","full_name":"Gebken, Bennet","last_name":"Gebken","first_name":"Bennet"},{"last_name":"Dellnitz","full_name":"Dellnitz, Michael","first_name":"Michael"},{"first_name":"Sebastian","last_name":"Peitz","orcid":"https://orcid.org/0000-0002-3389-793X","id":"47427","full_name":"Peitz, Sebastian"},{"first_name":"Stefan","full_name":"Volkwein, Stefan","last_name":"Volkwein"}],"date_created":"2020-03-13T12:45:31Z","publisher":"Springer","date_updated":"2022-03-14T13:04:51Z","oa":"1","doi":"10.1007/978-3-030-79393-7_3","main_file_link":[{"url":"https://arxiv.org/pdf/1906.09075.pdf","open_access":"1"}],"title":"ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical Continuation","publication":"Non-Smooth and Complementarity-Based Distributed Parameter Systems","type":"book_chapter","status":"public","abstract":[{"lang":"eng","text":"Multiobjective optimization plays an increasingly important role in modern\r\napplications, where several objectives are often of equal importance. The task\r\nin multiobjective optimization and multiobjective optimal control is therefore\r\nto compute the set of optimal compromises (the Pareto set) between the\r\nconflicting objectives. Since the Pareto set generally consists of an infinite\r\nnumber of solutions, the computational effort can quickly become challenging\r\nwhich is particularly problematic when the objectives are costly to evaluate as\r\nis the case for models governed by partial differential equations (PDEs). To\r\ndecrease the numerical effort to an affordable amount, surrogate models can be\r\nused to replace the expensive PDE evaluations. Existing multiobjective\r\noptimization methods using model reduction are limited either to low parameter\r\ndimensions or to few (ideally two) objectives. In this article, we present a\r\ncombination of the reduced basis model reduction method with a continuation\r\napproach using inexact gradients. The resulting approach can handle an\r\narbitrary number of objectives while yielding a significant reduction in\r\ncomputing time."}],"editor":[{"first_name":"Hintermüller","full_name":"Michael, Hintermüller","last_name":"Michael"},{"first_name":"Herzog","full_name":"Roland, Herzog","last_name":"Roland"},{"last_name":"Christian","full_name":"Christian, Kanzow","first_name":"Kanzow"},{"first_name":"Ulbrich","last_name":"Michael","full_name":"Michael, Ulbrich"},{"first_name":"Ulbrich","full_name":"Stefan, Ulbrich","last_name":"Stefan"}],"department":[{"_id":"101"},{"_id":"655"}],"user_id":"47427","_id":"16296","language":[{"iso":"eng"}]},{"year":"2022","title":"Efficient Virtual Design and Testing of Autonomous Vehicles","date_created":"2022-03-14T07:32:41Z","publisher":"Springer International Publishing","abstract":[{"lang":"eng","text":"With the ever increasing capabilities of sensors and controllers, autonomous driving is quickly becoming a reality. This disruptive change in the automotive industry poses major challenges for manufacturers as well as suppliers as entirely new design and testing strategies have to be developed to remain competitive. Most importantly, the complexity of autonomously driving vehicles in a complex, uncertain, and safety-critical environment requires new testing procedures to cover the almost infinite range of potential scenarios."}],"publication":"German Success Stories in Industrial Mathematics","language":[{"iso":"eng"}],"intvolume":"        35","citation":{"apa":"Peitz, S., Dellnitz, M., &#38; Bannenberg, S. (2022). Efficient Virtual Design and Testing of Autonomous Vehicles. In H. G. Bock, K.-H. Küfer, P. Maas, A. Milde, &#38; V. Schulz (Eds.), <i>German Success Stories in Industrial Mathematics</i> (Vol. 35). Springer International Publishing. <a href=\"https://doi.org/10.1007/978-3-030-81455-7_23\">https://doi.org/10.1007/978-3-030-81455-7_23</a>","short":"S. Peitz, M. Dellnitz, S. Bannenberg, in: H.G. Bock, K.-H. Küfer, P. Maas, A. Milde, V. Schulz (Eds.), German Success Stories in Industrial Mathematics, Springer International Publishing, Cham, 2022.","mla":"Peitz, Sebastian, et al. “Efficient Virtual Design and Testing of Autonomous Vehicles.” <i>German Success Stories in Industrial Mathematics</i>, edited by H. G. Bock et al., vol. 35, Springer International Publishing, 2022, doi:<a href=\"https://doi.org/10.1007/978-3-030-81455-7_23\">10.1007/978-3-030-81455-7_23</a>.","bibtex":"@inbook{Peitz_Dellnitz_Bannenberg_2022, place={Cham}, series={Mathematics in Industry}, title={Efficient Virtual Design and Testing of Autonomous Vehicles}, volume={35}, DOI={<a href=\"https://doi.org/10.1007/978-3-030-81455-7_23\">10.1007/978-3-030-81455-7_23</a>}, booktitle={German Success Stories in Industrial Mathematics}, publisher={Springer International Publishing}, author={Peitz, Sebastian and Dellnitz, Michael and Bannenberg, Sebastian}, editor={Bock, H. G. and Küfer, K.-H. and Maas, P. and Milde, A. and Schulz, V.}, year={2022}, collection={Mathematics in Industry} }","ieee":"S. Peitz, M. Dellnitz, and S. Bannenberg, “Efficient Virtual Design and Testing of Autonomous Vehicles,” in <i>German Success Stories in Industrial Mathematics</i>, vol. 35, H. G. Bock, K.-H. Küfer, P. Maas, A. Milde, and V. Schulz, Eds. Cham: Springer International Publishing, 2022.","chicago":"Peitz, Sebastian, Michael Dellnitz, and Sebastian Bannenberg. “Efficient Virtual Design and Testing of Autonomous Vehicles.” In <i>German Success Stories in Industrial Mathematics</i>, edited by H. G. Bock, K.-H. Küfer, P. Maas, A. Milde, and V. Schulz, Vol. 35. Mathematics in Industry. Cham: Springer International Publishing, 2022. <a href=\"https://doi.org/10.1007/978-3-030-81455-7_23\">https://doi.org/10.1007/978-3-030-81455-7_23</a>.","ama":"Peitz S, Dellnitz M, Bannenberg S. Efficient Virtual Design and Testing of Autonomous Vehicles. In: Bock HG, Küfer K-H, Maas P, Milde A, Schulz V, eds. <i>German Success Stories in Industrial Mathematics</i>. Vol 35. Mathematics in Industry. Springer International Publishing; 2022. doi:<a href=\"https://doi.org/10.1007/978-3-030-81455-7_23\">10.1007/978-3-030-81455-7_23</a>"},"place":"Cham","publication_identifier":{"isbn":["9783030814540","9783030814557"],"issn":["1612-3956","2198-3283"]},"publication_status":"published","doi":"10.1007/978-3-030-81455-7_23","volume":35,"author":[{"first_name":"Sebastian","orcid":"0000-0002-3389-793X","last_name":"Peitz","full_name":"Peitz, Sebastian","id":"47427"},{"full_name":"Dellnitz, Michael","last_name":"Dellnitz","first_name":"Michael"},{"full_name":"Bannenberg, Sebastian","last_name":"Bannenberg","first_name":"Sebastian"}],"date_updated":"2022-03-14T07:42:01Z","status":"public","editor":[{"full_name":"Bock, H. G.","last_name":"Bock","first_name":"H. G."},{"last_name":"Küfer","full_name":"Küfer, K.-H.","first_name":"K.-H."},{"full_name":"Maas, P.","last_name":"Maas","first_name":"P."},{"first_name":"A.","last_name":"Milde","full_name":"Milde, A."},{"first_name":"V.","last_name":"Schulz","full_name":"Schulz, V."}],"type":"book_chapter","department":[{"_id":"101"},{"_id":"655"}],"user_id":"47427","series_title":"Mathematics in Industry","_id":"30294"},{"publication_status":"published","citation":{"ieee":"S. Klus, F. Nüske, and S. Peitz, “Koopman analysis of quantum systems,” <i>Journal of Physics A: Mathematical and Theoretical</i>, vol. 55, no. 31, p. 314002, 2022, doi: <a href=\"https://doi.org/10.1088/1751-8121/ac7d22\">10.1088/1751-8121/ac7d22</a>.","chicago":"Klus, Stefan, Feliks Nüske, and Sebastian Peitz. “Koopman Analysis of Quantum Systems.” <i>Journal of Physics A: Mathematical and Theoretical</i> 55, no. 31 (2022): 314002. <a href=\"https://doi.org/10.1088/1751-8121/ac7d22\">https://doi.org/10.1088/1751-8121/ac7d22</a>.","ama":"Klus S, Nüske F, Peitz S. Koopman analysis of quantum systems. <i>Journal of Physics A: Mathematical and Theoretical</i>. 2022;55(31):314002. doi:<a href=\"https://doi.org/10.1088/1751-8121/ac7d22\">10.1088/1751-8121/ac7d22</a>","apa":"Klus, S., Nüske, F., &#38; Peitz, S. (2022). Koopman analysis of quantum systems. <i>Journal of Physics A: Mathematical and Theoretical</i>, <i>55</i>(31), 314002. <a href=\"https://doi.org/10.1088/1751-8121/ac7d22\">https://doi.org/10.1088/1751-8121/ac7d22</a>","mla":"Klus, Stefan, et al. “Koopman Analysis of Quantum Systems.” <i>Journal of Physics A: Mathematical and Theoretical</i>, vol. 55, no. 31, IOP Publishing Ltd., 2022, p. 314002, doi:<a href=\"https://doi.org/10.1088/1751-8121/ac7d22\">10.1088/1751-8121/ac7d22</a>.","bibtex":"@article{Klus_Nüske_Peitz_2022, title={Koopman analysis of quantum systems}, volume={55}, DOI={<a href=\"https://doi.org/10.1088/1751-8121/ac7d22\">10.1088/1751-8121/ac7d22</a>}, number={31}, journal={Journal of Physics A: Mathematical and Theoretical}, publisher={IOP Publishing Ltd.}, author={Klus, Stefan and Nüske, Feliks and Peitz, Sebastian}, year={2022}, pages={314002} }","short":"S. Klus, F. Nüske, S. Peitz, Journal of Physics A: Mathematical and Theoretical 55 (2022) 314002."},"page":"314002","intvolume":"        55","author":[{"last_name":"Klus","full_name":"Klus, Stefan","first_name":"Stefan"},{"first_name":"Feliks","full_name":"Nüske, Feliks","id":"81513","orcid":"0000-0003-2444-7889","last_name":"Nüske"},{"first_name":"Sebastian","full_name":"Peitz, Sebastian","id":"47427","orcid":"0000-0002-3389-793X","last_name":"Peitz"}],"volume":55,"oa":"1","date_updated":"2022-07-18T14:26:41Z","main_file_link":[{"url":"https://iopscience.iop.org/article/10.1088/1751-8121/ac7d22/pdf","open_access":"1"}],"doi":"10.1088/1751-8121/ac7d22","type":"journal_article","status":"public","user_id":"47427","department":[{"_id":"655"},{"_id":"101"}],"_id":"29673","issue":"31","year":"2022","date_created":"2022-01-31T09:49:40Z","publisher":"IOP Publishing Ltd.","title":"Koopman analysis of quantum systems","publication":"Journal of Physics A: Mathematical and Theoretical","abstract":[{"text":"Koopman operator theory has been successfully applied to problems from various research areas such as fluid dynamics, molecular dynamics, climate science, engineering, and biology. Applications include detecting metastable or coherent sets, coarse-graining, system identification, and control. There is an intricate connection between dynamical systems driven by stochastic differential equations and quantum mechanics. In this paper, we compare the ground-state transformation and Nelson's stochastic mechanics and demonstrate how data-driven methods developed for the approximation of the Koopman operator can be used to analyze quantum physics problems. Moreover, we exploit the relationship between Schrödinger operators and stochastic control problems to show that modern data-driven methods for stochastic control can be used to solve the stationary or imaginary-time Schrödinger equation. Our findings open up a new avenue towards solving Schrödinger's equation using recently developed tools from data science.","lang":"eng"}],"external_id":{"arxiv":["2201.12062"]},"language":[{"iso":"eng"}]},{"date_created":"2022-12-20T15:25:17Z","author":[{"id":"32643","full_name":"Gebken, Bennet","last_name":"Gebken","first_name":"Bennet"}],"oa":"1","date_updated":"2022-12-20T15:28:54Z","main_file_link":[{"open_access":"1","url":"https://arxiv.org/pdf/2210.04579"}],"title":"Using second-order information in gradient sampling methods for  nonsmooth optimization","citation":{"apa":"Gebken, B. (2022). Using second-order information in gradient sampling methods for  nonsmooth optimization. In <i>arXiv:2210.04579</i>.","short":"B. Gebken, ArXiv:2210.04579 (2022).","mla":"Gebken, Bennet. “Using Second-Order Information in Gradient Sampling Methods for  Nonsmooth Optimization.” <i>ArXiv:2210.04579</i>, 2022.","bibtex":"@article{Gebken_2022, title={Using second-order information in gradient sampling methods for  nonsmooth optimization}, journal={arXiv:2210.04579}, author={Gebken, Bennet}, year={2022} }","ama":"Gebken B. Using second-order information in gradient sampling methods for  nonsmooth optimization. <i>arXiv:221004579</i>. Published online 2022.","chicago":"Gebken, Bennet. “Using Second-Order Information in Gradient Sampling Methods for  Nonsmooth Optimization.” <i>ArXiv:2210.04579</i>, 2022.","ieee":"B. Gebken, “Using second-order information in gradient sampling methods for  nonsmooth optimization,” <i>arXiv:2210.04579</i>. 2022."},"year":"2022","department":[{"_id":"101"}],"user_id":"32643","_id":"34618","external_id":{"arxiv":["2210.04579"]},"language":[{"iso":"eng"}],"publication":"arXiv:2210.04579","type":"preprint","status":"public","abstract":[{"text":"In this article, we show how second-order derivative information can be\r\nincorporated into gradient sampling methods for nonsmooth optimization. The\r\nsecond-order information we consider is essentially the set of coefficients of\r\nall second-order Taylor expansions of the objective in a closed ball around a\r\ngiven point. Based on this concept, we define a model of the objective as the\r\nmaximum of these Taylor expansions. Iteratively minimizing this model\r\n(constrained to the closed ball) results in a simple descent method, for which\r\nwe prove convergence to minimal points in case the objective is convex. To\r\nobtain an implementable method, we construct an approximation scheme for the\r\nsecond-order information based on sampling objective values, gradients and\r\nHessian matrices at finitely many points. Using a set of test problems, we\r\ncompare the resulting method to five other available solvers. Considering the\r\nnumber of function evaluations, the results suggest that the method we propose\r\nis superior to the standard gradient sampling method, and competitive compared\r\nto other methods.","lang":"eng"}]},{"title":"Computation and analysis of Pareto critical sets in smooth and nonsmooth multiobjective optimization","main_file_link":[{"open_access":"1","url":"https://digital.ub.uni-paderborn.de/hs/download/pdf/6531779"}],"doi":"10.17619/UNIPB/1-1327","oa":"1","date_updated":"2022-06-01T07:13:09Z","author":[{"id":"32643","full_name":"Gebken, Bennet","last_name":"Gebken","first_name":"Bennet"}],"supervisor":[{"first_name":"Michael","full_name":"Dellnitz, Michael","last_name":"Dellnitz"}],"date_created":"2022-06-01T06:48:08Z","year":"2022","citation":{"mla":"Gebken, Bennet. <i>Computation and Analysis of Pareto Critical Sets in Smooth and Nonsmooth Multiobjective Optimization</i>. 2022, doi:<a href=\"https://doi.org/10.17619/UNIPB/1-1327\">10.17619/UNIPB/1-1327</a>.","bibtex":"@book{Gebken_2022, title={Computation and analysis of Pareto critical sets in smooth and nonsmooth multiobjective optimization}, DOI={<a href=\"https://doi.org/10.17619/UNIPB/1-1327\">10.17619/UNIPB/1-1327</a>}, author={Gebken, Bennet}, year={2022} }","short":"B. Gebken, Computation and Analysis of Pareto Critical Sets in Smooth and Nonsmooth Multiobjective Optimization, 2022.","apa":"Gebken, B. (2022). <i>Computation and analysis of Pareto critical sets in smooth and nonsmooth multiobjective optimization</i>. <a href=\"https://doi.org/10.17619/UNIPB/1-1327\">https://doi.org/10.17619/UNIPB/1-1327</a>","chicago":"Gebken, Bennet. <i>Computation and Analysis of Pareto Critical Sets in Smooth and Nonsmooth Multiobjective Optimization</i>, 2022. <a href=\"https://doi.org/10.17619/UNIPB/1-1327\">https://doi.org/10.17619/UNIPB/1-1327</a>.","ieee":"B. Gebken, <i>Computation and analysis of Pareto critical sets in smooth and nonsmooth multiobjective optimization</i>. 2022.","ama":"Gebken B. <i>Computation and Analysis of Pareto Critical Sets in Smooth and Nonsmooth Multiobjective Optimization</i>.; 2022. doi:<a href=\"https://doi.org/10.17619/UNIPB/1-1327\">10.17619/UNIPB/1-1327</a>"},"language":[{"iso":"eng"}],"_id":"31556","user_id":"32643","department":[{"_id":"101"}],"abstract":[{"lang":"ger","text":"Mehrzieloptimierung behandelt Probleme, bei denen mehrere skalare Zielfunktionen simultan optimiert werden sollen. Ein Punkt ist in diesem Fall optimal, wenn es keinen anderen Punkt gibt, der mindestens genauso gut ist in allen Zielfunktionen und besser in mindestens einer Zielfunktion. Ein notwendiges Optimalitätskriterium lässt sich über Ableitungsinformationen erster Ordnung der Zielfunktionen herleiten. Die Menge der Punkte, die dieses notwendige Kriterium erfüllen, wird als Pareto-kritische Menge bezeichnet. Diese Arbeit enthält neue Resultate über Pareto-kritische Mengen für glatte und nicht-glatte Mehrzieloptimierungsprobleme, sowohl was deren Berechnung betrifft als auch deren Struktur. Im glatten Fall erfolgt die Berechnung über ein Fortsetzungsverfahren, im nichtglatten Fall über ein Abstiegsverfahren. Anschließend wird die Struktur des Randes der Pareto-kritischen Menge analysiert, welcher aus Pareto-kritischen Mengen kleinerer Subprobleme besteht. Schlussendlich werden inverse Probleme betrachtet, bei denen zu einer gegebenen Datenmenge ein Zielfunktionsvektor gefunden werden soll, für den die Datenpunkte kritisch sind."},{"text":"Multiobjective optimization is concerned with the simultaneous optimization of multiple scalar-valued functions. In this case, a point is optimal if there is no other point that is at least as good in all objectives and better in at least one objective. A necessary condition for optimality can be derived based on first-order information of the objectives. The set of points that satisfy this necessary condition is called the Pareto critical set. This thesis presents new results about Pareto critical sets for smooth and nonsmooth multiobjective optimization problems, both in terms of their efficient computation and structural properties. In the smooth case they are computed via a continuation method and in the nonsmooth case via a descent method. Afterwards, the structure of the boundary of the Pareto critical set is analyzed, which consists of Pareto critical sets of smaller subproblems. Finally, inverse problems are considered, where a data set is given and an objective vector is sought for which the data points are critical.","lang":"eng"}],"status":"public","type":"dissertation"},{"year":"2022","citation":{"apa":"Berkemeier, M. B., &#38; Peitz, S. (2022). Multi-Objective Trust-Region Filter Method for Nonlinear Constraints using Inexact Gradients. In <i>arXiv:2208.12094</i>.","short":"M.B. Berkemeier, S. Peitz, ArXiv:2208.12094 (2022).","bibtex":"@article{Berkemeier_Peitz_2022, title={Multi-Objective Trust-Region Filter Method for Nonlinear Constraints using Inexact Gradients}, journal={arXiv:2208.12094}, author={Berkemeier, Manuel Bastian and Peitz, Sebastian}, year={2022} }","mla":"Berkemeier, Manuel Bastian, and Sebastian Peitz. “Multi-Objective Trust-Region Filter Method for Nonlinear Constraints Using Inexact Gradients.” <i>ArXiv:2208.12094</i>, 2022.","ama":"Berkemeier MB, Peitz S. Multi-Objective Trust-Region Filter Method for Nonlinear Constraints using Inexact Gradients. <i>arXiv:220812094</i>. Published online 2022.","ieee":"M. B. Berkemeier and S. Peitz, “Multi-Objective Trust-Region Filter Method for Nonlinear Constraints using Inexact Gradients,” <i>arXiv:2208.12094</i>. 2022.","chicago":"Berkemeier, Manuel Bastian, and Sebastian Peitz. “Multi-Objective Trust-Region Filter Method for Nonlinear Constraints Using Inexact Gradients.” <i>ArXiv:2208.12094</i>, 2022."},"title":"Multi-Objective Trust-Region Filter Method for Nonlinear Constraints using Inexact Gradients","main_file_link":[{"url":"https://arxiv.org/pdf/2208.12094","open_access":"1"}],"date_updated":"2022-08-26T06:12:10Z","oa":"1","date_created":"2022-08-26T06:08:06Z","author":[{"first_name":"Manuel Bastian","last_name":"Berkemeier","full_name":"Berkemeier, Manuel Bastian","id":"51701"},{"first_name":"Sebastian","id":"47427","full_name":"Peitz, Sebastian","orcid":"0000-0002-3389-793X","last_name":"Peitz"}],"abstract":[{"text":"In this article, we build on previous work to present an optimization algorithm for nonlinearly constrained multi-objective optimization problems. The algorithm combines a surrogate-assisted derivative-free trust-region approach with the filter method known from single-objective optimization. Instead of the true objective and constraint functions, so-called fully linear models are employed and we show how to deal with the gradient inexactness in the composite step setting, adapted from single-objective optimization as well. Under standard assumptions, we prove convergence of a subset of iterates to a quasi-stationary point and if constraint qualifications hold, then the limit point is also a KKT-point of the multi-objective problem.","lang":"eng"}],"status":"public","type":"preprint","publication":"arXiv:2208.12094","language":[{"iso":"eng"}],"_id":"33150","external_id":{"arxiv":["2208.12094"]},"user_id":"47427","department":[{"_id":"101"},{"_id":"655"}]},{"type":"journal_article","status":"public","_id":"20731","user_id":"47427","department":[{"_id":"101"},{"_id":"530"},{"_id":"655"}],"article_type":"original","file_date_updated":"2021-09-25T11:59:15Z","publication_status":"epub_ahead","has_accepted_license":"1","citation":{"apa":"Bieker, K., Gebken, B., &#38; Peitz, S. (2022). On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation. <i>IEEE Transactions on Pattern Analysis and Machine Intelligence</i>, <i>44</i>(11), 7797–7808. <a href=\"https://doi.org/10.1109/TPAMI.2021.3114962\">https://doi.org/10.1109/TPAMI.2021.3114962</a>","mla":"Bieker, Katharina, et al. “On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation.” <i>IEEE Transactions on Pattern Analysis and Machine Intelligence</i>, vol. 44, no. 11, IEEE, 2022, pp. 7797–808, doi:<a href=\"https://doi.org/10.1109/TPAMI.2021.3114962\">10.1109/TPAMI.2021.3114962</a>.","short":"K. Bieker, B. Gebken, S. Peitz, IEEE Transactions on Pattern Analysis and Machine Intelligence 44 (2022) 7797–7808.","bibtex":"@article{Bieker_Gebken_Peitz_2022, title={On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation}, volume={44}, DOI={<a href=\"https://doi.org/10.1109/TPAMI.2021.3114962\">10.1109/TPAMI.2021.3114962</a>}, number={11}, journal={IEEE Transactions on Pattern Analysis and Machine Intelligence}, publisher={IEEE}, author={Bieker, Katharina and Gebken, Bennet and Peitz, Sebastian}, year={2022}, pages={7797–7808} }","ieee":"K. Bieker, B. Gebken, and S. Peitz, “On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation,” <i>IEEE Transactions on Pattern Analysis and Machine Intelligence</i>, vol. 44, no. 11, pp. 7797–7808, 2022, doi: <a href=\"https://doi.org/10.1109/TPAMI.2021.3114962\">10.1109/TPAMI.2021.3114962</a>.","chicago":"Bieker, Katharina, Bennet Gebken, and Sebastian Peitz. “On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation.” <i>IEEE Transactions on Pattern Analysis and Machine Intelligence</i> 44, no. 11 (2022): 7797–7808. <a href=\"https://doi.org/10.1109/TPAMI.2021.3114962\">https://doi.org/10.1109/TPAMI.2021.3114962</a>.","ama":"Bieker K, Gebken B, Peitz S. On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation. <i>IEEE Transactions on Pattern Analysis and Machine Intelligence</i>. 2022;44(11):7797-7808. doi:<a href=\"https://doi.org/10.1109/TPAMI.2021.3114962\">10.1109/TPAMI.2021.3114962</a>"},"intvolume":"        44","page":"7797-7808","oa":"1","date_updated":"2022-10-21T12:27:16Z","author":[{"first_name":"Katharina","last_name":"Bieker","id":"32829","full_name":"Bieker, Katharina"},{"first_name":"Bennet","last_name":"Gebken","full_name":"Gebken, Bennet","id":"32643"},{"first_name":"Sebastian","id":"47427","full_name":"Peitz, Sebastian","last_name":"Peitz","orcid":"0000-0002-3389-793X"}],"volume":44,"main_file_link":[{"url":"https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=9547772","open_access":"1"}],"doi":"10.1109/TPAMI.2021.3114962","publication":"IEEE Transactions on Pattern Analysis and Machine Intelligence","abstract":[{"text":"We present a novel algorithm that allows us to gain detailed insight into the effects of sparsity in linear and nonlinear optimization, which is of great importance in many scientific areas such as image and signal processing, medical imaging, compressed sensing, and machine learning (e.g., for the training of neural networks). Sparsity is an important feature to ensure robustness against noisy data, but also to find models that are interpretable and easy to analyze due to the small number of relevant terms. It is common practice to enforce sparsity by adding the ℓ1-norm as a weighted penalty term. In order to gain a better understanding and to allow for an informed model selection, we directly solve the corresponding multiobjective optimization problem (MOP) that arises when we minimize the main objective and the ℓ1-norm simultaneously. As this MOP is in general non-convex for nonlinear objectives, the weighting method will fail to provide all optimal compromises. To avoid this issue, we present a continuation method which is specifically tailored to MOPs with two objective functions one of which is the ℓ1-norm. Our method can be seen as a generalization of well-known homotopy methods for linear regression problems to the nonlinear case. Several numerical examples - including neural network training - demonstrate our theoretical findings and the additional insight that can be gained by this multiobjective approach.","lang":"eng"}],"file":[{"creator":"speitz","date_created":"2021-09-25T11:59:15Z","date_updated":"2021-09-25T11:59:15Z","file_name":"On_the_Treatment_of_Optimization_Problems_with_L1_Penalty_Terms_via_Multiobjective_Continuation.pdf","access_level":"closed","file_id":"25040","file_size":7990831,"content_type":"application/pdf","relation":"main_file","success":1}],"ddc":["510"],"language":[{"iso":"eng"}],"issue":"11","year":"2022","publisher":"IEEE","date_created":"2020-12-15T07:46:36Z","title":"On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation"},{"citation":{"ama":"Nüske F, Gelß P, Klus S, Clementi C. Tensor-based computation of metastable and coherent sets. <i>Physica D: Nonlinear Phenomena</i>. Published online 2021. doi:<a href=\"https://doi.org/10.1016/j.physd.2021.133018\">10.1016/j.physd.2021.133018</a>","chicago":"Nüske, Feliks, Patrick Gelß, Stefan Klus, and Cecilia Clementi. “Tensor-Based Computation of Metastable and Coherent Sets.” <i>Physica D: Nonlinear Phenomena</i>, 2021. <a href=\"https://doi.org/10.1016/j.physd.2021.133018\">https://doi.org/10.1016/j.physd.2021.133018</a>.","ieee":"F. Nüske, P. Gelß, S. Klus, and C. Clementi, “Tensor-based computation of metastable and coherent sets,” <i>Physica D: Nonlinear Phenomena</i>, Art. no. 133018, 2021, doi: <a href=\"https://doi.org/10.1016/j.physd.2021.133018\">10.1016/j.physd.2021.133018</a>.","short":"F. Nüske, P. Gelß, S. Klus, C. Clementi, Physica D: Nonlinear Phenomena (2021).","bibtex":"@article{Nüske_Gelß_Klus_Clementi_2021, title={Tensor-based computation of metastable and coherent sets}, DOI={<a href=\"https://doi.org/10.1016/j.physd.2021.133018\">10.1016/j.physd.2021.133018</a>}, number={133018}, journal={Physica D: Nonlinear Phenomena}, author={Nüske, Feliks and Gelß, Patrick and Klus, Stefan and Clementi, Cecilia}, year={2021} }","mla":"Nüske, Feliks, et al. “Tensor-Based Computation of Metastable and Coherent Sets.” <i>Physica D: Nonlinear Phenomena</i>, 133018, 2021, doi:<a href=\"https://doi.org/10.1016/j.physd.2021.133018\">10.1016/j.physd.2021.133018</a>.","apa":"Nüske, F., Gelß, P., Klus, S., &#38; Clementi, C. (2021). Tensor-based computation of metastable and coherent sets. <i>Physica D: Nonlinear Phenomena</i>, Article 133018. <a href=\"https://doi.org/10.1016/j.physd.2021.133018\">https://doi.org/10.1016/j.physd.2021.133018</a>"},"year":"2021","publication_identifier":{"issn":["0167-2789"]},"publication_status":"published","doi":"10.1016/j.physd.2021.133018","title":"Tensor-based computation of metastable and coherent sets","date_created":"2021-09-12T08:51:24Z","author":[{"first_name":"Feliks","orcid":"0000-0003-2444-7889","last_name":"Nüske","full_name":"Nüske, Feliks","id":"81513"},{"full_name":"Gelß, Patrick","last_name":"Gelß","first_name":"Patrick"},{"first_name":"Stefan","full_name":"Klus, Stefan","last_name":"Klus"},{"full_name":"Clementi, Cecilia","last_name":"Clementi","first_name":"Cecilia"}],"date_updated":"2022-01-06T06:56:08Z","status":"public","publication":"Physica D: Nonlinear Phenomena","type":"journal_article","language":[{"iso":"eng"}],"article_number":"133018","department":[{"_id":"101"}],"user_id":"81513","_id":"24169"},{"language":[{"iso":"eng"}],"article_number":"045016","user_id":"81513","department":[{"_id":"101"}],"_id":"24170","status":"public","type":"journal_article","publication":"Machine Learning: Science and Technology","doi":"10.1088/2632-2153/ac14ad","title":"Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry","date_created":"2021-09-12T08:52:57Z","author":[{"full_name":"Klus, Stefan","last_name":"Klus","first_name":"Stefan"},{"full_name":"Gelß, Patrick","last_name":"Gelß","first_name":"Patrick"},{"first_name":"Feliks","orcid":"0000-0003-2444-7889","last_name":"Nüske","full_name":"Nüske, Feliks","id":"81513"},{"last_name":"Noé","full_name":"Noé, Frank","first_name":"Frank"}],"date_updated":"2022-01-06T06:56:08Z","citation":{"short":"S. Klus, P. Gelß, F. Nüske, F. Noé, Machine Learning: Science and Technology (2021).","bibtex":"@article{Klus_Gelß_Nüske_Noé_2021, title={Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry}, DOI={<a href=\"https://doi.org/10.1088/2632-2153/ac14ad\">10.1088/2632-2153/ac14ad</a>}, number={045016}, journal={Machine Learning: Science and Technology}, author={Klus, Stefan and Gelß, Patrick and Nüske, Feliks and Noé, Frank}, year={2021} }","mla":"Klus, Stefan, et al. “Symmetric and Antisymmetric Kernels for Machine Learning Problems in Quantum Physics and Chemistry.” <i>Machine Learning: Science and Technology</i>, 045016, 2021, doi:<a href=\"https://doi.org/10.1088/2632-2153/ac14ad\">10.1088/2632-2153/ac14ad</a>.","apa":"Klus, S., Gelß, P., Nüske, F., &#38; Noé, F. (2021). Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry. <i>Machine Learning: Science and Technology</i>, Article 045016. <a href=\"https://doi.org/10.1088/2632-2153/ac14ad\">https://doi.org/10.1088/2632-2153/ac14ad</a>","ama":"Klus S, Gelß P, Nüske F, Noé F. Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry. <i>Machine Learning: Science and Technology</i>. Published online 2021. doi:<a href=\"https://doi.org/10.1088/2632-2153/ac14ad\">10.1088/2632-2153/ac14ad</a>","chicago":"Klus, Stefan, Patrick Gelß, Feliks Nüske, and Frank Noé. “Symmetric and Antisymmetric Kernels for Machine Learning Problems in Quantum Physics and Chemistry.” <i>Machine Learning: Science and Technology</i>, 2021. <a href=\"https://doi.org/10.1088/2632-2153/ac14ad\">https://doi.org/10.1088/2632-2153/ac14ad</a>.","ieee":"S. Klus, P. Gelß, F. Nüske, and F. Noé, “Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry,” <i>Machine Learning: Science and Technology</i>, Art. no. 045016, 2021, doi: <a href=\"https://doi.org/10.1088/2632-2153/ac14ad\">10.1088/2632-2153/ac14ad</a>."},"year":"2021","publication_status":"published","publication_identifier":{"issn":["2632-2153"]}},{"_id":"21195","user_id":"32643","department":[{"_id":"101"}],"language":[{"iso":"eng"}],"type":"journal_article","publication":"Cognitive Neurodynamics","status":"public","date_updated":"2022-01-06T06:54:49Z","date_created":"2021-02-08T13:16:07Z","author":[{"first_name":"Christian","full_name":"Goelz, Christian","last_name":"Goelz"},{"full_name":"Mora, Karin","last_name":"Mora","first_name":"Karin"},{"last_name":"Stroehlein","full_name":"Stroehlein, Julia Kristin","first_name":"Julia Kristin"},{"full_name":"Haase, Franziska Katharina","last_name":"Haase","first_name":"Franziska Katharina"},{"full_name":"Dellnitz, Michael","last_name":"Dellnitz","first_name":"Michael"},{"first_name":"Claus","full_name":"Reinsberger, Claus","last_name":"Reinsberger"},{"last_name":"Vieluf","full_name":"Vieluf, Solveig","first_name":"Solveig"}],"title":"Electrophysiological signatures of dedifferentiation differ between fit and less fit older adults","main_file_link":[{"url":"https://link.springer.com/content/pdf/10.1007/s11571-020-09656-9.pdf"}],"doi":"10.1007/s11571-020-09656-9","year":"2021","citation":{"ieee":"C. Goelz <i>et al.</i>, “Electrophysiological signatures of dedifferentiation differ between fit and less fit older adults,” <i>Cognitive Neurodynamics</i>, 2021.","chicago":"Goelz, Christian, Karin Mora, Julia Kristin Stroehlein, Franziska Katharina Haase, Michael Dellnitz, Claus Reinsberger, and Solveig Vieluf. “Electrophysiological Signatures of Dedifferentiation Differ between Fit and Less Fit Older Adults.” <i>Cognitive Neurodynamics</i>, 2021. <a href=\"https://doi.org/10.1007/s11571-020-09656-9\">https://doi.org/10.1007/s11571-020-09656-9</a>.","ama":"Goelz C, Mora K, Stroehlein JK, et al. Electrophysiological signatures of dedifferentiation differ between fit and less fit older adults. <i>Cognitive Neurodynamics</i>. 2021. doi:<a href=\"https://doi.org/10.1007/s11571-020-09656-9\">10.1007/s11571-020-09656-9</a>","mla":"Goelz, Christian, et al. “Electrophysiological Signatures of Dedifferentiation Differ between Fit and Less Fit Older Adults.” <i>Cognitive Neurodynamics</i>, 2021, doi:<a href=\"https://doi.org/10.1007/s11571-020-09656-9\">10.1007/s11571-020-09656-9</a>.","bibtex":"@article{Goelz_Mora_Stroehlein_Haase_Dellnitz_Reinsberger_Vieluf_2021, title={Electrophysiological signatures of dedifferentiation differ between fit and less fit older adults}, DOI={<a href=\"https://doi.org/10.1007/s11571-020-09656-9\">10.1007/s11571-020-09656-9</a>}, journal={Cognitive Neurodynamics}, author={Goelz, Christian and Mora, Karin and Stroehlein, Julia Kristin and Haase, Franziska Katharina and Dellnitz, Michael and Reinsberger, Claus and Vieluf, Solveig}, year={2021} }","short":"C. Goelz, K. Mora, J.K. Stroehlein, F.K. Haase, M. Dellnitz, C. Reinsberger, S. Vieluf, Cognitive Neurodynamics (2021).","apa":"Goelz, C., Mora, K., Stroehlein, J. K., Haase, F. K., Dellnitz, M., Reinsberger, C., &#38; Vieluf, S. (2021). Electrophysiological signatures of dedifferentiation differ between fit and less fit older adults. <i>Cognitive Neurodynamics</i>. <a href=\"https://doi.org/10.1007/s11571-020-09656-9\">https://doi.org/10.1007/s11571-020-09656-9</a>"}},{"abstract":[{"lang":"eng","text":"We present a flexible trust region descend algorithm for unconstrained and\r\nconvexly constrained multiobjective optimization problems. It is targeted at\r\nheterogeneous and expensive problems, i.e., problems that have at least one\r\nobjective function that is computationally expensive. The method is\r\nderivative-free in the sense that neither need derivative information be\r\navailable for the expensive objectives nor are gradients approximated using\r\nrepeated function evaluations as is the case in finite-difference methods.\r\nInstead, a multiobjective trust region approach is used that works similarly to\r\nits well-known scalar pendants. Local surrogate models constructed from\r\nevaluation data of the true objective functions are employed to compute\r\npossible descent directions. In contrast to existing multiobjective trust\r\nregion algorithms, these surrogates are not polynomial but carefully\r\nconstructed radial basis function networks. This has the important advantage\r\nthat the number of data points scales linearly with the parameter space\r\ndimension. The local models qualify as fully linear and the corresponding\r\ngeneral scalar framework is adapted for problems with multiple objectives.\r\nConvergence to Pareto critical points is proven and numerical examples\r\nillustrate our findings."}],"publication":"Mathematical and Computational Applications","language":[{"iso":"eng"}],"year":"2021","issue":"2","title":"Derivative-Free Multiobjective Trust Region Descent Method Using Radial  Basis Function Surrogate Models","date_created":"2021-03-01T10:46:48Z","status":"public","type":"journal_article","article_number":"31","department":[{"_id":"101"},{"_id":"655"}],"user_id":"47427","_id":"21337","intvolume":"        26","citation":{"bibtex":"@article{Berkemeier_Peitz_2021, title={Derivative-Free Multiobjective Trust Region Descent Method Using Radial  Basis Function Surrogate Models}, volume={26}, DOI={<a href=\"https://doi.org/10.3390/mca26020031\">10.3390/mca26020031</a>}, number={231}, journal={Mathematical and Computational Applications}, author={Berkemeier, Manuel Bastian and Peitz, Sebastian}, year={2021} }","mla":"Berkemeier, Manuel Bastian, and Sebastian Peitz. “Derivative-Free Multiobjective Trust Region Descent Method Using Radial  Basis Function Surrogate Models.” <i>Mathematical and Computational Applications</i>, vol. 26, no. 2, 31, 2021, doi:<a href=\"https://doi.org/10.3390/mca26020031\">10.3390/mca26020031</a>.","short":"M.B. Berkemeier, S. Peitz, Mathematical and Computational Applications 26 (2021).","apa":"Berkemeier, M. B., &#38; Peitz, S. (2021). Derivative-Free Multiobjective Trust Region Descent Method Using Radial  Basis Function Surrogate Models. <i>Mathematical and Computational Applications</i>, <i>26</i>(2). <a href=\"https://doi.org/10.3390/mca26020031\">https://doi.org/10.3390/mca26020031</a>","chicago":"Berkemeier, Manuel Bastian, and Sebastian Peitz. “Derivative-Free Multiobjective Trust Region Descent Method Using Radial  Basis Function Surrogate Models.” <i>Mathematical and Computational Applications</i> 26, no. 2 (2021). <a href=\"https://doi.org/10.3390/mca26020031\">https://doi.org/10.3390/mca26020031</a>.","ieee":"M. B. Berkemeier and S. Peitz, “Derivative-Free Multiobjective Trust Region Descent Method Using Radial  Basis Function Surrogate Models,” <i>Mathematical and Computational Applications</i>, vol. 26, no. 2, 2021.","ama":"Berkemeier MB, Peitz S. Derivative-Free Multiobjective Trust Region Descent Method Using Radial  Basis Function Surrogate Models. <i>Mathematical and Computational Applications</i>. 2021;26(2). doi:<a href=\"https://doi.org/10.3390/mca26020031\">10.3390/mca26020031</a>"},"publication_identifier":{"eissn":["2297-8747"]},"publication_status":"published","doi":"10.3390/mca26020031","main_file_link":[{"url":"https://www.mdpi.com/2297-8747/26/2/31/pdf","open_access":"1"}],"volume":26,"author":[{"full_name":"Berkemeier, Manuel Bastian","id":"51701","last_name":"Berkemeier","first_name":"Manuel Bastian"},{"id":"47427","full_name":"Peitz, Sebastian","last_name":"Peitz","orcid":"0000-0002-3389-793X","first_name":"Sebastian"}],"date_updated":"2022-01-06T06:54:55Z","oa":"1"},{"user_id":"81513","department":[{"_id":"101"}],"_id":"21820","language":[{"iso":"eng"}],"article_number":"134","type":"journal_article","publication":"Entropy","status":"public","abstract":[{"lang":"eng","text":"<jats:p>The reduction of high-dimensional systems to effective models on a smaller set of variables is an essential task in many areas of science. For stochastic dynamics governed by diffusion processes, a general procedure to find effective equations is the conditioning approach. In this paper, we are interested in the spectrum of the generator of the resulting effective dynamics, and how it compares to the spectrum of the full generator. We prove a new relative error bound in terms of the eigenfunction approximation error for reversible systems. We also present numerical examples indicating that, if Kramers–Moyal (KM) type approximations are used to compute the spectrum of the reduced generator, it seems largely insensitive to the time window used for the KM estimators. We analyze the implications of these observations for systems driven by underdamped Langevin dynamics, and show how meaningful effective dynamics can be defined in this setting.</jats:p>"}],"date_created":"2021-04-28T18:07:56Z","author":[{"orcid":"0000-0003-2444-7889","last_name":"Nüske","full_name":"Nüske, Feliks","id":"81513","first_name":"Feliks"},{"full_name":"Koltai, Péter","last_name":"Koltai","first_name":"Péter"},{"last_name":"Boninsegna","full_name":"Boninsegna, Lorenzo","first_name":"Lorenzo"},{"first_name":"Cecilia","full_name":"Clementi, Cecilia","last_name":"Clementi"}],"oa":"1","date_updated":"2022-01-06T06:55:16Z","main_file_link":[{"url":"https://www.mdpi.com/1099-4300/23/2/134","open_access":"1"}],"doi":"10.3390/e23020134","title":"Spectral Properties of Effective Dynamics from Conditional Expectations","publication_status":"published","publication_identifier":{"issn":["1099-4300"]},"citation":{"bibtex":"@article{Nüske_Koltai_Boninsegna_Clementi_2021, title={Spectral Properties of Effective Dynamics from Conditional Expectations}, DOI={<a href=\"https://doi.org/10.3390/e23020134\">10.3390/e23020134</a>}, number={134}, journal={Entropy}, author={Nüske, Feliks and Koltai, Péter and Boninsegna, Lorenzo and Clementi, Cecilia}, year={2021} }","short":"F. Nüske, P. Koltai, L. Boninsegna, C. Clementi, Entropy (2021).","mla":"Nüske, Feliks, et al. “Spectral Properties of Effective Dynamics from Conditional Expectations.” <i>Entropy</i>, 134, 2021, doi:<a href=\"https://doi.org/10.3390/e23020134\">10.3390/e23020134</a>.","apa":"Nüske, F., Koltai, P., Boninsegna, L., &#38; Clementi, C. (2021). Spectral Properties of Effective Dynamics from Conditional Expectations. <i>Entropy</i>. <a href=\"https://doi.org/10.3390/e23020134\">https://doi.org/10.3390/e23020134</a>","ieee":"F. Nüske, P. Koltai, L. Boninsegna, and C. Clementi, “Spectral Properties of Effective Dynamics from Conditional Expectations,” <i>Entropy</i>, 2021.","chicago":"Nüske, Feliks, Péter Koltai, Lorenzo Boninsegna, and Cecilia Clementi. “Spectral Properties of Effective Dynamics from Conditional Expectations.” <i>Entropy</i>, 2021. <a href=\"https://doi.org/10.3390/e23020134\">https://doi.org/10.3390/e23020134</a>.","ama":"Nüske F, Koltai P, Boninsegna L, Clementi C. Spectral Properties of Effective Dynamics from Conditional Expectations. <i>Entropy</i>. 2021. doi:<a href=\"https://doi.org/10.3390/e23020134\">10.3390/e23020134</a>"},"year":"2021"},{"_id":"16867","department":[{"_id":"101"}],"user_id":"47427","language":[{"iso":"eng"}],"publication":"Journal of Optimization Theory and Applications","type":"journal_article","abstract":[{"lang":"eng","text":"In this article, we present an efficient descent method for locally Lipschitz\r\ncontinuous multiobjective optimization problems (MOPs). The method is realized\r\nby combining a theoretical result regarding the computation of descent\r\ndirections for nonsmooth MOPs with a practical method to approximate the\r\nsubdifferentials of the objective functions. We show convergence to points\r\nwhich satisfy a necessary condition for Pareto optimality. Using a set of test\r\nproblems, we compare our method to the multiobjective proximal bundle method by\r\nM\\\"akel\\\"a. The results indicate that our method is competitive while being\r\neasier to implement. While the number of objective function evaluations is\r\nlarger, the overall number of subgradient evaluations is lower. Finally, we\r\nshow that our method can be combined with a subdivision algorithm to compute\r\nentire Pareto sets of nonsmooth MOPs."}],"status":"public","oa":"1","date_updated":"2022-01-06T06:52:57Z","volume":188,"date_created":"2020-04-27T09:11:22Z","author":[{"first_name":"Bennet","last_name":"Gebken","full_name":"Gebken, Bennet","id":"32643"},{"last_name":"Peitz","orcid":"0000-0002-3389-793X","full_name":"Peitz, Sebastian","id":"47427","first_name":"Sebastian"}],"title":"An efficient descent method for locally Lipschitz multiobjective optimization problems","doi":"10.1007/s10957-020-01803-w","main_file_link":[{"url":"https://link.springer.com/content/pdf/10.1007/s10957-020-01803-w.pdf","open_access":"1"}],"publication_status":"published","year":"2021","intvolume":"       188","page":"696-723","citation":{"short":"B. Gebken, S. Peitz, Journal of Optimization Theory and Applications 188 (2021) 696–723.","mla":"Gebken, Bennet, and Sebastian Peitz. “An Efficient Descent Method for Locally Lipschitz Multiobjective Optimization Problems.” <i>Journal of Optimization Theory and Applications</i>, vol. 188, 2021, pp. 696–723, doi:<a href=\"https://doi.org/10.1007/s10957-020-01803-w\">10.1007/s10957-020-01803-w</a>.","bibtex":"@article{Gebken_Peitz_2021, title={An efficient descent method for locally Lipschitz multiobjective optimization problems}, volume={188}, DOI={<a href=\"https://doi.org/10.1007/s10957-020-01803-w\">10.1007/s10957-020-01803-w</a>}, journal={Journal of Optimization Theory and Applications}, author={Gebken, Bennet and Peitz, Sebastian}, year={2021}, pages={696–723} }","apa":"Gebken, B., &#38; Peitz, S. (2021). An efficient descent method for locally Lipschitz multiobjective optimization problems. <i>Journal of Optimization Theory and Applications</i>, <i>188</i>, 696–723. <a href=\"https://doi.org/10.1007/s10957-020-01803-w\">https://doi.org/10.1007/s10957-020-01803-w</a>","chicago":"Gebken, Bennet, and Sebastian Peitz. “An Efficient Descent Method for Locally Lipschitz Multiobjective Optimization Problems.” <i>Journal of Optimization Theory and Applications</i> 188 (2021): 696–723. <a href=\"https://doi.org/10.1007/s10957-020-01803-w\">https://doi.org/10.1007/s10957-020-01803-w</a>.","ieee":"B. Gebken and S. Peitz, “An efficient descent method for locally Lipschitz multiobjective optimization problems,” <i>Journal of Optimization Theory and Applications</i>, vol. 188, pp. 696–723, 2021.","ama":"Gebken B, Peitz S. An efficient descent method for locally Lipschitz multiobjective optimization problems. <i>Journal of Optimization Theory and Applications</i>. 2021;188:696-723. doi:<a href=\"https://doi.org/10.1007/s10957-020-01803-w\">10.1007/s10957-020-01803-w</a>"}},{"language":[{"iso":"eng"}],"_id":"16295","user_id":"47427","department":[{"_id":"101"}],"abstract":[{"lang":"eng","text":"It is a challenging task to identify the objectives on which a certain decision was based, in particular if several, potentially conflicting criteria are equally important and a continuous set of optimal compromise decisions exists. This task can be understood as the inverse problem of multiobjective optimization, where the goal is to find the objective function vector of a given Pareto set. To this end, we present a method to construct the objective function vector of an unconstrained multiobjective optimization problem (MOP) such that the Pareto critical set contains a given set of data points with prescribed KKT multipliers. If such an MOP can not be found, then the method instead produces an MOP whose Pareto critical set is at least close to the data points. The key idea is to consider the objective function vector in the multiobjective KKT conditions as variable and then search for the objectives that minimize the Euclidean norm of the resulting system of equations. By expressing the objectives in a finite-dimensional basis, we transform this problem into a homogeneous, linear system of equations that can be solved efficiently. Potential applications of this approach include the identification of objectives (both from clean and noisy data) and the construction of surrogate models for expensive MOPs."}],"status":"public","type":"journal_article","publication":"Journal of Global Optimization","title":"Inverse multiobjective optimization: Inferring decision criteria from data","main_file_link":[{"open_access":"1","url":"https://link.springer.com/content/pdf/10.1007/s10898-020-00983-z.pdf"}],"doi":"10.1007/s10898-020-00983-z","publisher":"Springer","oa":"1","date_updated":"2022-01-06T06:52:48Z","author":[{"id":"32643","full_name":"Gebken, Bennet","last_name":"Gebken","first_name":"Bennet"},{"first_name":"Sebastian","orcid":"https://orcid.org/0000-0002-3389-793X","last_name":"Peitz","id":"47427","full_name":"Peitz, Sebastian"}],"date_created":"2020-03-13T12:45:05Z","volume":80,"year":"2021","citation":{"bibtex":"@article{Gebken_Peitz_2021, title={Inverse multiobjective optimization: Inferring decision criteria from data}, volume={80}, DOI={<a href=\"https://doi.org/10.1007/s10898-020-00983-z\">10.1007/s10898-020-00983-z</a>}, journal={Journal of Global Optimization}, publisher={Springer}, author={Gebken, Bennet and Peitz, Sebastian}, year={2021}, pages={3–29} }","mla":"Gebken, Bennet, and Sebastian Peitz. “Inverse Multiobjective Optimization: Inferring Decision Criteria from Data.” <i>Journal of Global Optimization</i>, vol. 80, Springer, 2021, pp. 3–29, doi:<a href=\"https://doi.org/10.1007/s10898-020-00983-z\">10.1007/s10898-020-00983-z</a>.","short":"B. Gebken, S. Peitz, Journal of Global Optimization 80 (2021) 3–29.","apa":"Gebken, B., &#38; Peitz, S. (2021). Inverse multiobjective optimization: Inferring decision criteria from data. <i>Journal of Global Optimization</i>, <i>80</i>, 3–29. <a href=\"https://doi.org/10.1007/s10898-020-00983-z\">https://doi.org/10.1007/s10898-020-00983-z</a>","ieee":"B. Gebken and S. Peitz, “Inverse multiobjective optimization: Inferring decision criteria from data,” <i>Journal of Global Optimization</i>, vol. 80, pp. 3–29, 2021.","chicago":"Gebken, Bennet, and Sebastian Peitz. “Inverse Multiobjective Optimization: Inferring Decision Criteria from Data.” <i>Journal of Global Optimization</i> 80 (2021): 3–29. <a href=\"https://doi.org/10.1007/s10898-020-00983-z\">https://doi.org/10.1007/s10898-020-00983-z</a>.","ama":"Gebken B, Peitz S. Inverse multiobjective optimization: Inferring decision criteria from data. <i>Journal of Global Optimization</i>. 2021;80:3-29. doi:<a href=\"https://doi.org/10.1007/s10898-020-00983-z\">10.1007/s10898-020-00983-z</a>"},"page":"3-29","intvolume":"        80"},{"oa":"1","date_updated":"2022-06-20T13:40:30Z","date_created":"2022-06-20T09:54:24Z","author":[{"last_name":"Gerlach","full_name":"Gerlach, Raphael","id":"32655","first_name":"Raphael"}],"supervisor":[{"full_name":"Dellnitz , Michael","last_name":"Dellnitz ","first_name":"Michael"},{"last_name":"Koltai","full_name":"Koltai, Péter","first_name":"Péter"}],"title":"The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems","main_file_link":[{"open_access":"1","url":"https://digital.ub.uni-paderborn.de/hs/download/pdf/6214949"}],"doi":"10.17619/UNIPB/1-1278","year":"2021","citation":{"chicago":"Gerlach, Raphael. <i>The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems</i>, 2021. <a href=\"https://doi.org/10.17619/UNIPB/1-1278\">https://doi.org/10.17619/UNIPB/1-1278</a>.","ieee":"R. Gerlach, <i>The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems</i>. 2021.","ama":"Gerlach R. <i>The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems</i>.; 2021. doi:<a href=\"https://doi.org/10.17619/UNIPB/1-1278\">10.17619/UNIPB/1-1278</a>","short":"R. Gerlach, The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems, 2021.","mla":"Gerlach, Raphael. <i>The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems</i>. 2021, doi:<a href=\"https://doi.org/10.17619/UNIPB/1-1278\">10.17619/UNIPB/1-1278</a>.","bibtex":"@book{Gerlach_2021, title={The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems}, DOI={<a href=\"https://doi.org/10.17619/UNIPB/1-1278\">10.17619/UNIPB/1-1278</a>}, author={Gerlach, Raphael}, year={2021} }","apa":"Gerlach, R. (2021). <i>The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems</i>. <a href=\"https://doi.org/10.17619/UNIPB/1-1278\">https://doi.org/10.17619/UNIPB/1-1278</a>"},"_id":"32057","user_id":"32643","department":[{"_id":"101"}],"language":[{"iso":"eng"}],"type":"dissertation","abstract":[{"lang":"ger","text":"Ein zentraler Aspekt bei der Untersuchung dynamischer Systeme ist die Analyse ihrer invarianten Mengen wie des globalen Attraktors und (in)stabiler Mannigfaltigkeiten. Insbesondere wenn das zugrunde liegende System von einem Parameter abhängt, ist es entscheidend, sie im Bezug auf diesen Parameter effizient zu verfolgen. Für die Berechnung invarianter Mengen stützen wir uns für ihre Approximation auf numerische Algorithmen. Typischerweise können diese Methoden jedoch nur auf endlich-dimensionale dynamische Systeme angewendet werden. In dieser Arbeit präsentieren wir daher einen numerischen Rahmen für die globale dynamische Analyse unendlich-dimensionaler Systeme. Wir werden Einbettungstechniken verwenden, um das core dynamical system (CDS) zu definieren, welches ein dynamisch äquivalentes endlich-dimensionales System ist.Das CDS wird dann verwendet, um eingebettete invariante Mengen, also eins-zu-eins Bilder, mittels Mengen-orientierten numerischen Methoden zu approximieren. Bei der Konstruktion des CDS ist es entscheidend, eine geeignete Beobachtungsabbildung auszuwählen und die geeignete inverse Abbildung zu entwerfen. Dazu werden wir geeignete numerische Implementierungen des CDS für DDEs und PDEs vorstellen. Für eine nachfolgende geometrische Analyse der eingebetteten invarianten Menge betrachten wir eine Lerntechnik namens diffusion maps, die ihre intrinsische Geometrie enthüllt sowie ihre Dimension schätzt. Schließlich wenden wir unsere entwickelten numerischen Methoden an einigen bekannten unendlich-dimensionale dynamischen Systeme an, wie die Mackey-Glass-Gleichung, die Kuramoto-Sivashinsky-Gleichung und die Navier-Stokes-Gleichung."},{"lang":"eng","text":"One central aspect in the study of dynamical systems is the analysis of its invariant sets such as the global attractor and (un)stable manifolds. In particular, when the underlying system depends on a parameter it is crucial to efficiently track those set with respect to this parameter. For the computation of invariant sets we rely on numerical algorithms for their approximation but typically those tools can only be applied to finite-dimensional dynamical systems. Thus, in thesis we present a numerical framework for the global dynamical analysis of infinite-dimensional systems. We will use embedding techniques for the definition of the core dynamical system (CDS) which is a dynamically equivalent finite-dimensional system. The CDS is then used for the approximation of related embedded invariant sets, i.e, one-to-one images, by set-oriented numerical methods. For the construction of the CDS it is crucial to choose an appropriate observation map and to design its corresponding inverse. Therefore, we will present suitable numerical realizations of the CDS for DDEs and PDEs. For a subsequent geometric analysis of the embedded invariant set we will consider a manifold learning technique called diffusion maps which reveals its intrinsic geometry and estimates its dimension. Finally, we apply our develop numerical tools on some well-known infinite-dimensional dynamical systems such as the Mackey-Glass equation, the Kuramoto-Sivashinsky equation and the Navier-Stokes equation."}],"status":"public"},{"oa":"1","date_updated":"2022-01-24T13:27:50Z","volume":"31(2)","date_created":"2020-03-13T12:44:36Z","author":[{"first_name":"Sina","full_name":"Ober-Blöbaum, Sina","id":"16494","last_name":"Ober-Blöbaum"},{"orcid":"https://orcid.org/0000-0002-3389-793X","last_name":"Peitz","full_name":"Peitz, Sebastian","id":"47427","first_name":"Sebastian"}],"title":"Explicit multiobjective model predictive control for nonlinear systems  with symmetries","doi":"10.1002/rnc.5281","main_file_link":[{"url":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/rnc.5281","open_access":"1"}],"year":"2021","page":"380-403","citation":{"ama":"Ober-Blöbaum S, Peitz S. Explicit multiobjective model predictive control for nonlinear systems  with symmetries. <i>International Journal of Robust and Nonlinear Control</i>. 2021;31(2):380-403. doi:<a href=\"https://doi.org/10.1002/rnc.5281\">10.1002/rnc.5281</a>","ieee":"S. Ober-Blöbaum and S. Peitz, “Explicit multiobjective model predictive control for nonlinear systems  with symmetries,” <i>International Journal of Robust and Nonlinear Control</i>, vol. 31(2), pp. 380–403, 2021, doi: <a href=\"https://doi.org/10.1002/rnc.5281\">10.1002/rnc.5281</a>.","chicago":"Ober-Blöbaum, Sina, and Sebastian Peitz. “Explicit Multiobjective Model Predictive Control for Nonlinear Systems  with Symmetries.” <i>International Journal of Robust and Nonlinear Control</i> 31(2) (2021): 380–403. <a href=\"https://doi.org/10.1002/rnc.5281\">https://doi.org/10.1002/rnc.5281</a>.","apa":"Ober-Blöbaum, S., &#38; Peitz, S. (2021). Explicit multiobjective model predictive control for nonlinear systems  with symmetries. <i>International Journal of Robust and Nonlinear Control</i>, <i>31(2)</i>, 380–403. <a href=\"https://doi.org/10.1002/rnc.5281\">https://doi.org/10.1002/rnc.5281</a>","bibtex":"@article{Ober-Blöbaum_Peitz_2021, title={Explicit multiobjective model predictive control for nonlinear systems  with symmetries}, volume={31(2)}, DOI={<a href=\"https://doi.org/10.1002/rnc.5281\">10.1002/rnc.5281</a>}, journal={International Journal of Robust and Nonlinear Control}, author={Ober-Blöbaum, Sina and Peitz, Sebastian}, year={2021}, pages={380–403} }","short":"S. Ober-Blöbaum, S. Peitz, International Journal of Robust and Nonlinear Control 31(2) (2021) 380–403.","mla":"Ober-Blöbaum, Sina, and Sebastian Peitz. “Explicit Multiobjective Model Predictive Control for Nonlinear Systems  with Symmetries.” <i>International Journal of Robust and Nonlinear Control</i>, vol. 31(2), 2021, pp. 380–403, doi:<a href=\"https://doi.org/10.1002/rnc.5281\">10.1002/rnc.5281</a>."},"_id":"16294","project":[{"_id":"52","name":"Computing Resources Provided by the Paderborn Center for Parallel Computing"}],"department":[{"_id":"101"}],"user_id":"15694","language":[{"iso":"eng"}],"publication":"International Journal of Robust and Nonlinear Control","type":"journal_article","abstract":[{"lang":"eng","text":"Model predictive control is a prominent approach to construct a feedback\r\ncontrol loop for dynamical systems. Due to real-time constraints, the major\r\nchallenge in MPC is to solve model-based optimal control problems in a very\r\nshort amount of time. For linear-quadratic problems, Bemporad et al. have\r\nproposed an explicit formulation where the underlying optimization problems are\r\nsolved a priori in an offline phase. In this article, we present an extension\r\nof this concept in two significant ways. We consider nonlinear problems and -\r\nmore importantly - problems with multiple conflicting objective functions. In\r\nthe offline phase, we build a library of Pareto optimal solutions from which we\r\nthen obtain a valid compromise solution in the online phase according to a\r\ndecision maker's preference. Since the standard multi-parametric programming\r\napproach is no longer valid in this situation, we instead use interpolation\r\nbetween different entries of the library. To reduce the number of problems that\r\nhave to be solved in the offline phase, we exploit symmetries in the dynamical\r\nsystem and the corresponding multiobjective optimal control problem. The\r\nresults are verified using two different examples from autonomous driving."}],"status":"public"},{"year":"2020","place":"Cham","citation":{"ama":"Flaßkamp K, Ober-Blöbaum S, Peitz S. Symmetry in Optimal Control: A Multiobjective Model Predictive Control Approach. In: Junge O, Schütze O, Froyland G, Ober-Blöbaum S, Padberg-Gehle K, eds. <i>Advances in Dynamics, Optimization and Computation</i>. Cham: Springer; 2020. doi:<a href=\"https://doi.org/10.1007/978-3-030-51264-4_9\">10.1007/978-3-030-51264-4_9</a>","ieee":"K. Flaßkamp, S. Ober-Blöbaum, and S. Peitz, “Symmetry in Optimal Control: A Multiobjective Model Predictive Control Approach,” in <i>Advances in Dynamics, Optimization and Computation</i>, O. Junge, O. Schütze, G. Froyland, S. Ober-Blöbaum, and K. Padberg-Gehle, Eds. Cham: Springer, 2020.","chicago":"Flaßkamp, Kathrin, Sina Ober-Blöbaum, and Sebastian Peitz. “Symmetry in Optimal Control: A Multiobjective Model Predictive Control Approach.” In <i>Advances in Dynamics, Optimization and Computation</i>, edited by Oliver Junge, Oliver Schütze, Gary Froyland, Sina Ober-Blöbaum, and Kathrin Padberg-Gehle. Cham: Springer, 2020. <a href=\"https://doi.org/10.1007/978-3-030-51264-4_9\">https://doi.org/10.1007/978-3-030-51264-4_9</a>.","mla":"Flaßkamp, Kathrin, et al. “Symmetry in Optimal Control: A Multiobjective Model Predictive Control Approach.” <i>Advances in Dynamics, Optimization and Computation</i>, edited by Oliver Junge et al., Springer, 2020, doi:<a href=\"https://doi.org/10.1007/978-3-030-51264-4_9\">10.1007/978-3-030-51264-4_9</a>.","bibtex":"@inbook{Flaßkamp_Ober-Blöbaum_Peitz_2020, place={Cham}, title={Symmetry in Optimal Control: A Multiobjective Model Predictive Control Approach}, DOI={<a href=\"https://doi.org/10.1007/978-3-030-51264-4_9\">10.1007/978-3-030-51264-4_9</a>}, booktitle={Advances in Dynamics, Optimization and Computation}, publisher={Springer}, author={Flaßkamp, Kathrin and Ober-Blöbaum, Sina and Peitz, Sebastian}, editor={Junge, Oliver and Schütze, Oliver and Froyland, Gary and Ober-Blöbaum, Sina and Padberg-Gehle, KathrinEditors}, year={2020} }","short":"K. Flaßkamp, S. Ober-Blöbaum, S. Peitz, in: O. Junge, O. Schütze, G. Froyland, S. Ober-Blöbaum, K. Padberg-Gehle (Eds.), Advances in Dynamics, Optimization and Computation, Springer, Cham, 2020.","apa":"Flaßkamp, K., Ober-Blöbaum, S., &#38; Peitz, S. (2020). Symmetry in Optimal Control: A Multiobjective Model Predictive Control Approach. In O. Junge, O. Schütze, G. Froyland, S. Ober-Blöbaum, &#38; K. Padberg-Gehle (Eds.), <i>Advances in Dynamics, Optimization and Computation</i>. Cham: Springer. <a href=\"https://doi.org/10.1007/978-3-030-51264-4_9\">https://doi.org/10.1007/978-3-030-51264-4_9</a>"},"publication_status":"published","publication_identifier":{"isbn":["9783030512637","9783030512644"],"issn":["2198-4182","2198-4190"]},"title":"Symmetry in Optimal Control: A Multiobjective Model Predictive Control Approach","doi":"10.1007/978-3-030-51264-4_9","date_updated":"2022-01-06T06:53:11Z","publisher":"Springer","author":[{"full_name":"Flaßkamp, Kathrin","last_name":"Flaßkamp","first_name":"Kathrin"},{"last_name":"Ober-Blöbaum","full_name":"Ober-Blöbaum, Sina","first_name":"Sina"},{"first_name":"Sebastian","id":"47427","full_name":"Peitz, Sebastian","last_name":"Peitz","orcid":"0000-0002-3389-793X"}],"date_created":"2020-07-27T09:50:19Z","abstract":[{"lang":"eng","text":"Many dynamical systems possess symmetries, e.g. rotational and translational invariances of mechanical systems. These can be beneficially exploited in the design of numerical optimal control methods. We present a model predictive control scheme which is based on a library of precomputed motion primitives. The primitives are equivalence classes w.r.t. the symmetry of the optimal control problems. Trim primitives as relative equilibria w.r.t. this symmetry, play a crucial role in the algorithm. The approach is illustrated using an academic mobile robot example."}],"editor":[{"last_name":"Junge","full_name":"Junge, Oliver","first_name":"Oliver"},{"first_name":"Oliver","last_name":"Schütze","full_name":"Schütze, Oliver"},{"full_name":"Froyland, Gary","last_name":"Froyland","first_name":"Gary"},{"first_name":"Sina","full_name":"Ober-Blöbaum, Sina","last_name":"Ober-Blöbaum"},{"full_name":"Padberg-Gehle, Kathrin","last_name":"Padberg-Gehle","first_name":"Kathrin"}],"status":"public","type":"book_chapter","publication":"Advances in Dynamics, Optimization and Computation","language":[{"iso":"eng"}],"_id":"17411","user_id":"47427","department":[{"_id":"101"}]},{"_id":"21819","department":[{"_id":"101"}],"user_id":"81513","article_number":"722","language":[{"iso":"eng"}],"publication":"Entropy","type":"journal_article","abstract":[{"lang":"eng","text":"<jats:p>Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems.</jats:p>"}],"status":"public","oa":"1","date_updated":"2022-01-06T06:55:16Z","date_created":"2021-04-28T18:06:35Z","author":[{"first_name":"Stefan","last_name":"Klus","full_name":"Klus, Stefan"},{"id":"81513","full_name":"Nüske, Feliks","last_name":"Nüske","orcid":"0000-0003-2444-7889","first_name":"Feliks"},{"last_name":"Hamzi","full_name":"Hamzi, Boumediene","first_name":"Boumediene"}],"title":"Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator","doi":"10.3390/e22070722","main_file_link":[{"open_access":"1","url":"https://www.mdpi.com/1099-4300/22/7/722"}],"publication_identifier":{"issn":["1099-4300"]},"publication_status":"published","year":"2020","citation":{"mla":"Klus, Stefan, et al. “Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator.” <i>Entropy</i>, 722, 2020, doi:<a href=\"https://doi.org/10.3390/e22070722\">10.3390/e22070722</a>.","bibtex":"@article{Klus_Nüske_Hamzi_2020, title={Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator}, DOI={<a href=\"https://doi.org/10.3390/e22070722\">10.3390/e22070722</a>}, number={722}, journal={Entropy}, author={Klus, Stefan and Nüske, Feliks and Hamzi, Boumediene}, year={2020} }","short":"S. Klus, F. Nüske, B. Hamzi, Entropy (2020).","apa":"Klus, S., Nüske, F., &#38; Hamzi, B. (2020). Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator. <i>Entropy</i>. <a href=\"https://doi.org/10.3390/e22070722\">https://doi.org/10.3390/e22070722</a>","ama":"Klus S, Nüske F, Hamzi B. Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator. <i>Entropy</i>. 2020. doi:<a href=\"https://doi.org/10.3390/e22070722\">10.3390/e22070722</a>","chicago":"Klus, Stefan, Feliks Nüske, and Boumediene Hamzi. “Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator.” <i>Entropy</i>, 2020. <a href=\"https://doi.org/10.3390/e22070722\">https://doi.org/10.3390/e22070722</a>.","ieee":"S. Klus, F. Nüske, and B. Hamzi, “Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator,” <i>Entropy</i>, 2020."}},{"abstract":[{"text":"Multi-objective optimization is an active field of research that has many applications. Owing to its success and because decision-making processes are becoming more and more complex, there is a recent trend for incorporating many objectives into such problems. The challenge with such problems, however, is that the dimensions of the solution sets—the so-called Pareto sets and fronts—grow with the number of objectives. It is thus no longer possible to compute or to approximate the entire solution set of a given problem that contains many (e.g. more than three) objectives. On the other hand, the computation of single solutions (e.g. via scalarization methods) leads to unsatisfying results in many cases, even if user preferences are incorporated. In this article, the Pareto Explorer tool is presented—a global/local exploration tool for the treatment of many-objective optimization problems (MaOPs). In the first step, a solution of the problem is computed via a global search algorithm that ideally already includes user preferences. In the second step, a local search along the Pareto set/front of the given MaOP is performed in user specified directions. For this, several continuation-like procedures are proposed that can incorporate preferences defined in decision, objective, or in weight space. The applicability and usefulness of Pareto Explorer is demonstrated on benchmark problems as well as on an application from industrial laundry design.","lang":"eng"}],"status":"public","type":"journal_article","publication":"Engineering Optimization","article_type":"original","language":[{"iso":"eng"}],"_id":"10596","user_id":"47427","department":[{"_id":"101"}],"year":"2020","citation":{"chicago":"Schütze, Oliver, Oliver Cuate, Adanay Martín, Sebastian Peitz, and Michael Dellnitz. “Pareto Explorer: A Global/Local Exploration Tool for Many-Objective Optimization Problems.” <i>Engineering Optimization</i> 52, no. 5 (2020): 832–55. <a href=\"https://doi.org/10.1080/0305215x.2019.1617286\">https://doi.org/10.1080/0305215x.2019.1617286</a>.","ieee":"O. Schütze, O. Cuate, A. Martín, S. Peitz, and M. Dellnitz, “Pareto Explorer: a global/local exploration tool for many-objective optimization problems,” <i>Engineering Optimization</i>, vol. 52, no. 5, pp. 832–855, 2020.","ama":"Schütze O, Cuate O, Martín A, Peitz S, Dellnitz M. Pareto Explorer: a global/local exploration tool for many-objective optimization problems. <i>Engineering Optimization</i>. 2020;52(5):832-855. doi:<a href=\"https://doi.org/10.1080/0305215x.2019.1617286\">10.1080/0305215x.2019.1617286</a>","apa":"Schütze, O., Cuate, O., Martín, A., Peitz, S., &#38; Dellnitz, M. (2020). Pareto Explorer: a global/local exploration tool for many-objective optimization problems. <i>Engineering Optimization</i>, <i>52</i>(5), 832–855. <a href=\"https://doi.org/10.1080/0305215x.2019.1617286\">https://doi.org/10.1080/0305215x.2019.1617286</a>","mla":"Schütze, Oliver, et al. “Pareto Explorer: A Global/Local Exploration Tool for Many-Objective Optimization Problems.” <i>Engineering Optimization</i>, vol. 52, no. 5, 2020, pp. 832–55, doi:<a href=\"https://doi.org/10.1080/0305215x.2019.1617286\">10.1080/0305215x.2019.1617286</a>.","short":"O. Schütze, O. Cuate, A. Martín, S. Peitz, M. Dellnitz, Engineering Optimization 52 (2020) 832–855.","bibtex":"@article{Schütze_Cuate_Martín_Peitz_Dellnitz_2020, title={Pareto Explorer: a global/local exploration tool for many-objective optimization problems}, volume={52}, DOI={<a href=\"https://doi.org/10.1080/0305215x.2019.1617286\">10.1080/0305215x.2019.1617286</a>}, number={5}, journal={Engineering Optimization}, author={Schütze, Oliver and Cuate, Oliver and Martín, Adanay and Peitz, Sebastian and Dellnitz, Michael}, year={2020}, pages={832–855} }"},"page":"832-855","intvolume":"        52","publication_status":"published","publication_identifier":{"issn":["0305-215X","1029-0273"]},"issue":"5","title":"Pareto Explorer: a global/local exploration tool for many-objective optimization problems","doi":"10.1080/0305215x.2019.1617286","date_updated":"2022-01-06T06:50:46Z","author":[{"last_name":"Schütze","full_name":"Schütze, Oliver","first_name":"Oliver"},{"first_name":"Oliver","full_name":"Cuate, Oliver","last_name":"Cuate"},{"last_name":"Martín","full_name":"Martín, Adanay","first_name":"Adanay"},{"id":"47427","full_name":"Peitz, Sebastian","last_name":"Peitz","orcid":"https://orcid.org/0000-0002-3389-793X","first_name":"Sebastian"},{"last_name":"Dellnitz","full_name":"Dellnitz, Michael","first_name":"Michael"}],"date_created":"2019-07-10T08:14:39Z","volume":52},{"_id":"16288","user_id":"47427","department":[{"_id":"101"}],"article_number":"132416","language":[{"iso":"eng"}],"type":"journal_article","publication":"Physica D: Nonlinear Phenomena","abstract":[{"lang":"eng","text":"We derive a data-driven method for the approximation of the Koopman generator called gEDMD, which can be regarded as a straightforward extension of EDMD (extended dynamic mode decomposition). This approach is applicable to deterministic and stochastic dynamical systems. It can be used for computing eigenvalues, eigenfunctions, and modes of the generator and for system identification. In addition to learning the governing equations of deterministic systems, which then reduces to SINDy (sparse identification of nonlinear dynamics), it is possible to identify the drift and diffusion terms of stochastic differential equations from data. Moreover, we apply gEDMD to derive coarse-grained models of high-dimensional systems, and also to determine efficient model predictive control strategies. We highlight relationships with other methods and demonstrate the efficacy of the proposed methods using several guiding examples and prototypical molecular dynamics problems."}],"status":"public","date_updated":"2022-01-06T06:52:48Z","author":[{"first_name":"Stefan","full_name":"Klus, Stefan","last_name":"Klus"},{"first_name":"Feliks","full_name":"Nüske, Feliks","id":"81513","orcid":"0000-0003-2444-7889","last_name":"Nüske"},{"last_name":"Peitz","orcid":"https://orcid.org/0000-0002-3389-793X","id":"47427","full_name":"Peitz, Sebastian","first_name":"Sebastian"},{"last_name":"Niemann","full_name":"Niemann, Jan-Hendrik","first_name":"Jan-Hendrik"},{"last_name":"Clementi","full_name":"Clementi, Cecilia","first_name":"Cecilia"},{"last_name":"Schütte","full_name":"Schütte, Christof","first_name":"Christof"}],"date_created":"2020-03-13T12:35:40Z","volume":406,"title":"Data-driven approximation of the Koopman generator: Model reduction, system identification, and control","doi":"10.1016/j.physd.2020.132416","publication_status":"published","publication_identifier":{"issn":["0167-2789"]},"year":"2020","citation":{"ieee":"S. Klus, F. Nüske, S. Peitz, J.-H. Niemann, C. Clementi, and C. Schütte, “Data-driven approximation of the Koopman generator: Model reduction, system identification, and control,” <i>Physica D: Nonlinear Phenomena</i>, vol. 406, 2020.","chicago":"Klus, Stefan, Feliks Nüske, Sebastian Peitz, Jan-Hendrik Niemann, Cecilia Clementi, and Christof Schütte. “Data-Driven Approximation of the Koopman Generator: Model Reduction, System Identification, and Control.” <i>Physica D: Nonlinear Phenomena</i> 406 (2020). <a href=\"https://doi.org/10.1016/j.physd.2020.132416\">https://doi.org/10.1016/j.physd.2020.132416</a>.","ama":"Klus S, Nüske F, Peitz S, Niemann J-H, Clementi C, Schütte C. Data-driven approximation of the Koopman generator: Model reduction, system identification, and control. <i>Physica D: Nonlinear Phenomena</i>. 2020;406. doi:<a href=\"https://doi.org/10.1016/j.physd.2020.132416\">10.1016/j.physd.2020.132416</a>","mla":"Klus, Stefan, et al. “Data-Driven Approximation of the Koopman Generator: Model Reduction, System Identification, and Control.” <i>Physica D: Nonlinear Phenomena</i>, vol. 406, 132416, 2020, doi:<a href=\"https://doi.org/10.1016/j.physd.2020.132416\">10.1016/j.physd.2020.132416</a>.","bibtex":"@article{Klus_Nüske_Peitz_Niemann_Clementi_Schütte_2020, title={Data-driven approximation of the Koopman generator: Model reduction, system identification, and control}, volume={406}, DOI={<a href=\"https://doi.org/10.1016/j.physd.2020.132416\">10.1016/j.physd.2020.132416</a>}, number={132416}, journal={Physica D: Nonlinear Phenomena}, author={Klus, Stefan and Nüske, Feliks and Peitz, Sebastian and Niemann, Jan-Hendrik and Clementi, Cecilia and Schütte, Christof}, year={2020} }","short":"S. Klus, F. Nüske, S. Peitz, J.-H. Niemann, C. Clementi, C. Schütte, Physica D: Nonlinear Phenomena 406 (2020).","apa":"Klus, S., Nüske, F., Peitz, S., Niemann, J.-H., Clementi, C., &#38; Schütte, C. (2020). Data-driven approximation of the Koopman generator: Model reduction, system identification, and control. <i>Physica D: Nonlinear Phenomena</i>, <i>406</i>. <a href=\"https://doi.org/10.1016/j.physd.2020.132416\">https://doi.org/10.1016/j.physd.2020.132416</a>"},"intvolume":"       406"}]