[{"publication_identifier":{"isbn":["9783031261039","9783031261046"]},"title":"The Digital Twin of Humans","year":"2023","status":"public","publication_status":"published","date_updated":"2024-03-25T09:07:55Z","_id":"45191","language":[{"iso":"eng"}],"publisher":"Springer International Publishing","alternative_title":["An Interdisciplinary Concept of Digital Working Environments in Industry 4.0"],"editor":[{"id":"47565","orcid":"0000-0001-5765-971X","first_name":"Iris","last_name":"Gräßler","full_name":"Gräßler, Iris"},{"full_name":"Maier, Günter W.","last_name":"Maier","first_name":"Günter W."},{"orcid":"0000-0002-9808-7401","first_name":"Eckhard","last_name":"Steffen","full_name":"Steffen, Eckhard","id":"15548"},{"id":"54680","full_name":"Roesmann, Daniel","first_name":"Daniel","last_name":"Roesmann"}],"user_id":"5905","doi":"10.1007/978-3-031-26104-6","citation":{"bibtex":"@book{Gräßler_Maier_Steffen_Roesmann_2023, place={Cham}, title={The Digital Twin of Humans}, DOI={<a href=\"https://doi.org/10.1007/978-3-031-26104-6\">10.1007/978-3-031-26104-6</a>}, publisher={Springer International Publishing}, year={2023} }","ama":"Gräßler I, Maier GW, Steffen E, Roesmann D, eds. <i>The Digital Twin of Humans</i>. Springer International Publishing; 2023. doi:<a href=\"https://doi.org/10.1007/978-3-031-26104-6\">10.1007/978-3-031-26104-6</a>","mla":"Gräßler, Iris, et al., editors. <i>The Digital Twin of Humans</i>. Springer International Publishing, 2023, doi:<a href=\"https://doi.org/10.1007/978-3-031-26104-6\">10.1007/978-3-031-26104-6</a>.","short":"I. Gräßler, G.W. Maier, E. Steffen, D. Roesmann, eds., The Digital Twin of Humans, Springer International Publishing, Cham, 2023.","chicago":"Gräßler, Iris, Günter W. Maier, Eckhard Steffen, and Daniel Roesmann, eds. <i>The Digital Twin of Humans</i>. Cham: Springer International Publishing, 2023. <a href=\"https://doi.org/10.1007/978-3-031-26104-6\">https://doi.org/10.1007/978-3-031-26104-6</a>.","ieee":"I. Gräßler, G. W. Maier, E. Steffen, and D. Roesmann, Eds., <i>The Digital Twin of Humans</i>. Cham: Springer International Publishing, 2023.","apa":"Gräßler, I., Maier, G. W., Steffen, E., &#38; Roesmann, D. (Eds.). (2023). <i>The Digital Twin of Humans</i>. Springer International Publishing. <a href=\"https://doi.org/10.1007/978-3-031-26104-6\">https://doi.org/10.1007/978-3-031-26104-6</a>"},"quality_controlled":"1","date_created":"2023-05-22T10:24:10Z","place":"Cham","department":[{"_id":"542"},{"_id":"152"}],"type":"book_editor"},{"abstract":[{"lang":"eng","text":"<jats:title>Abstract</jats:title>\r\n               <jats:p>An error estimate for a canonical discretization of the harmonic map heat flow into spheres is derived. The numerical scheme uses standard finite elements with a nodal treatment of linearized unit-length constraints. The analysis is based on elementary approximation results and only uses the discrete weak formulation.</jats:p>"}],"publication":"IMA Journal of Numerical Analysis","citation":{"bibtex":"@article{Bartels_Kovács_Wang_2023, title={Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints}, DOI={<a href=\"https://doi.org/10.1093/imanum/drad037\">10.1093/imanum/drad037</a>}, journal={IMA Journal of Numerical Analysis}, publisher={Oxford University Press (OUP)}, author={Bartels, Sören and Kovács, Balázs and Wang, Zhangxian}, year={2023} }","ama":"Bartels S, Kovács B, Wang Z. Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints. <i>IMA Journal of Numerical Analysis</i>. Published online 2023. doi:<a href=\"https://doi.org/10.1093/imanum/drad037\">10.1093/imanum/drad037</a>","mla":"Bartels, Sören, et al. “Error Analysis for the Numerical Approximation of the Harmonic Map Heat Flow with Nodal Constraints.” <i>IMA Journal of Numerical Analysis</i>, Oxford University Press (OUP), 2023, doi:<a href=\"https://doi.org/10.1093/imanum/drad037\">10.1093/imanum/drad037</a>.","chicago":"Bartels, Sören, Balázs Kovács, and Zhangxian Wang. “Error Analysis for the Numerical Approximation of the Harmonic Map Heat Flow with Nodal Constraints.” <i>IMA Journal of Numerical Analysis</i>, 2023. <a href=\"https://doi.org/10.1093/imanum/drad037\">https://doi.org/10.1093/imanum/drad037</a>.","short":"S. Bartels, B. Kovács, Z. Wang, IMA Journal of Numerical Analysis (2023).","ieee":"S. Bartels, B. Kovács, and Z. Wang, “Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints,” <i>IMA Journal of Numerical Analysis</i>, 2023, doi: <a href=\"https://doi.org/10.1093/imanum/drad037\">10.1093/imanum/drad037</a>.","apa":"Bartels, S., Kovács, B., &#38; Wang, Z. (2023). Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints. <i>IMA Journal of Numerical Analysis</i>. <a href=\"https://doi.org/10.1093/imanum/drad037\">https://doi.org/10.1093/imanum/drad037</a>"},"keyword":["Applied Mathematics","Computational Mathematics","General Mathematics"],"type":"journal_article","department":[{"_id":"841"}],"date_created":"2023-07-10T12:32:10Z","publication_status":"published","date_updated":"2024-04-03T09:15:27Z","year":"2023","status":"public","title":"Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints","publication_identifier":{"issn":["0272-4979","1464-3642"]},"author":[{"last_name":"Bartels","first_name":"Sören","full_name":"Bartels, Sören"},{"id":"100441","full_name":"Kovács, Balázs","last_name":"Kovács","orcid":"0000-0001-9872-3474","first_name":"Balázs"},{"last_name":"Wang","first_name":"Zhangxian","full_name":"Wang, Zhangxian"}],"user_id":"100441","doi":"10.1093/imanum/drad037","language":[{"iso":"eng"}],"_id":"45971","publisher":"Oxford University Press (OUP)"},{"date_updated":"2024-04-03T09:12:47Z","author":[{"full_name":"Contri, Alessandro","last_name":"Contri","first_name":"Alessandro"},{"id":"100441","orcid":"0000-0001-9872-3474","last_name":"Kovács","first_name":"Balázs","full_name":"Kovács, Balázs"},{"last_name":"Massing","first_name":"André","full_name":"Massing, André"}],"year":"2023","status":"public","title":"Error analysis of BDF 1-6 time-stepping methods for the transient Stokes problem: velocity and pressure estimates","user_id":"100441","doi":"10.48550/ARXIV.2312.05511","_id":"53140","language":[{"iso":"eng"}],"abstract":[{"text":"We present a new stability and error analysis of fully discrete approximation schemes for the transient Stokes equation. For the spatial discretization, we consider a wide class of Galerkin finite element methods which includes both inf-sup stable spaces and symmetric pressure stabilized formulations. We extend the results from Burman and Fernández [\\textit{SIAM J. Numer. Anal.}, 47 (2009), pp. 409-439] and provide a unified theoretical analysis of backward difference formulae (BDF methods) of order 1 to 6. The main novelty of our approach lies in the use of Dahlquist's G-stability concept together with multiplier techniques introduced by Nevannlina-Odeh and recently by Akrivis et al. [\\textit{SIAM J. Numer. Anal.}, 59 (2021), pp. 2449-2472] to derive optimal stability and error estimates for both the velocity and the pressure. When combined with a method dependent Ritz projection for the initial data, unconditional stability can be shown while for arbitrary interpolation, pressure stability is subordinate to the fulfillment of a mild inverse CFL-type condition between space and time discretizations.","lang":"eng"}],"citation":{"mla":"Contri, Alessandro, et al. “Error Analysis of BDF 1-6 Time-Stepping Methods for the Transient Stokes Problem: Velocity and Pressure Estimates.” <i>ArXiv</i>, 2023, doi:<a href=\"https://doi.org/10.48550/ARXIV.2312.05511\">10.48550/ARXIV.2312.05511</a>.","ama":"Contri A, Kovács B, Massing A. Error analysis of BDF 1-6 time-stepping methods for the transient Stokes problem: velocity and pressure estimates. <i>arXiv</i>. Published online 2023. doi:<a href=\"https://doi.org/10.48550/ARXIV.2312.05511\">10.48550/ARXIV.2312.05511</a>","bibtex":"@article{Contri_Kovács_Massing_2023, title={Error analysis of BDF 1-6 time-stepping methods for the transient Stokes problem: velocity and pressure estimates}, DOI={<a href=\"https://doi.org/10.48550/ARXIV.2312.05511\">10.48550/ARXIV.2312.05511</a>}, journal={arXiv}, author={Contri, Alessandro and Kovács, Balázs and Massing, André}, year={2023} }","apa":"Contri, A., Kovács, B., &#38; Massing, A. (2023). Error analysis of BDF 1-6 time-stepping methods for the transient Stokes problem: velocity and pressure estimates. <i>ArXiv</i>. <a href=\"https://doi.org/10.48550/ARXIV.2312.05511\">https://doi.org/10.48550/ARXIV.2312.05511</a>","ieee":"A. Contri, B. Kovács, and A. Massing, “Error analysis of BDF 1-6 time-stepping methods for the transient Stokes problem: velocity and pressure estimates,” <i>arXiv</i>, 2023, doi: <a href=\"https://doi.org/10.48550/ARXIV.2312.05511\">10.48550/ARXIV.2312.05511</a>.","chicago":"Contri, Alessandro, Balázs Kovács, and André Massing. “Error Analysis of BDF 1-6 Time-Stepping Methods for the Transient Stokes Problem: Velocity and Pressure Estimates.” <i>ArXiv</i>, 2023. <a href=\"https://doi.org/10.48550/ARXIV.2312.05511\">https://doi.org/10.48550/ARXIV.2312.05511</a>.","short":"A. Contri, B. Kovács, A. Massing, ArXiv (2023)."},"publication":"arXiv","department":[{"_id":"841"}],"type":"journal_article","date_created":"2024-04-03T09:08:38Z"},{"status":"public","volume":175,"user_id":"46953","publisher":"Cambridge University Press (CUP)","_id":"53533","page":"129-160","citation":{"mla":"Ernst, Alena, and Kai-Uwe Schmidt. “Intersection Theorems for Finite General Linear Groups.” <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>, vol. 175, no. 1, Cambridge University Press (CUP), 2023, pp. 129–60, doi:<a href=\"https://doi.org/10.1017/s0305004123000075\">10.1017/s0305004123000075</a>.","ama":"Ernst A, Schmidt K-U. Intersection theorems for finite general linear groups. <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>. 2023;175(1):129-160. doi:<a href=\"https://doi.org/10.1017/s0305004123000075\">10.1017/s0305004123000075</a>","bibtex":"@article{Ernst_Schmidt_2023, title={Intersection theorems for finite general linear groups}, volume={175}, DOI={<a href=\"https://doi.org/10.1017/s0305004123000075\">10.1017/s0305004123000075</a>}, number={1}, journal={Mathematical Proceedings of the Cambridge Philosophical Society}, publisher={Cambridge University Press (CUP)}, author={Ernst, Alena and Schmidt, Kai-Uwe}, year={2023}, pages={129–160} }","apa":"Ernst, A., &#38; Schmidt, K.-U. (2023). Intersection theorems for finite general linear groups. <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>, <i>175</i>(1), 129–160. <a href=\"https://doi.org/10.1017/s0305004123000075\">https://doi.org/10.1017/s0305004123000075</a>","ieee":"A. Ernst and K.-U. Schmidt, “Intersection theorems for finite general linear groups,” <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>, vol. 175, no. 1, pp. 129–160, 2023, doi: <a href=\"https://doi.org/10.1017/s0305004123000075\">10.1017/s0305004123000075</a>.","short":"A. Ernst, K.-U. Schmidt, Mathematical Proceedings of the Cambridge Philosophical Society 175 (2023) 129–160.","chicago":"Ernst, Alena, and Kai-Uwe Schmidt. “Intersection Theorems for Finite General Linear Groups.” <i>Mathematical Proceedings of the Cambridge Philosophical Society</i> 175, no. 1 (2023): 129–60. <a href=\"https://doi.org/10.1017/s0305004123000075\">https://doi.org/10.1017/s0305004123000075</a>."},"intvolume":"       175","date_updated":"2024-05-07T08:29:59Z","publication_status":"published","author":[{"full_name":"Ernst, Alena","last_name":"Ernst","first_name":"Alena","id":"46953"},{"last_name":"Schmidt","first_name":"Kai-Uwe","full_name":"Schmidt, Kai-Uwe"}],"publication_identifier":{"issn":["0305-0041","1469-8064"]},"title":"Intersection theorems for finite general linear groups","year":"2023","doi":"10.1017/s0305004123000075","language":[{"iso":"eng"}],"issue":"1","publication":"Mathematical Proceedings of the Cambridge Philosophical Society","department":[{"_id":"100"}],"type":"journal_article","keyword":["General Mathematics"],"date_created":"2024-04-17T12:23:18Z"},{"type":"journal_article","department":[{"_id":"102"}],"date_created":"2024-07-16T11:09:01Z","extern":"1","issue":"10","publication":"Int. J. Number Theory","citation":{"short":"B. Klahn, M. Technau, Int. J. Number Theory 19 (2023) 2443–2450.","chicago":"Klahn, B., and Marc Technau. “Galois Groups of (<sup>n</sup>₀)+(<sup>n</sup>₁)X+…+(<sup>n</sup>₆)X<sup>6</sup>.” <i>Int. J. Number Theory</i> 19, no. 10 (2023): 2443–2450. <a href=\"https://doi.org/10.1142/S1793042123501208\">https://doi.org/10.1142/S1793042123501208</a>.","ieee":"B. Klahn and M. Technau, “Galois groups of (<sup>n</sup>₀)+(<sup>n</sup>₁)X+…+(<sup>n</sup>₆)X<sup>6</sup>,” <i>Int. J. Number Theory</i>, vol. 19, no. 10, pp. 2443–2450, 2023, doi: <a href=\"https://doi.org/10.1142/S1793042123501208\">10.1142/S1793042123501208</a>.","apa":"Klahn, B., &#38; Technau, M. (2023). Galois groups of (<sup>n</sup>₀)+(<sup>n</sup>₁)X+…+(<sup>n</sup>₆)X<sup>6</sup>. <i>Int. J. Number Theory</i>, <i>19</i>(10), 2443–2450. <a href=\"https://doi.org/10.1142/S1793042123501208\">https://doi.org/10.1142/S1793042123501208</a>","bibtex":"@article{Klahn_Technau_2023, title={Galois groups of (<sup>n</sup>₀)+(<sup>n</sup>₁)X+…+(<sup>n</sup>₆)X<sup>6</sup>}, volume={19}, DOI={<a href=\"https://doi.org/10.1142/S1793042123501208\">10.1142/S1793042123501208</a>}, number={10}, journal={Int. J. Number Theory}, author={Klahn, B. and Technau, Marc}, year={2023}, pages={2443–2450} }","ama":"Klahn B, Technau M. Galois groups of (<sup>n</sup>₀)+(<sup>n</sup>₁)X+…+(<sup>n</sup>₆)X<sup>6</sup>. <i>Int J Number Theory</i>. 2023;19(10):2443–2450. doi:<a href=\"https://doi.org/10.1142/S1793042123501208\">10.1142/S1793042123501208</a>","mla":"Klahn, B., and Marc Technau. “Galois Groups of (<sup>n</sup>₀)+(<sup>n</sup>₁)X+…+(<sup>n</sup>₆)X<sup>6</sup>.” <i>Int. J. Number Theory</i>, vol. 19, no. 10, 2023, pp. 2443–2450, doi:<a href=\"https://doi.org/10.1142/S1793042123501208\">10.1142/S1793042123501208</a>."},"user_id":"106108","doi":"10.1142/S1793042123501208","volume":19,"page":"2443–2450","_id":"55277","language":[{"iso":"eng"}],"date_updated":"2024-07-24T07:23:33Z","intvolume":"        19","title":"Galois groups of (ⁿ₀)+(ⁿ₁)X+…+(ⁿ₆)X⁶","status":"public","year":"2023","author":[{"full_name":"Klahn, B.","first_name":"B.","last_name":"Klahn"},{"full_name":"Technau, Marc","first_name":"Marc","orcid":"0000-0001-9650-2459","last_name":"Technau","id":"106108"}]},{"extern":"1","publication":"Math. Ann.","citation":{"ama":"Minelli P, Sourmelidis A, Technau M. Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sums. <i>Math Ann</i>. 2023;387:291–320. doi:<a href=\"https://doi.org/10.1007/s00208-022-02452-2\">10.1007/s00208-022-02452-2</a>","bibtex":"@article{Minelli_Sourmelidis_Technau_2023, title={Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sums}, volume={387}, DOI={<a href=\"https://doi.org/10.1007/s00208-022-02452-2\">10.1007/s00208-022-02452-2</a>}, journal={Math. Ann.}, author={Minelli, P. and Sourmelidis, A. and Technau, Marc}, year={2023}, pages={291–320} }","mla":"Minelli, P., et al. “Bias in the Number of Steps in the Euclidean Algorithm and a Conjecture of Ito on Dedekind Sums.” <i>Math. Ann.</i>, vol. 387, 2023, pp. 291–320, doi:<a href=\"https://doi.org/10.1007/s00208-022-02452-2\">10.1007/s00208-022-02452-2</a>.","chicago":"Minelli, P., A. Sourmelidis, and Marc Technau. “Bias in the Number of Steps in the Euclidean Algorithm and a Conjecture of Ito on Dedekind Sums.” <i>Math. Ann.</i> 387 (2023): 291–320. <a href=\"https://doi.org/10.1007/s00208-022-02452-2\">https://doi.org/10.1007/s00208-022-02452-2</a>.","short":"P. Minelli, A. Sourmelidis, M. Technau, Math. Ann. 387 (2023) 291–320.","apa":"Minelli, P., Sourmelidis, A., &#38; Technau, M. (2023). Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sums. <i>Math. Ann.</i>, <i>387</i>, 291–320. <a href=\"https://doi.org/10.1007/s00208-022-02452-2\">https://doi.org/10.1007/s00208-022-02452-2</a>","ieee":"P. Minelli, A. Sourmelidis, and M. Technau, “Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sums,” <i>Math. Ann.</i>, vol. 387, pp. 291–320, 2023, doi: <a href=\"https://doi.org/10.1007/s00208-022-02452-2\">10.1007/s00208-022-02452-2</a>."},"type":"journal_article","department":[{"_id":"102"}],"date_created":"2024-07-16T11:09:01Z","date_updated":"2024-07-24T07:26:05Z","intvolume":"       387","year":"2023","title":"Bias in the number of steps in the Euclidean algorithm and a conjecture of Ito on Dedekind sums","status":"public","author":[{"full_name":"Minelli, P.","last_name":"Minelli","first_name":"P."},{"full_name":"Sourmelidis, A.","first_name":"A.","last_name":"Sourmelidis"},{"id":"106108","first_name":"Marc","orcid":"0000-0001-9650-2459","last_name":"Technau","full_name":"Technau, Marc"}],"user_id":"106108","doi":"10.1007/s00208-022-02452-2","volume":387,"page":"291–320","_id":"55279","language":[{"iso":"eng"}]},{"user_id":"85279","ddc":["510"],"volume":14071,"editor":[{"full_name":"Nielsen, F","first_name":"F","last_name":"Nielsen"},{"first_name":"F","last_name":"Barbaresco","full_name":"Barbaresco, F"}],"page":"569-579","_id":"42163","publisher":"Springer, Cham.","has_accepted_license":"1","status":"public","conference":{"name":"  GSI'23 6th International Conference on Geometric Science of Information","start_date":"2023-08-30","location":"Saint-Malo, Palais du Grand Large, France","end_date":"2023-09-01"},"oa":"1","external_id":{"arxiv":["2302.08232 "]},"quality_controlled":"1","project":[{"_id":"52","name":"PC2: Computing Resources Provided by the Paderborn Center for Parallel Computing"}],"file_date_updated":"2023-08-02T12:04:17Z","citation":{"ieee":"C. Offen and S. Ober-Blöbaum, “Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves,” in <i>Geometric Science of Information</i>, Saint-Malo, Palais du Grand Large, France, 2023, vol. 14071, pp. 569–579, doi: <a href=\"https://doi.org/10.1007/978-3-031-38271-0_57\">10.1007/978-3-031-38271-0_57</a>.","apa":"Offen, C., &#38; Ober-Blöbaum, S. (2023). Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves. In F. Nielsen &#38; F. Barbaresco (Eds.), <i>Geometric Science of Information</i> (Vol. 14071, pp. 569–579). Springer, Cham. <a href=\"https://doi.org/10.1007/978-3-031-38271-0_57\">https://doi.org/10.1007/978-3-031-38271-0_57</a>","chicago":"Offen, Christian, and Sina Ober-Blöbaum. “Learning Discrete Lagrangians for Variational PDEs from Data and Detection of Travelling Waves.” In <i>Geometric Science of Information</i>, edited by F Nielsen and F Barbaresco, 14071:569–79. Lecture Notes in Computer Science (LNCS). Springer, Cham., 2023. <a href=\"https://doi.org/10.1007/978-3-031-38271-0_57\">https://doi.org/10.1007/978-3-031-38271-0_57</a>.","short":"C. Offen, S. Ober-Blöbaum, in: F. Nielsen, F. Barbaresco (Eds.), Geometric Science of Information, Springer, Cham., 2023, pp. 569–579.","mla":"Offen, Christian, and Sina Ober-Blöbaum. “Learning Discrete Lagrangians for Variational PDEs from Data and Detection of Travelling Waves.” <i>Geometric Science of Information</i>, edited by F Nielsen and F Barbaresco, vol. 14071, Springer, Cham., 2023, pp. 569–79, doi:<a href=\"https://doi.org/10.1007/978-3-031-38271-0_57\">10.1007/978-3-031-38271-0_57</a>.","bibtex":"@inproceedings{Offen_Ober-Blöbaum_2023, series={Lecture Notes in Computer Science (LNCS)}, title={Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves}, volume={14071}, DOI={<a href=\"https://doi.org/10.1007/978-3-031-38271-0_57\">10.1007/978-3-031-38271-0_57</a>}, booktitle={Geometric Science of Information}, publisher={Springer, Cham.}, author={Offen, Christian and Ober-Blöbaum, Sina}, editor={Nielsen, F and Barbaresco, F}, year={2023}, pages={569–579}, collection={Lecture Notes in Computer Science (LNCS)} }","ama":"Offen C, Ober-Blöbaum S. Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves. In: Nielsen F, Barbaresco F, eds. <i>Geometric Science of Information</i>. Vol 14071. Lecture Notes in Computer Science (LNCS). Springer, Cham.; 2023:569-579. doi:<a href=\"https://doi.org/10.1007/978-3-031-38271-0_57\">10.1007/978-3-031-38271-0_57</a>"},"doi":"10.1007/978-3-031-38271-0_57","language":[{"iso":"eng"}],"series_title":"Lecture Notes in Computer Science (LNCS)","publication_status":"published","date_updated":"2024-08-12T13:46:29Z","intvolume":"     14071","year":"2023","title":"Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves","publication_identifier":{"eisbn":["978-3-031-38271-0"]},"author":[{"id":"85279","full_name":"Offen, Christian","first_name":"Christian","orcid":"0000-0002-5940-8057","last_name":"Offen"},{"full_name":"Ober-Blöbaum, Sina","last_name":"Ober-Blöbaum","first_name":"Sina","id":"16494"}],"type":"conference","keyword":["System identification","discrete Lagrangians","travelling waves"],"department":[{"_id":"636"}],"file":[{"date_created":"2023-08-02T12:04:17Z","description":"The article shows how to learn models of dynamical systems\nfrom data which are governed by an unknown variational PDE. Rather\nthan employing reduction techniques, we learn a discrete field theory\ngoverned by a discrete Lagrangian density Ld that is modelled as a neural network. Careful regularisation of the loss function for training Ld is\nnecessary to obtain a field theory that is suitable for numerical computations: we derive a regularisation term which optimises the solvability of\nthe discrete Euler–Lagrange equations. Secondly, we develop a method to\nfind solutions to machine learned discrete field theories which constitute\ntravelling waves of the underlying continuous PDE.","creator":"coffen","file_id":"46273","content_type":"application/pdf","title":"Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves","file_name":"LDensityLearning.pdf","file_size":1938962,"access_level":"open_access","relation":"main_file","date_updated":"2023-08-02T12:04:17Z"}],"date_created":"2023-02-16T11:32:48Z","related_material":{"link":[{"description":"GitHub","url":"https://github.com/Christian-Offen/LagrangianDensityML","relation":"software"}]},"abstract":[{"lang":"eng","text":"The article shows how to learn models of dynamical systems from data which are governed by an unknown variational PDE. Rather than employing reduction techniques, we learn a discrete field theory governed by a discrete Lagrangian density $L_d$ that is modelled as a neural network. Careful regularisation of the loss function for training $L_d$ is necessary to obtain a field theory that is suitable for numerical computations: we derive a regularisation term which optimises the solvability of the discrete Euler--Lagrange equations. Secondly, we develop a method to find solutions to machine learned discrete field theories which constitute travelling waves of the underlying continuous PDE."}],"publication":"Geometric Science of Information"},{"main_file_link":[{"open_access":"1","url":"https://link.springer.com/content/pdf/10.1007/s10898-022-01223-2.pdf"}],"language":[{"iso":"eng"}],"doi":"10.1007/s10898-022-01223-2","title":"On the structure of regularization paths for piecewise differentiable regularization terms","year":"2023","author":[{"last_name":"Gebken","first_name":"Bennet","full_name":"Gebken, Bennet","id":"32643"},{"last_name":"Bieker","first_name":"Katharina","full_name":"Bieker, Katharina","id":"32829"},{"id":"47427","orcid":"0000-0002-3389-793X","last_name":"Peitz","first_name":"Sebastian","full_name":"Peitz, Sebastian"}],"date_updated":"2023-03-11T17:16:33Z","intvolume":"        85","date_created":"2021-11-15T09:24:59Z","type":"journal_article","department":[{"_id":"101"},{"_id":"655"}],"publication":"Journal of Global Optimization","issue":"3","abstract":[{"text":"Regularization is used in many different areas of optimization when solutions\r\nare sought which not only minimize a given function, but also possess a certain\r\ndegree of regularity. Popular applications are image denoising, sparse\r\nregression and machine learning. Since the choice of the regularization\r\nparameter is crucial but often difficult, path-following methods are used to\r\napproximate the entire regularization path, i.e., the set of all possible\r\nsolutions for all regularization parameters. Due to their nature, the\r\ndevelopment of these methods requires structural results about the\r\nregularization path. The goal of this article is to derive these results for\r\nthe case of a smooth objective function which is penalized by a piecewise\r\ndifferentiable regularization term. We do this by treating regularization as a\r\nmultiobjective optimization problem. Our results suggest that even in this\r\ngeneral case, the regularization path is piecewise smooth. Moreover, our theory\r\nallows for a classification of the nonsmooth features that occur in between\r\nsmooth parts. This is demonstrated in two applications, namely support-vector\r\nmachines and exact penalty methods.","lang":"eng"}],"page":"709-741","_id":"27426","user_id":"47427","volume":85,"status":"public","oa":"1","citation":{"ieee":"B. Gebken, K. Bieker, and S. Peitz, “On the structure of regularization paths for piecewise differentiable regularization terms,” <i>Journal of Global Optimization</i>, vol. 85, no. 3, pp. 709–741, 2023, doi: <a href=\"https://doi.org/10.1007/s10898-022-01223-2\">10.1007/s10898-022-01223-2</a>.","apa":"Gebken, B., Bieker, K., &#38; Peitz, S. (2023). On the structure of regularization paths for piecewise differentiable regularization terms. <i>Journal of Global Optimization</i>, <i>85</i>(3), 709–741. <a href=\"https://doi.org/10.1007/s10898-022-01223-2\">https://doi.org/10.1007/s10898-022-01223-2</a>","chicago":"Gebken, Bennet, Katharina Bieker, and Sebastian Peitz. “On the Structure of Regularization Paths for Piecewise Differentiable Regularization Terms.” <i>Journal of Global Optimization</i> 85, no. 3 (2023): 709–41. <a href=\"https://doi.org/10.1007/s10898-022-01223-2\">https://doi.org/10.1007/s10898-022-01223-2</a>.","short":"B. Gebken, K. Bieker, S. Peitz, Journal of Global Optimization 85 (2023) 709–741.","mla":"Gebken, Bennet, et al. “On the Structure of Regularization Paths for Piecewise Differentiable Regularization Terms.” <i>Journal of Global Optimization</i>, vol. 85, no. 3, 2023, pp. 709–41, doi:<a href=\"https://doi.org/10.1007/s10898-022-01223-2\">10.1007/s10898-022-01223-2</a>.","bibtex":"@article{Gebken_Bieker_Peitz_2023, title={On the structure of regularization paths for piecewise differentiable regularization terms}, volume={85}, DOI={<a href=\"https://doi.org/10.1007/s10898-022-01223-2\">10.1007/s10898-022-01223-2</a>}, number={3}, journal={Journal of Global Optimization}, author={Gebken, Bennet and Bieker, Katharina and Peitz, Sebastian}, year={2023}, pages={709–741} }","ama":"Gebken B, Bieker K, Peitz S. On the structure of regularization paths for piecewise differentiable regularization terms. <i>Journal of Global Optimization</i>. 2023;85(3):709-741. doi:<a href=\"https://doi.org/10.1007/s10898-022-01223-2\">10.1007/s10898-022-01223-2</a>"}},{"citation":{"chicago":"Silva, Helmuth O.M., Diego P. Rubert, Eloi Araujo, Eckhard Steffen, Daniel Doerr, and Fábio V. Martinez. “Algorithms for the Genome Median under a Restricted Measure of Rearrangement.” <i>RAIRO - Operations Research</i> 57, no. 3 (2023): 1045–58. <a href=\"https://doi.org/10.1051/ro/2023052\">https://doi.org/10.1051/ro/2023052</a>.","short":"H.O.M. Silva, D.P. Rubert, E. Araujo, E. Steffen, D. Doerr, F.V. Martinez, RAIRO - Operations Research 57 (2023) 1045–1058.","apa":"Silva, H. O. M., Rubert, D. P., Araujo, E., Steffen, E., Doerr, D., &#38; Martinez, F. V. (2023). Algorithms for the genome median under a restricted measure of rearrangement. <i>RAIRO - Operations Research</i>, <i>57</i>(3), 1045–1058. <a href=\"https://doi.org/10.1051/ro/2023052\">https://doi.org/10.1051/ro/2023052</a>","ieee":"H. O. M. Silva, D. P. Rubert, E. Araujo, E. Steffen, D. Doerr, and F. V. Martinez, “Algorithms for the genome median under a restricted measure of rearrangement,” <i>RAIRO - Operations Research</i>, vol. 57, no. 3, pp. 1045–1058, 2023, doi: <a href=\"https://doi.org/10.1051/ro/2023052\">10.1051/ro/2023052</a>.","ama":"Silva HOM, Rubert DP, Araujo E, Steffen E, Doerr D, Martinez FV. Algorithms for the genome median under a restricted measure of rearrangement. <i>RAIRO - Operations Research</i>. 2023;57(3):1045-1058. doi:<a href=\"https://doi.org/10.1051/ro/2023052\">10.1051/ro/2023052</a>","bibtex":"@article{Silva_Rubert_Araujo_Steffen_Doerr_Martinez_2023, title={Algorithms for the genome median under a restricted measure of rearrangement}, volume={57}, DOI={<a href=\"https://doi.org/10.1051/ro/2023052\">10.1051/ro/2023052</a>}, number={3}, journal={RAIRO - Operations Research}, publisher={EDP Sciences}, author={Silva, Helmuth O.M. and Rubert, Diego P. and Araujo, Eloi and Steffen, Eckhard and Doerr, Daniel and Martinez, Fábio V.}, year={2023}, pages={1045–1058} }","mla":"Silva, Helmuth O. M., et al. “Algorithms for the Genome Median under a Restricted Measure of Rearrangement.” <i>RAIRO - Operations Research</i>, vol. 57, no. 3, EDP Sciences, 2023, pp. 1045–58, doi:<a href=\"https://doi.org/10.1051/ro/2023052\">10.1051/ro/2023052</a>."},"page":"1045-1058","publisher":"EDP Sciences","_id":"44857","user_id":"15540","volume":57,"status":"public","date_created":"2023-05-16T08:48:22Z","type":"journal_article","keyword":["Management Science and Operations Research","Computer Science Applications","Theoretical Computer Science"],"department":[{"_id":"542"}],"publication":"RAIRO - Operations Research","issue":"3","abstract":[{"lang":"eng","text":"Ancestral reconstruction is a classic task in comparative genomics. Here, we study the genome median problem, a related computational problem which, given a set of three or more genomes, asks to find a new genome that minimizes the sum of pairwise distances between it and the given genomes. The distance stands for the amount of evolution observed at the genome level, for which we determine the minimum number of rearrangement operations necessary to transform one genome into the other. For almost all rearrangement operations the median problem is NP-hard, with the exception of the breakpoint median that can be constructed efficiently for multichromosomal circular and mixed genomes. In this work, we study the median problem under a restricted rearrangement measure called c4-distance, which is closely related to the breakpoint and the DCJ distance. We identify tight bounds and decomposers of the c4-median and develop algorithms for its construction, one exact ILP-based and three combinatorial heuristics. Subsequently, we perform experiments on simulated data sets. Our results suggest that the c4-distance is useful for the study the genome median problem, from theoretical and practical perspectives."}],"language":[{"iso":"eng"}],"doi":"10.1051/ro/2023052","title":"Algorithms for the genome median under a restricted measure of rearrangement","year":"2023","publication_identifier":{"issn":["0399-0559","2804-7303"]},"author":[{"full_name":"Silva, Helmuth O.M.","last_name":"Silva","first_name":"Helmuth O.M."},{"first_name":"Diego P.","last_name":"Rubert","full_name":"Rubert, Diego P."},{"first_name":"Eloi","last_name":"Araujo","full_name":"Araujo, Eloi"},{"id":"15548","full_name":"Steffen, Eckhard","first_name":"Eckhard","last_name":"Steffen","orcid":"0000-0002-9808-7401"},{"first_name":"Daniel","last_name":"Doerr","full_name":"Doerr, Daniel"},{"full_name":"Martinez, Fábio V.","first_name":"Fábio V.","last_name":"Martinez"}],"date_updated":"2023-05-16T08:49:30Z","publication_status":"published","intvolume":"        57"},{"citation":{"apa":"Ma, Y., Mattiolo, D., Steffen, E., &#38; Wolf, I. H. (2023). Sets of r-graphs that color all r-graphs. In <i>arXiv:2305.08619</i>.","ieee":"Y. Ma, D. Mattiolo, E. Steffen, and I. H. Wolf, “Sets of r-graphs that color all r-graphs,” <i>arXiv:2305.08619</i>. 2023.","chicago":"Ma, Yulai, Davide Mattiolo, Eckhard Steffen, and Isaak Hieronymus Wolf. “Sets of R-Graphs That Color All r-Graphs.” <i>ArXiv:2305.08619</i>, 2023.","short":"Y. Ma, D. Mattiolo, E. Steffen, I.H. Wolf, ArXiv:2305.08619 (2023).","mla":"Ma, Yulai, et al. “Sets of R-Graphs That Color All r-Graphs.” <i>ArXiv:2305.08619</i>, 2023.","ama":"Ma Y, Mattiolo D, Steffen E, Wolf IH. Sets of r-graphs that color all r-graphs. <i>arXiv:230508619</i>. Published online 2023.","bibtex":"@article{Ma_Mattiolo_Steffen_Wolf_2023, title={Sets of r-graphs that color all r-graphs}, journal={arXiv:2305.08619}, author={Ma, Yulai and Mattiolo, Davide and Steffen, Eckhard and Wolf, Isaak Hieronymus}, year={2023} }"},"publication":"arXiv:2305.08619","date_created":"2023-05-16T10:07:47Z","external_id":{"arxiv":["2305.08619"]},"department":[{"_id":"542"}],"type":"preprint","author":[{"id":"92748","full_name":"Ma, Yulai","last_name":"Ma","first_name":"Yulai"},{"full_name":"Mattiolo, Davide","last_name":"Mattiolo","first_name":"Davide"},{"id":"15548","orcid":"0000-0002-9808-7401","first_name":"Eckhard","last_name":"Steffen","full_name":"Steffen, Eckhard"},{"full_name":"Wolf, Isaak Hieronymus","last_name":"Wolf","first_name":"Isaak Hieronymus","id":"88145"}],"status":"public","title":"Sets of r-graphs that color all r-graphs","year":"2023","date_updated":"2023-05-16T11:17:26Z","language":[{"iso":"eng"}],"_id":"44859","user_id":"15540"},{"_id":"45498","language":[{"iso":"eng"}],"main_file_link":[{"url":"https://arxiv.org/pdf/2306.03320"}],"page":"29","user_id":"97359","author":[{"first_name":"Sören","last_name":"von der Gracht","orcid":"0000-0002-8054-2058","full_name":"von der Gracht, Sören","id":"97359"},{"full_name":"Nijholt, Eddie","last_name":"Nijholt","first_name":"Eddie"},{"first_name":"Bob","last_name":"Rink","full_name":"Rink, Bob"}],"year":"2023","status":"public","title":"A parametrisation method for high-order phase reduction in coupled  oscillator networks","publication_status":"submitted","date_updated":"2023-06-07T07:59:06Z","date_created":"2023-06-07T07:57:28Z","external_id":{"arxiv":["2306.03320"]},"department":[{"_id":"101"}],"type":"preprint","citation":{"ama":"von der Gracht S, Nijholt E, Rink B. A parametrisation method for high-order phase reduction in coupled  oscillator networks. <i>arXiv:230603320</i>.","bibtex":"@article{von der Gracht_Nijholt_Rink, title={A parametrisation method for high-order phase reduction in coupled  oscillator networks}, journal={arXiv:2306.03320}, author={von der Gracht, Sören and Nijholt, Eddie and Rink, Bob} }","mla":"von der Gracht, Sören, et al. “A Parametrisation Method for High-Order Phase Reduction in Coupled  Oscillator Networks.” <i>ArXiv:2306.03320</i>.","short":"S. von der Gracht, E. Nijholt, B. Rink, ArXiv:2306.03320 (n.d.).","chicago":"Gracht, Sören von der, Eddie Nijholt, and Bob Rink. “A Parametrisation Method for High-Order Phase Reduction in Coupled  Oscillator Networks.” <i>ArXiv:2306.03320</i>, n.d.","apa":"von der Gracht, S., Nijholt, E., &#38; Rink, B. (n.d.). A parametrisation method for high-order phase reduction in coupled  oscillator networks. In <i>arXiv:2306.03320</i>.","ieee":"S. von der Gracht, E. Nijholt, and B. Rink, “A parametrisation method for high-order phase reduction in coupled  oscillator networks,” <i>arXiv:2306.03320</i>. ."},"publication":"arXiv:2306.03320","abstract":[{"text":"We present a novel method for high-order phase reduction in networks of\r\nweakly coupled oscillators and, more generally, perturbations of reducible\r\nnormally hyperbolic (quasi-)periodic tori. Our method works by computing an\r\nasymptotic expansion for an embedding of the perturbed invariant torus, as well\r\nas for the reduced phase dynamics in local coordinates. Both can be determined\r\nto arbitrary degrees of accuracy, and we show that the phase dynamics may\r\ndirectly be obtained in normal form. We apply the method to predict remote\r\nsynchronisation in a chain of coupled Stuart-Landau oscillators.","lang":"eng"}]},{"date_created":"2023-08-01T10:08:32Z","type":"journal_article","keyword":["General Mathematics"],"department":[{"_id":"542"}],"publication":"SIAM Journal on Discrete Mathematics","issue":"3","language":[{"iso":"eng"}],"doi":"10.1137/22m1500654","title":"Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs","year":"2023","publication_identifier":{"issn":["0895-4801","1095-7146"]},"author":[{"first_name":"Yulai","last_name":"Ma","full_name":"Ma, Yulai","id":"92748"},{"first_name":"Davide","last_name":"Mattiolo","full_name":"Mattiolo, Davide"},{"id":"15548","full_name":"Steffen, Eckhard","last_name":"Steffen","first_name":"Eckhard","orcid":"0000-0002-9808-7401"},{"id":"88145","first_name":"Isaak Hieronymus","last_name":"Wolf","full_name":"Wolf, Isaak Hieronymus"}],"date_updated":"2023-08-01T10:09:35Z","publication_status":"published","intvolume":"        37","citation":{"mla":"Ma, Yulai, et al. “Pairwise Disjoint Perfect Matchings in R-Edge-Connected r-Regular Graphs.” <i>SIAM Journal on Discrete Mathematics</i>, vol. 37, no. 3, Society for Industrial &#38; Applied Mathematics (SIAM), 2023, pp. 1548–65, doi:<a href=\"https://doi.org/10.1137/22m1500654\">10.1137/22m1500654</a>.","bibtex":"@article{Ma_Mattiolo_Steffen_Wolf_2023, title={Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs}, volume={37}, DOI={<a href=\"https://doi.org/10.1137/22m1500654\">10.1137/22m1500654</a>}, number={3}, journal={SIAM Journal on Discrete Mathematics}, publisher={Society for Industrial &#38; Applied Mathematics (SIAM)}, author={Ma, Yulai and Mattiolo, Davide and Steffen, Eckhard and Wolf, Isaak Hieronymus}, year={2023}, pages={1548–1565} }","ama":"Ma Y, Mattiolo D, Steffen E, Wolf IH. Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs. <i>SIAM Journal on Discrete Mathematics</i>. 2023;37(3):1548-1565. doi:<a href=\"https://doi.org/10.1137/22m1500654\">10.1137/22m1500654</a>","ieee":"Y. Ma, D. Mattiolo, E. Steffen, and I. H. Wolf, “Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs,” <i>SIAM Journal on Discrete Mathematics</i>, vol. 37, no. 3, pp. 1548–1565, 2023, doi: <a href=\"https://doi.org/10.1137/22m1500654\">10.1137/22m1500654</a>.","apa":"Ma, Y., Mattiolo, D., Steffen, E., &#38; Wolf, I. H. (2023). Pairwise Disjoint Perfect Matchings in r-Edge-Connected r-Regular Graphs. <i>SIAM Journal on Discrete Mathematics</i>, <i>37</i>(3), 1548–1565. <a href=\"https://doi.org/10.1137/22m1500654\">https://doi.org/10.1137/22m1500654</a>","short":"Y. Ma, D. Mattiolo, E. Steffen, I.H. Wolf, SIAM Journal on Discrete Mathematics 37 (2023) 1548–1565.","chicago":"Ma, Yulai, Davide Mattiolo, Eckhard Steffen, and Isaak Hieronymus Wolf. “Pairwise Disjoint Perfect Matchings in R-Edge-Connected r-Regular Graphs.” <i>SIAM Journal on Discrete Mathematics</i> 37, no. 3 (2023): 1548–65. <a href=\"https://doi.org/10.1137/22m1500654\">https://doi.org/10.1137/22m1500654</a>."},"page":"1548-1565","_id":"46256","publisher":"Society for Industrial & Applied Mathematics (SIAM)","user_id":"15540","volume":37,"status":"public"},{"volume":421,"ddc":["510"],"user_id":"85279","publisher":"Elsevier","_id":"29240","page":"114780","has_accepted_license":"1","status":"public","oa":"1","external_id":{"arxiv":["2112.12619"]},"quality_controlled":"1","citation":{"mla":"Ober-Blöbaum, Sina, and Christian Offen. “Variational Learning of Euler–Lagrange Dynamics from Data.” <i>Journal of Computational and Applied Mathematics</i>, vol. 421, Elsevier, 2023, p. 114780, doi:<a href=\"https://doi.org/10.1016/j.cam.2022.114780\">10.1016/j.cam.2022.114780</a>.","ama":"Ober-Blöbaum S, Offen C. Variational Learning of Euler–Lagrange Dynamics from Data. <i>Journal of Computational and Applied Mathematics</i>. 2023;421:114780. doi:<a href=\"https://doi.org/10.1016/j.cam.2022.114780\">10.1016/j.cam.2022.114780</a>","bibtex":"@article{Ober-Blöbaum_Offen_2023, title={Variational Learning of Euler–Lagrange Dynamics from Data}, volume={421}, DOI={<a href=\"https://doi.org/10.1016/j.cam.2022.114780\">10.1016/j.cam.2022.114780</a>}, journal={Journal of Computational and Applied Mathematics}, publisher={Elsevier}, author={Ober-Blöbaum, Sina and Offen, Christian}, year={2023}, pages={114780} }","apa":"Ober-Blöbaum, S., &#38; Offen, C. (2023). Variational Learning of Euler–Lagrange Dynamics from Data. <i>Journal of Computational and Applied Mathematics</i>, <i>421</i>, 114780. <a href=\"https://doi.org/10.1016/j.cam.2022.114780\">https://doi.org/10.1016/j.cam.2022.114780</a>","ieee":"S. Ober-Blöbaum and C. Offen, “Variational Learning of Euler–Lagrange Dynamics from Data,” <i>Journal of Computational and Applied Mathematics</i>, vol. 421, p. 114780, 2023, doi: <a href=\"https://doi.org/10.1016/j.cam.2022.114780\">10.1016/j.cam.2022.114780</a>.","short":"S. Ober-Blöbaum, C. Offen, Journal of Computational and Applied Mathematics 421 (2023) 114780.","chicago":"Ober-Blöbaum, Sina, and Christian Offen. “Variational Learning of Euler–Lagrange Dynamics from Data.” <i>Journal of Computational and Applied Mathematics</i> 421 (2023): 114780. <a href=\"https://doi.org/10.1016/j.cam.2022.114780\">https://doi.org/10.1016/j.cam.2022.114780</a>."},"file_date_updated":"2022-06-28T15:25:50Z","doi":"10.1016/j.cam.2022.114780","language":[{"iso":"eng"}],"intvolume":"       421","article_type":"original","date_updated":"2023-08-10T08:42:39Z","publication_status":"epub_ahead","publication_identifier":{"issn":["0377-0427"]},"author":[{"id":"16494","full_name":"Ober-Blöbaum, Sina","last_name":"Ober-Blöbaum","first_name":"Sina"},{"full_name":"Offen, Christian","orcid":"0000-0002-5940-8057","last_name":"Offen","first_name":"Christian","id":"85279"}],"year":"2023","title":"Variational Learning of Euler–Lagrange Dynamics from Data","department":[{"_id":"636"}],"keyword":["Lagrangian learning","variational backward error analysis","modified Lagrangian","variational integrators","physics informed learning"],"type":"journal_article","date_created":"2022-01-11T13:24:00Z","file":[{"creator":"coffen","description":"The principle of least action is one of the most fundamental physical principle. It says that among all possible motions\nconnecting two points in a phase space, the system will exhibit those motions which extremise an action functional.\nMany qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equa-\ntions, are related to the existence of an action functional. Incorporating variational structure into learning algorithms\nfor dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features\nwith the exact physical system. In this paper we show how to incorporate variational principles into trajectory predic-\ntions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position\ndata of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no\nprior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward\nerror analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the\nlearned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this,\nwe introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of\nvariational backward error analysis. (3) Finally, we introduce a method to perform system identification from position\nobservations only, based on variational backward error analysis.","date_created":"2022-06-28T15:25:50Z","relation":"main_file","date_updated":"2022-06-28T15:25:50Z","file_name":"ShadowLagrangian_revision1_journal_style_arxiv.pdf","access_level":"open_access","file_size":3640770,"title":"Variational Learning of Euler–Lagrange Dynamics from Data","file_id":"32274","content_type":"application/pdf"}],"related_material":{"link":[{"url":"https://github.com/Christian-Offen/LagrangianShadowIntegration","relation":"software"}]},"abstract":[{"text":"The principle of least action is one of the most fundamental physical principle. It says that among all possible motions connecting two points in a phase space, the system will exhibit those motions which extremise an action functional. Many qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equations, are related to the existence of an action functional. Incorporating variational structure into learning algorithms for dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features with the exact physical system. In this paper we show how to incorporate variational principles into trajectory predictions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position data of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no prior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward error analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the learned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this,\r\nwe introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of variational backward error analysis. (3) Finally, we introduce a method to perform system identification from position observations only, based on variational backward error analysis.","lang":"eng"}],"publication":"Journal of Computational and Applied Mathematics"},{"external_id":{"arxiv":["2201.03911"]},"oa":"1","file_date_updated":"2022-08-12T16:48:59Z","citation":{"bibtex":"@article{McLachlan_Offen_2023, title={Backward error analysis for conjugate symplectic methods}, volume={15}, DOI={<a href=\"https://doi.org/10.3934/jgm.2023005\">10.3934/jgm.2023005</a>}, number={1}, journal={Journal of Geometric Mechanics}, publisher={AIMS Press}, author={McLachlan, Robert and Offen, Christian}, year={2023}, pages={98–115} }","ama":"McLachlan R, Offen C. Backward error analysis for conjugate symplectic methods. <i>Journal of Geometric Mechanics</i>. 2023;15(1):98-115. doi:<a href=\"https://doi.org/10.3934/jgm.2023005\">10.3934/jgm.2023005</a>","mla":"McLachlan, Robert, and Christian Offen. “Backward Error Analysis for Conjugate Symplectic Methods.” <i>Journal of Geometric Mechanics</i>, vol. 15, no. 1, AIMS Press, 2023, pp. 98–115, doi:<a href=\"https://doi.org/10.3934/jgm.2023005\">10.3934/jgm.2023005</a>.","short":"R. McLachlan, C. Offen, Journal of Geometric Mechanics 15 (2023) 98–115.","chicago":"McLachlan, Robert, and Christian Offen. “Backward Error Analysis for Conjugate Symplectic Methods.” <i>Journal of Geometric Mechanics</i> 15, no. 1 (2023): 98–115. <a href=\"https://doi.org/10.3934/jgm.2023005\">https://doi.org/10.3934/jgm.2023005</a>.","ieee":"R. McLachlan and C. Offen, “Backward error analysis for conjugate symplectic methods,” <i>Journal of Geometric Mechanics</i>, vol. 15, no. 1, pp. 98–115, 2023, doi: <a href=\"https://doi.org/10.3934/jgm.2023005\">10.3934/jgm.2023005</a>.","apa":"McLachlan, R., &#38; Offen, C. (2023). Backward error analysis for conjugate symplectic methods. <i>Journal of Geometric Mechanics</i>, <i>15</i>(1), 98–115. <a href=\"https://doi.org/10.3934/jgm.2023005\">https://doi.org/10.3934/jgm.2023005</a>"},"quality_controlled":"1","page":"98-115","publisher":"AIMS Press","_id":"29236","user_id":"85279","ddc":["510"],"volume":15,"status":"public","has_accepted_license":"1","file":[{"description":"The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.","date_created":"2022-08-12T16:48:59Z","creator":"coffen","title":"Backward error analysis for conjugate symplectic methods","content_type":"application/pdf","file_id":"32801","date_updated":"2022-08-12T16:48:59Z","relation":"main_file","file_size":827030,"access_level":"open_access","file_name":"BEA_MultiStep_Matrix.pdf"}],"date_created":"2022-01-11T12:48:39Z","keyword":["variational integrators","backward error analysis","Euler--Lagrange equations","multistep methods","conjugate symplectic methods"],"type":"journal_article","department":[{"_id":"636"}],"issue":"1","publication":"Journal of Geometric Mechanics","related_material":{"link":[{"relation":"software","url":"https://github.com/Christian-Offen/BEAConjugateSymplectic"}]},"abstract":[{"text":"The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.","lang":"eng"}],"language":[{"iso":"eng"}],"doi":"10.3934/jgm.2023005","title":"Backward error analysis for conjugate symplectic methods","year":"2023","author":[{"first_name":"Robert","last_name":"McLachlan","full_name":"McLachlan, Robert"},{"full_name":"Offen, Christian","last_name":"Offen","orcid":"0000-0002-5940-8057","first_name":"Christian","id":"85279"}],"publication_status":"published","date_updated":"2023-08-10T08:40:30Z","article_type":"original","intvolume":"        15"},{"file_date_updated":"2023-04-26T16:20:56Z","citation":{"apa":"Dierkes, E., Offen, C., Ober-Blöbaum, S., &#38; Flaßkamp, K. (2023). Hamiltonian Neural Networks with Automatic Symmetry Detection. <i>Chaos</i>, <i>33</i>(6), Article 063115. <a href=\"https://doi.org/10.1063/5.0142969\">https://doi.org/10.1063/5.0142969</a>","ieee":"E. Dierkes, C. Offen, S. Ober-Blöbaum, and K. Flaßkamp, “Hamiltonian Neural Networks with Automatic Symmetry Detection,” <i>Chaos</i>, vol. 33, no. 6, Art. no. 063115, 2023, doi: <a href=\"https://doi.org/10.1063/5.0142969\">10.1063/5.0142969</a>.","chicago":"Dierkes, Eva, Christian Offen, Sina Ober-Blöbaum, and Kathrin Flaßkamp. “Hamiltonian Neural Networks with Automatic Symmetry Detection.” <i>Chaos</i> 33, no. 6 (2023). <a href=\"https://doi.org/10.1063/5.0142969\">https://doi.org/10.1063/5.0142969</a>.","short":"E. Dierkes, C. Offen, S. Ober-Blöbaum, K. Flaßkamp, Chaos 33 (2023).","mla":"Dierkes, Eva, et al. “Hamiltonian Neural Networks with Automatic Symmetry Detection.” <i>Chaos</i>, vol. 33, no. 6, 063115, AIP Publishing, 2023, doi:<a href=\"https://doi.org/10.1063/5.0142969\">10.1063/5.0142969</a>.","ama":"Dierkes E, Offen C, Ober-Blöbaum S, Flaßkamp K. Hamiltonian Neural Networks with Automatic Symmetry Detection. <i>Chaos</i>. 2023;33(6). doi:<a href=\"https://doi.org/10.1063/5.0142969\">10.1063/5.0142969</a>","bibtex":"@article{Dierkes_Offen_Ober-Blöbaum_Flaßkamp_2023, title={Hamiltonian Neural Networks with Automatic Symmetry Detection}, volume={33}, DOI={<a href=\"https://doi.org/10.1063/5.0142969\">10.1063/5.0142969</a>}, number={6063115}, journal={Chaos}, publisher={AIP Publishing}, author={Dierkes, Eva and Offen, Christian and Ober-Blöbaum, Sina and Flaßkamp, Kathrin}, year={2023} }"},"oa":"1","external_id":{"arxiv":["2301.07928"]},"has_accepted_license":"1","status":"public","ddc":["510"],"user_id":"85279","volume":33,"_id":"37654","publisher":"AIP Publishing","abstract":[{"text":"Recently, Hamiltonian neural networks (HNN) have been introduced to incorporate prior physical knowledge when\r\nlearning the dynamical equations of Hamiltonian systems. Hereby, the symplectic system structure is preserved despite\r\nthe data-driven modeling approach. However, preserving symmetries requires additional attention. In this research, we\r\nenhance the HNN with a Lie algebra framework to detect and embed symmetries in the neural network. This approach\r\nallows to simultaneously learn the symmetry group action and the total energy of the system. As illustrating examples,\r\na pendulum on a cart and a two-body problem from astrodynamics are considered.","lang":"eng"}],"related_material":{"link":[{"description":"GitHub","relation":"software","url":"https://github.com/eva-dierkes/HNN_withSymmetries"}]},"issue":"6","publication":"Chaos","type":"journal_article","department":[{"_id":"636"}],"file":[{"description":"Incorporating physical system knowledge into data-driven\nsystem identification has been shown to be beneficial. The\napproach presented in this article combines learning of an\nenergy-conserving model from data with detecting a Lie\ngroup representation of the unknown system symmetry.\nThe proposed approach can improve the learned model\nand reveal underlying symmetry simultaneously.","date_created":"2023-04-26T16:20:56Z","creator":"coffen","title":"Hamiltonian Neural Networks with Automatic Symmetry Detection","file_id":"44205","content_type":"application/pdf","relation":"main_file","date_updated":"2023-04-26T16:20:56Z","file_name":"JournalPaper_main.pdf","file_size":5200111,"access_level":"open_access"}],"date_created":"2023-01-20T09:10:06Z","date_updated":"2023-08-10T08:37:01Z","publication_status":"published","intvolume":"        33","article_type":"original","title":"Hamiltonian Neural Networks with Automatic Symmetry Detection","year":"2023","author":[{"last_name":"Dierkes","first_name":"Eva","full_name":"Dierkes, Eva"},{"orcid":"0000-0002-5940-8057","last_name":"Offen","first_name":"Christian","full_name":"Offen, Christian","id":"85279"},{"full_name":"Ober-Blöbaum, Sina","first_name":"Sina","last_name":"Ober-Blöbaum","id":"16494"},{"full_name":"Flaßkamp, Kathrin","first_name":"Kathrin","last_name":"Flaßkamp"}],"publication_identifier":{"issn":["1054-1500"]},"doi":"10.1063/5.0142969","article_number":"063115","language":[{"iso":"eng"}]},{"citation":{"mla":"Nüske, Feliks, et al. “Finite-Data Error Bounds for Koopman-Based Prediction and Control.” <i>Journal of Nonlinear Science</i>, vol. 33, 14, 2023, doi:<a href=\"https://doi.org/10.1007/s00332-022-09862-1\">10.1007/s00332-022-09862-1</a>.","bibtex":"@article{Nüske_Peitz_Philipp_Schaller_Worthmann_2023, title={Finite-data error bounds for Koopman-based prediction and control}, volume={33}, DOI={<a href=\"https://doi.org/10.1007/s00332-022-09862-1\">10.1007/s00332-022-09862-1</a>}, number={14}, journal={Journal of Nonlinear Science}, author={Nüske, Feliks and Peitz, Sebastian and Philipp, Friedrich and Schaller, Manuel and Worthmann, Karl}, year={2023} }","ama":"Nüske F, Peitz S, Philipp F, Schaller M, Worthmann K. Finite-data error bounds for Koopman-based prediction and control. <i>Journal of Nonlinear Science</i>. 2023;33. doi:<a href=\"https://doi.org/10.1007/s00332-022-09862-1\">10.1007/s00332-022-09862-1</a>","ieee":"F. Nüske, S. Peitz, F. Philipp, M. Schaller, and K. Worthmann, “Finite-data error bounds for Koopman-based prediction and control,” <i>Journal of Nonlinear Science</i>, vol. 33, Art. no. 14, 2023, doi: <a href=\"https://doi.org/10.1007/s00332-022-09862-1\">10.1007/s00332-022-09862-1</a>.","apa":"Nüske, F., Peitz, S., Philipp, F., Schaller, M., &#38; Worthmann, K. (2023). Finite-data error bounds for Koopman-based prediction and control. <i>Journal of Nonlinear Science</i>, <i>33</i>, Article 14. <a href=\"https://doi.org/10.1007/s00332-022-09862-1\">https://doi.org/10.1007/s00332-022-09862-1</a>","short":"F. Nüske, S. Peitz, F. Philipp, M. Schaller, K. Worthmann, Journal of Nonlinear Science 33 (2023).","chicago":"Nüske, Feliks, Sebastian Peitz, Friedrich Philipp, Manuel Schaller, and Karl Worthmann. “Finite-Data Error Bounds for Koopman-Based Prediction and Control.” <i>Journal of Nonlinear Science</i> 33 (2023). <a href=\"https://doi.org/10.1007/s00332-022-09862-1\">https://doi.org/10.1007/s00332-022-09862-1</a>."},"oa":"1","status":"public","_id":"23428","user_id":"47427","volume":33,"publication":"Journal of Nonlinear Science","abstract":[{"lang":"eng","text":"The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems in recent years, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still quite scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points; for both ordinary and stochastic differential equations. Moreover, we extend our analysis to nonlinear control-affine systems using either ergodic trajectories or i.i.d.\r\nsamples. Here, we exploit the linearity of the Koopman generator to obtain a bilinear system and, thus, circumvent the curse of dimensionality since we do not autonomize the system by augmenting the state by the control inputs. To the\r\nbest of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the proposed approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled."}],"date_created":"2021-08-17T12:25:09Z","type":"journal_article","department":[{"_id":"101"},{"_id":"655"}],"year":"2023","title":"Finite-data error bounds for Koopman-based prediction and control","author":[{"full_name":"Nüske, Feliks","orcid":"0000-0003-2444-7889","first_name":"Feliks","last_name":"Nüske","id":"81513"},{"id":"47427","full_name":"Peitz, Sebastian","first_name":"Sebastian","orcid":"0000-0002-3389-793X","last_name":"Peitz"},{"full_name":"Philipp, Friedrich","first_name":"Friedrich","last_name":"Philipp"},{"full_name":"Schaller, Manuel","first_name":"Manuel","last_name":"Schaller"},{"first_name":"Karl","last_name":"Worthmann","full_name":"Worthmann, Karl"}],"publication_status":"published","date_updated":"2023-08-24T07:50:12Z","intvolume":"        33","article_number":"14","main_file_link":[{"open_access":"1","url":"https://link.springer.com/content/pdf/10.1007/s00332-022-09862-1.pdf"}],"language":[{"iso":"eng"}],"doi":"10.1007/s00332-022-09862-1"},{"type":"journal_article","department":[{"_id":"101"},{"_id":"636"},{"_id":"355"},{"_id":"655"}],"date_created":"2021-04-09T07:59:19Z","abstract":[{"lang":"eng","text":"Many problems in science and engineering require an efficient numerical approximation of integrals or solutions to differential equations. For systems with rapidly changing dynamics, an equidistant discretization is often inadvisable as it results in prohibitively large errors or computational effort. To this end, adaptive schemes, such as solvers based on Runge–Kutta pairs, have been developed which adapt the step size based on local error estimations at each step. While the classical schemes apply very generally and are highly efficient on regular systems, they can behave suboptimally when an inefficient step rejection mechanism is triggered by structurally complex systems such as chaotic systems. To overcome these issues, we propose a method to tailor numerical schemes to the problem class at hand. This is achieved by combining simple, classical quadrature rules or ODE solvers with data-driven time-stepping controllers. Compared with learning solution operators to ODEs directly, it generalizes better to unseen initial data as our approach employs classical numerical schemes as base methods. At the same time it can make use of identified structures of a problem class and, therefore, outperforms state-of-the-art adaptive schemes. Several examples demonstrate superior efficiency. Source code is available at https://github.com/lueckem/quadrature-ML."}],"related_material":{"link":[{"description":"GitHub","url":"https://github.com/lueckem/quadrature-ML","relation":"software"}]},"issue":"2","publication":"SIAM Journal on Scientific Computing","doi":"10.1137/21M1412682","main_file_link":[{"url":"https://epubs.siam.org/doi/reader/10.1137/21M1412682"}],"language":[{"iso":"eng"}],"publication_status":"published","date_updated":"2023-08-25T09:24:50Z","intvolume":"        45","title":"Efficient time stepping for numerical integration using reinforcement  learning","year":"2023","author":[{"last_name":"Dellnitz","first_name":"Michael","full_name":"Dellnitz, Michael"},{"last_name":"Hüllermeier","first_name":"Eyke","full_name":"Hüllermeier, Eyke","id":"48129"},{"last_name":"Lücke","first_name":"Marvin","full_name":"Lücke, Marvin"},{"last_name":"Ober-Blöbaum","first_name":"Sina","full_name":"Ober-Blöbaum, Sina","id":"16494"},{"full_name":"Offen, Christian","last_name":"Offen","orcid":"0000-0002-5940-8057","first_name":"Christian","id":"85279"},{"id":"47427","full_name":"Peitz, Sebastian","last_name":"Peitz","orcid":"0000-0002-3389-793X","first_name":"Sebastian"},{"id":"13472","full_name":"Pfannschmidt, Karlson","last_name":"Pfannschmidt","orcid":"0000-0001-9407-7903","first_name":"Karlson"}],"external_id":{"arxiv":["arXiv:2104.03562"]},"citation":{"apa":"Dellnitz, M., Hüllermeier, E., Lücke, M., Ober-Blöbaum, S., Offen, C., Peitz, S., &#38; Pfannschmidt, K. (2023). Efficient time stepping for numerical integration using reinforcement  learning. <i>SIAM Journal on Scientific Computing</i>, <i>45</i>(2), A579–A595. <a href=\"https://doi.org/10.1137/21M1412682\">https://doi.org/10.1137/21M1412682</a>","ieee":"M. Dellnitz <i>et al.</i>, “Efficient time stepping for numerical integration using reinforcement  learning,” <i>SIAM Journal on Scientific Computing</i>, vol. 45, no. 2, pp. A579–A595, 2023, doi: <a href=\"https://doi.org/10.1137/21M1412682\">10.1137/21M1412682</a>.","chicago":"Dellnitz, Michael, Eyke Hüllermeier, Marvin Lücke, Sina Ober-Blöbaum, Christian Offen, Sebastian Peitz, and Karlson Pfannschmidt. “Efficient Time Stepping for Numerical Integration Using Reinforcement  Learning.” <i>SIAM Journal on Scientific Computing</i> 45, no. 2 (2023): A579–95. <a href=\"https://doi.org/10.1137/21M1412682\">https://doi.org/10.1137/21M1412682</a>.","short":"M. Dellnitz, E. Hüllermeier, M. Lücke, S. Ober-Blöbaum, C. Offen, S. Peitz, K. Pfannschmidt, SIAM Journal on Scientific Computing 45 (2023) A579–A595.","mla":"Dellnitz, Michael, et al. “Efficient Time Stepping for Numerical Integration Using Reinforcement  Learning.” <i>SIAM Journal on Scientific Computing</i>, vol. 45, no. 2, 2023, pp. A579–95, doi:<a href=\"https://doi.org/10.1137/21M1412682\">10.1137/21M1412682</a>.","ama":"Dellnitz M, Hüllermeier E, Lücke M, et al. Efficient time stepping for numerical integration using reinforcement  learning. <i>SIAM Journal on Scientific Computing</i>. 2023;45(2):A579-A595. doi:<a href=\"https://doi.org/10.1137/21M1412682\">10.1137/21M1412682</a>","bibtex":"@article{Dellnitz_Hüllermeier_Lücke_Ober-Blöbaum_Offen_Peitz_Pfannschmidt_2023, title={Efficient time stepping for numerical integration using reinforcement  learning}, volume={45}, DOI={<a href=\"https://doi.org/10.1137/21M1412682\">10.1137/21M1412682</a>}, number={2}, journal={SIAM Journal on Scientific Computing}, author={Dellnitz, Michael and Hüllermeier, Eyke and Lücke, Marvin and Ober-Blöbaum, Sina and Offen, Christian and Peitz, Sebastian and Pfannschmidt, Karlson}, year={2023}, pages={A579–A595} }"},"user_id":"47427","ddc":["510"],"volume":45,"page":"A579-A595","_id":"21600","has_accepted_license":"1","status":"public"},{"publisher":"Springer","_id":"16296","page":"43-76","editor":[{"full_name":"Michael, Hintermüller","last_name":"Michael","first_name":"Hintermüller"},{"full_name":"Roland, Herzog","first_name":"Herzog","last_name":"Roland"},{"first_name":"Kanzow","last_name":"Christian","full_name":"Christian, Kanzow"},{"first_name":"Ulbrich","last_name":"Michael","full_name":"Michael, Ulbrich"},{"full_name":"Stefan, Ulbrich","last_name":"Stefan","first_name":"Ulbrich"}],"user_id":"47427","status":"public","place":"Cham","oa":"1","citation":{"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>","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.","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.","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>.","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>.","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>","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} }"},"language":[{"iso":"eng"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/pdf/1906.09075.pdf"}],"doi":"10.1007/978-3-030-79393-7_3","author":[{"last_name":"Banholzer","first_name":"Stefan","full_name":"Banholzer, Stefan"},{"full_name":"Gebken, Bennet","first_name":"Bennet","last_name":"Gebken","id":"32643"},{"first_name":"Michael","last_name":"Dellnitz","full_name":"Dellnitz, Michael"},{"orcid":"https://orcid.org/0000-0002-3389-793X","first_name":"Sebastian","last_name":"Peitz","full_name":"Peitz, Sebastian","id":"47427"},{"full_name":"Volkwein, Stefan","last_name":"Volkwein","first_name":"Stefan"}],"publication_identifier":{"isbn":["978-3-030-79392-0"]},"year":"2022","title":"ROM-Based Multiobjective Optimization of Elliptic PDEs via Numerical Continuation","date_updated":"2022-03-14T13:04:51Z","date_created":"2020-03-13T12:45:31Z","department":[{"_id":"101"},{"_id":"655"}],"type":"book_chapter","publication":"Non-Smooth and Complementarity-Based Distributed Parameter Systems","abstract":[{"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.","lang":"eng"}]},{"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>","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>.","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>.","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>","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} }"},"place":"Cham","status":"public","user_id":"47427","editor":[{"full_name":"Bock, H. G.","first_name":"H. G.","last_name":"Bock"},{"first_name":"K.-H.","last_name":"Küfer","full_name":"Küfer, K.-H."},{"full_name":"Maas, P.","last_name":"Maas","first_name":"P."},{"last_name":"Milde","first_name":"A.","full_name":"Milde, A."},{"last_name":"Schulz","first_name":"V.","full_name":"Schulz, V."}],"volume":35,"_id":"30294","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","type":"book_chapter","department":[{"_id":"101"},{"_id":"655"}],"date_created":"2022-03-14T07:32:41Z","date_updated":"2022-03-14T07:42:01Z","publication_status":"published","intvolume":"        35","title":"Efficient Virtual Design and Testing of Autonomous Vehicles","year":"2022","author":[{"full_name":"Peitz, Sebastian","first_name":"Sebastian","orcid":"0000-0002-3389-793X","last_name":"Peitz","id":"47427"},{"first_name":"Michael","last_name":"Dellnitz","full_name":"Dellnitz, Michael"},{"full_name":"Bannenberg, Sebastian","last_name":"Bannenberg","first_name":"Sebastian"}],"publication_identifier":{"issn":["1612-3956","2198-3283"],"isbn":["9783030814540","9783030814557"]},"doi":"10.1007/978-3-030-81455-7_23","language":[{"iso":"eng"}],"series_title":"Mathematics in Industry"},{"author":[{"full_name":"Cresson, Jacky","first_name":"Jacky","last_name":"Cresson"},{"full_name":"Jiménez, Fernando","first_name":"Fernando","last_name":"Jiménez"},{"id":"16494","first_name":"Sina","last_name":"Ober-Blöbaum","full_name":"Ober-Blöbaum, Sina"}],"year":"2022","status":"public","title":"Continuous and discrete Noether's fractional conserved quantities for restricted calculus of variations","date_updated":"2022-03-24T12:26:32Z","language":[{"iso":"eng"}],"_id":"30490","page":"57-89","volume":"14(1)","user_id":"15694","citation":{"ieee":"J. Cresson, F. Jiménez, and S. Ober-Blöbaum, “Continuous and discrete Noether’s fractional conserved quantities for restricted calculus of variations,” <i>AIMS</i>, vol. 14(1), pp. 57–89, 2022.","apa":"Cresson, J., Jiménez, F., &#38; Ober-Blöbaum, S. (2022). Continuous and discrete Noether’s fractional conserved quantities for restricted calculus of variations. <i>AIMS</i>, <i>14(1)</i>, 57–89.","mla":"Cresson, Jacky, et al. “Continuous and Discrete Noether’s Fractional Conserved Quantities for Restricted Calculus of Variations.” <i>AIMS</i>, vol. 14(1), 2022, pp. 57–89.","bibtex":"@article{Cresson_Jiménez_Ober-Blöbaum_2022, title={Continuous and discrete Noether’s fractional conserved quantities for restricted calculus of variations}, volume={14(1)}, journal={AIMS}, author={Cresson, Jacky and Jiménez, Fernando and Ober-Blöbaum, Sina}, year={2022}, pages={57–89} }","chicago":"Cresson, Jacky, Fernando Jiménez, and Sina Ober-Blöbaum. “Continuous and Discrete Noether’s Fractional Conserved Quantities for Restricted Calculus of Variations.” <i>AIMS</i> 14(1) (2022): 57–89.","ama":"Cresson J, Jiménez F, Ober-Blöbaum S. Continuous and discrete Noether’s fractional conserved quantities for restricted calculus of variations. <i>AIMS</i>. 2022;14(1):57-89.","short":"J. Cresson, F. Jiménez, S. Ober-Blöbaum, AIMS 14(1) (2022) 57–89."},"publication":"AIMS","date_created":"2022-03-24T12:26:10Z","department":[{"_id":"636"}],"type":"journal_article"}]