{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,10]],"date-time":"2026-07-10T15:31:05Z","timestamp":1783697465675,"version":"3.55.0"},"reference-count":66,"publisher":"American Mathematical Society (AMS)","issue":"347","license":[{"start":{"date-parts":[[2024,8,24]],"date-time":"2024-08-24T00:00:00Z","timestamp":1724457600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-17-1-9241"],"award-info":[{"award-number":["FA9550-17-1-9241"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-17-1-9241"],"award-info":[{"award-number":["FA9550-17-1-9241"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-17-1-9241"],"award-info":[{"award-number":["FA9550-17-1-9241"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-17-1-9241"],"award-info":[{"award-number":["FA9550-17-1-9241"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>High-resolution simulations of particle-based kinetic plasma models typically require a high number of particles and thus often become computationally intractable. This is exacerbated in multi-query simulations, where the problem depends on a set of parameters. In this work, we derive reduced order models for the semi-discrete Hamiltonian system resulting from a geometric particle-in-cell approximation of the parametric Vlasov\u2013Poisson equations. Since the problem\u2019s nondissipative and highly nonlinear nature makes it reducible only locally in time, we adopt a nonlinear reduced basis approach where the reduced phase space evolves in time. This strategy allows a significant reduction in the number of simulated particles, but the evaluation of the nonlinear operators associated with the Vlasov\u2013Poisson coupling remains computationally expensive. We propose a novel reduction of the nonlinear terms that combines adaptive parameter sampling and hyper-reduction techniques to address this. The proposed approach allows decoupling the operations having a cost dependent on the number of particles from those that depend on the instances of the required parameters. In particular, in each time step, the electric potential is approximated via dynamic mode decomposition (DMD) and the particle-to-grid map via a discrete empirical interpolation method (DEIM). These approximations are constructed from data obtained from a past temporal window at a few selected values of the parameters to guarantee a computationally efficient adaptation. The resulting DMD-DEIM reduced dynamical system retains the Hamiltonian structure of the full model, provides good approximations of the solution, and can be solved at a reduced computational cost.<\/p>","DOI":"10.1090\/mcom\/3885","type":"journal-article","created":{"date-parts":[[2023,8,24]],"date-time":"2023-08-24T14:22:52Z","timestamp":1692886972000},"page":"1153-1202","source":"Crossref","is-referenced-by-count":14,"title":["Adaptive symplectic model order reduction of parametric particle-based Vlasov\u2013Poisson equation"],"prefix":"10.1090","volume":"93","author":[{"given":"Jan","family":"Hesthaven","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Cecilia","family":"Pagliantini","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nicol\u00f2","family":"Ripamonti","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2023,8,24]]},"reference":[{"issue":"6","key":"1","doi-asserted-by":"publisher","first-page":"A2616--A2644","DOI":"10.1137\/17M1111991","article-title":"Structure preserving model reduction of parametric Hamiltonian systems","volume":"39","author":"Afkham, Babak Maboudi","year":"2017","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"key":"2","doi-asserted-by":"crossref","unstructured":"D. Anderson, R. Fedele, and M. Lisak, A tutorial presentation of the two stream instability and Landau damping, Amer. J. Phys. 69 (2001), no. 12, 1262\u20131266.","DOI":"10.1119\/1.1407252"},{"issue":"9","key":"3","doi-asserted-by":"publisher","first-page":"667","DOI":"10.1016\/j.crma.2004.08.006","article-title":"An \u2018empirical interpolation\u2019 method: application to efficient reduced-basis discretization of partial differential equations","volume":"339","author":"Barrault, Maxime","year":"2004","journal-title":"C. R. Math. Acad. Sci. Paris","ISSN":"https:\/\/id.crossref.org\/issn\/1631-073X","issn-type":"print"},{"key":"4","unstructured":"G. Berge, Landau damping in a plasma, lecture notes, University of Bergen, 1969."},{"key":"5","first-page":"968","article-title":"An improved approximation algorithm for the column subset selection problem","author":"Boutsidis, Christos","year":"2009"},{"key":"6","unstructured":"D. S. Broomhead and D. Lowe, Radial basis functions, multi-variable functional interpolation and adaptive networks, Tech. report, Royal Signals and Radar Establishment Malvern (United Kingdom), 1988."},{"key":"7","doi-asserted-by":"crossref","unstructured":"T. Bui-Thanh, M. Damodaran, and K. Willcox, Proper orthogonal decomposition extensions for parametric applications in compressible aerodynamics, 21st AIAA Applied Aerodynamics Conference, 2003, p. 4213.","DOI":"10.2514\/6.2003-4213"},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"769","DOI":"10.1007\/s00211-016-0816-z","article-title":"High-order Hamiltonian splitting for the Vlasov-Poisson equations","volume":"135","author":"Casas, Fernando","year":"2017","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"5","key":"9","doi-asserted-by":"publisher","first-page":"2737","DOI":"10.1137\/090766498","article-title":"Nonlinear model reduction via discrete empirical interpolation","volume":"32","author":"Chaturantabut, Saifon","year":"2010","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"key":"10","series-title":"SIAM Spotlights","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611973860","volume-title":"Active subspaces","volume":"2","author":"Constantine, Paul G.","year":"2015","ISBN":"https:\/\/id.crossref.org\/isbn\/9781611973853"},{"issue":"8","key":"11","doi-asserted-by":"publisher","first-page":"497","DOI":"10.1002\/fld.3946","article-title":"Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations","volume":"76","author":"\u015etef\u0103nescu, R\u0103zvan","year":"2014","journal-title":"Internat. J. Numer. Methods Fluids","ISSN":"https:\/\/id.crossref.org\/issn\/0271-2091","issn-type":"print"},{"key":"12","doi-asserted-by":"publisher","first-page":"194","DOI":"10.1016\/j.jcp.2016.08.004","article-title":"Nonlinear model-order reduction for compressible flow solvers using the discrete empirical interpolation method","volume":"324","author":"Fosas de Pando, Miguel","year":"2016","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"13","unstructured":"P. Druault, D\u00e9veloppement d\u2019interfaces exp\u00e9rience\/simulation. Application \u00e0 l\u2019\u00e9coulement de couche de m\u00e9lange plane turbulente, Ph.D. Thesis, Universit\u00e9 de Poitiers, France, 1999."},{"key":"14","doi-asserted-by":"crossref","unstructured":"D. Dylewsky, M. Tao, and J. Nathan Kutz, Dynamic mode decomposition for multiscale nonlinear physics, Phys. Rev. E 99 (2019), no. 6, 063311.","DOI":"10.1103\/PhysRevE.99.063311"},{"key":"15","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1016\/j.jcp.2017.03.015","article-title":"A dynamical adaptive tensor method for the Vlasov-Poisson system","volume":"339","author":"Ehrlacher, Virginie","year":"2017","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"16","doi-asserted-by":"publisher","first-page":"Paper No. 110495, 16","DOI":"10.1016\/j.jcp.2021.110495","article-title":"A mass, momentum, and energy conservative dynamical low-rank scheme for the Vlasov equation","volume":"443","author":"Einkemmer, Lukas","year":"2021","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"5","key":"17","doi-asserted-by":"publisher","first-page":"B1330--B1360","DOI":"10.1137\/18M116383X","article-title":"A low-rank projector-splitting integrator for the Vlasov-Poisson equation","volume":"40","author":"Einkemmer, Lukas","year":"2018","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"key":"18","doi-asserted-by":"crossref","unstructured":"L. Einkemmer and C. Lubich, A quasi-conservative dynamical low-rank algorithm for the Vlasov equation, SIAM J. Sci. Comput. 41 (2019), no. 5, B1061\u2013B1081.","DOI":"10.1137\/18M1218686"},{"key":"19","doi-asserted-by":"publisher","first-page":"376","DOI":"10.1016\/j.jcp.2013.03.006","article-title":"Variational formulation of particle algorithms for kinetic plasma simulations","volume":"245","author":"Evstatiev, E. G.","year":"2013","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"20","doi-asserted-by":"crossref","unstructured":"C. Gillot, G. Dif-Pradalier, X. Garbet, P. Ghendrih, V. Grandgirard, and Y. Sarazin, Model order reduction approach to the one-dimensional collisionless closure problem, Phys. Plasmas 28 (2021), no. 2, 022111.","DOI":"10.1063\/5.0023407"},{"key":"21","unstructured":"J. Grosek and J. N. Kutz, Dynamic mode decomposition for real-time background\/foreground separation in video, Preprint,  arXiv:1404.7592, 2014."},{"key":"22","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","volume-title":"Geometric numerical integration","volume":"31","author":"Hairer, Ernst","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/3540306633","edition":"2"},{"issue":"2","key":"23","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1137\/090771806","article-title":"Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions","volume":"53","author":"Halko, N.","year":"2011","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"},{"key":"24","doi-asserted-by":"crossref","unstructured":"Y. He, Y. Sun, H. Qin, and J. Liu, Hamiltonian particle-in-cell methods for Vlasov-Maxwell equations, Phys. Plasmas 23 (2016), no. 9, 092108.","DOI":"10.1063\/1.4962573"},{"key":"25","doi-asserted-by":"crossref","unstructured":"M. S. Hemati, M. O. Williams, and C. W. Rowley, Dynamic mode decomposition for large and streaming datasets, Phys. Fluids 26 (2014), no. 11, 111701.","DOI":"10.1063\/1.4901016"},{"key":"26","unstructured":"H. G. Hereward, Landau damping, Technical Report, 1977."},{"issue":"330","key":"27","doi-asserted-by":"publisher","first-page":"1701","DOI":"10.1090\/mcom\/3618","article-title":"Structure-preserving reduced basis methods for Poisson systems","volume":"90","author":"Hesthaven, Jan S.","year":"2021","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"28","doi-asserted-by":"publisher","first-page":"617","DOI":"10.1051\/m2an\/2022013","article-title":"Rank-adaptive structure-preserving model order reduction of Hamiltonian systems","volume":"56","author":"Hesthaven, Jan S.","year":"2022","journal-title":"ESAIM Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/2822-7840","issn-type":"print"},{"key":"29","doi-asserted-by":"crossref","unstructured":"R. W. Hockney, Characteristics of noise in a two-dimensional computer plasma, Phys. Fluids 11 (1968), no. 6, 1381\u20131383.","DOI":"10.1063\/1.1692116"},{"issue":"2","key":"30","doi-asserted-by":"publisher","first-page":"434","DOI":"10.1137\/050639703","article-title":"Dynamical low-rank approximation","volume":"29","author":"Koch, Othmar","year":"2007","journal-title":"SIAM J. Matrix Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0895-4798","issn-type":"print"},{"key":"31","doi-asserted-by":"crossref","unstructured":"B. O. Koopman, Hamiltonian systems and transformation in Hilbert space, Proc. Natl. Acad Sci. USA 17 (1931), no. 5, 315.","DOI":"10.1073\/pnas.17.5.315"},{"issue":"4","key":"32","doi-asserted-by":"publisher","first-page":"B613--B632","DOI":"10.1137\/140971270","article-title":"A semi-Lagrangian Vlasov solver in tensor train format","volume":"37","author":"Kormann, Katharina","year":"2015","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"key":"33","isbn-type":"print","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1007\/978-3-319-28262-6_7","article-title":"Sparse grids for the Vlasov-Poisson equation","author":"Kormann, Katharina","year":"2016","ISBN":"https:\/\/id.crossref.org\/isbn\/9783319282626"},{"key":"34","doi-asserted-by":"crossref","unstructured":"M. Kraus, K. Kormann, P. J. Morrison, and E. Sonnendr\u00fccker, GEMPIC: geometric electromagnetic particle-in-cell methods, J. Plasma Phys. 83 (2017), no. 4, 905830401.","DOI":"10.1017\/S002237781700040X"},{"key":"35","doi-asserted-by":"crossref","unstructured":"J. A. Krommes, Nonequilibrium gyrokinetic fluctuation theory and sampling noise in gyrokinetic particle-in-cell simulations, Phys. Plasmas 14 (2007), no. 9, 090501.","DOI":"10.1063\/1.2759879"},{"key":"36","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611974508","volume-title":"Dynamic mode decomposition","author":"Kutz, J. Nathan","year":"2016","ISBN":"https:\/\/id.crossref.org\/isbn\/9781611974492"},{"key":"37","unstructured":"A. B. Langdon and C. K. Birdsall, Plasma Physics Via Computer Simulation, 1st ed., Springer Series in Computational Mathematics, CRC Press, 1991."},{"key":"38","doi-asserted-by":"crossref","unstructured":"H. R. Lewis, Energy-conserving numerical approximations for Vlasov plasmas, J. Comput. Phys. 6 (1970), no. 1, 136\u2013141.","DOI":"10.1016\/0021-9991(70)90012-4"},{"key":"39","doi-asserted-by":"publisher","first-page":"282","DOI":"10.1098\/rspa.1958.0244","article-title":"A Lagrangian formulation of the Boltzmann-Vlasow equation for plasmas","volume":"248","author":"Low, F. E.","year":"1958","journal-title":"Proc. Roy. Soc. London Ser. A","ISSN":"https:\/\/id.crossref.org\/issn\/0962-8444","issn-type":"print"},{"key":"40","doi-asserted-by":"crossref","unstructured":"G. Manfredi, Long-time behavior of nonlinear Landau damping, Phys. Rev. Lett. 79 (1997), no. 15, 2815.","DOI":"10.1103\/PhysRevLett.79.2815"},{"key":"41","doi-asserted-by":"crossref","unstructured":"S. Mariappan, A. Donald Gardner, K. Richter, and M. Raffel, Analysis of dynamic stall using dynamic mode decomposition technique, AIAA J. 52 (2014), no. 11, 2427\u20132439.","DOI":"10.2514\/1.J052858"},{"issue":"3","key":"42","doi-asserted-by":"publisher","first-page":"394","DOI":"10.1016\/0167-2789(82)90043-4","article-title":"The Hamiltonian structure of the Maxwell-Vlasov equations","volume":"4","author":"Marsden, Jerrold E.","year":"1981","journal-title":"Phys. D","ISSN":"https:\/\/id.crossref.org\/issn\/0167-2789","issn-type":"print"},{"key":"43","unstructured":"D. Matsumoto and T. Indinger, On-the-fly algorithm for dynamic mode decomposition using incremental singular value decomposition and total least squares, Preprint,  arXiv:1703.11004, 2017."},{"key":"44","doi-asserted-by":"crossref","unstructured":"P. J. Morrison, Hamiltonian field description of the one-dimensional Poisson-Vlasov equations, 1981, DOI 10.2172\/6423520.","DOI":"10.2172\/6423520"},{"key":"45","unstructured":"E. Musharbash, F. Nobile, and E. Vidli\u010dkov\u00e1, Symplectic dynamical low rank approximation of wave equations with random parameters, BIT Numer. Math. (2017), 1\u201349."},{"key":"46","doi-asserted-by":"publisher","first-page":"Paper No. 110671, 27","DOI":"10.1016\/j.jcp.2021.110671","article-title":"Detection and prediction of equilibrium states in kinetic plasma simulations via mode tracking using reduced-order dynamic mode decomposition","volume":"447","author":"Nayak, Indranil","year":"2021","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"issue":"2","key":"47","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1007\/s00211-021-01211-w","article-title":"Dynamical reduced basis methods for Hamiltonian systems","volume":"148","author":"Pagliantini, Cecilia","year":"2021","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"4","key":"48","doi-asserted-by":"publisher","first-page":"A2123--A2150","DOI":"10.1137\/140989169","article-title":"Online adaptive model reduction for nonlinear systems via low-rank updates","volume":"37","author":"Peherstorfer, Benjamin","year":"2015","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"1","key":"49","doi-asserted-by":"publisher","first-page":"A1--A27","DOI":"10.1137\/140978922","article-title":"Symplectic model reduction of Hamiltonian systems","volume":"38","author":"Peng, Liqian","year":"2016","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"key":"50","series-title":"Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-69894-1","volume-title":"$n$-widths in approximation theory","volume":"7","author":"Pinkus, Allan","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/354013638X"},{"key":"51","doi-asserted-by":"crossref","unstructured":"J. L. Proctor and P. A. Eckhoff, Discovering dynamic patterns from infectious disease data using dynamic mode decomposition, Int. Health 7 (2015), no. 2, 139\u2013145.","DOI":"10.1093\/inthealth\/ihv009"},{"key":"52","doi-asserted-by":"crossref","unstructured":"H. Qin, J. Liu, J. Xiao, R. Zhang, Y. He, Y. Wang, Y. Sun, J. W. Burby, L. Ellison, and Y. Zhou, Canonical symplectic particle-in-cell method for long-term large-scale simulations of the Vlasov\u2013Maxwell equations, Nucl. Fusion 56 (2015), no. 1, DOI 10.1088\/0029-5515\/56\/1\/014001.","DOI":"10.1088\/0029-5515\/56\/1\/014001"},{"key":"53","series-title":"Unitext","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-15431-2","volume-title":"Reduced basis methods for partial differential equations","volume":"92","author":"Quarteroni, Alfio","year":"2016","ISBN":"https:\/\/id.crossref.org\/isbn\/9783319154305"},{"key":"54","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1017\/S0022112009992059","article-title":"Spectral analysis of nonlinear flows","volume":"641","author":"Rowley, Clarence W.","year":"2009","journal-title":"J. Fluid Mech.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-1120","issn-type":"print"},{"issue":"23-24","key":"55","doi-asserted-by":"publisher","first-page":"2347","DOI":"10.1016\/j.physd.2009.09.017","article-title":"Dynamically orthogonal field equations for continuous stochastic dynamical systems","volume":"238","author":"Sapsis, Themistoklis P.","year":"2009","journal-title":"Phys. D","ISSN":"https:\/\/id.crossref.org\/issn\/0167-2789","issn-type":"print"},{"key":"56","doi-asserted-by":"crossref","unstructured":"T. Sayadi, P. J. Schmid, F. Richecoeur, and D. Durox, Parametrized data-driven decomposition for bifurcation analysis, with application to thermo-acoustically unstable systems, Phys. Fluids 27 (2015), no. 3, 037102.","DOI":"10.1063\/1.4913868"},{"key":"57","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1017\/S0022112010001217","article-title":"Dynamic mode decomposition of numerical and experimental data","volume":"656","author":"Schmid, Peter J.","year":"2010","journal-title":"J. Fluid Mech.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-1120","issn-type":"print"},{"key":"58","doi-asserted-by":"crossref","unstructured":"P. J. Schmid, Application of the dynamic mode decomposition to experimental data, Exp. Fluids 50 (2011), no. 4, 1123\u20131130.","DOI":"10.1007\/s00348-010-0911-3"},{"issue":"5","key":"59","doi-asserted-by":"publisher","first-page":"683","DOI":"10.1002\/cpa.3160470505","article-title":"Finite-dimensional Lie-Poisson approximations to Vlasov-Poisson equations","volume":"47","author":"Scovel, Clint","year":"1994","journal-title":"Comm. Pure Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0010-3640","issn-type":"print"},{"key":"60","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1016\/j.laa.2020.09.015","article-title":"Column subset selection is NP-complete","volume":"610","author":"Shitov, Yaroslav","year":"2021","journal-title":"Linear Algebra Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0024-3795","issn-type":"print"},{"key":"61","doi-asserted-by":"crossref","unstructured":"J. Squire, H. Qin, and W. M. Tang, Geometric integration of the Vlasov-Maxwell system with a variational particle-in-cell scheme, Phys. Plasmas 19 (2012), no. 8, 084501.","DOI":"10.1063\/1.4742985"},{"key":"62","doi-asserted-by":"crossref","unstructured":"R. D. Sydora, Low-noise electromagnetic and relativistic particle-in-cell plasma simulation models, J. Comput. Appl. Math. 109 (1999), no. 1\u20132, 243\u2013259.","DOI":"10.1016\/S0377-0427(99)00161-2"},{"issue":"4","key":"63","doi-asserted-by":"publisher","first-page":"043303","DOI":"10.1103\/physreve.94.043303","article-title":"Explicit symplectic approximation of nonseparable Hamiltonians: algorithm and long time performance","volume":"94","author":"Tao, Molei","year":"2016","journal-title":"Phys. Rev. E","ISSN":"https:\/\/id.crossref.org\/issn\/2470-0045","issn-type":"print"},{"issue":"2","key":"64","doi-asserted-by":"publisher","first-page":"391","DOI":"10.3934\/jcd.2014.1.391","article-title":"On dynamic mode decomposition: theory and applications","volume":"1","author":"Tu, Jonathan H.","year":"2014","journal-title":"J. Comput. Dyn.","ISSN":"https:\/\/id.crossref.org\/issn\/2158-2491","issn-type":"print"},{"key":"65","unstructured":"T. M. Tyranowski and M. Kraus, Symplectic model reduction methods for the Vlasov equation, Preprint,  arXiv:1910.06026, 2019."},{"key":"66","doi-asserted-by":"crossref","unstructured":"J. Xiao, H. Qin, J. Liu, Y. He, R. Zhang, and Y. Sun, Explicit high-order non-canonical symplectic particle-in-cell algorithms for Vlasov-Maxwell systems, Phys. Plasmas 22 (2015), no. 11, 112504.","DOI":"10.1063\/1.4935904"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2024-93-347\/S0025-5718-2023-03885-6\/S0025-5718-2023-03885-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:15:53Z","timestamp":1776834953000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2024-93-347\/S0025-5718-2023-03885-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,24]]},"references-count":66,"journal-issue":{"issue":"347","published-print":{"date-parts":[[2024,5]]}},"alternative-id":["S0025-5718-2023-03885-6"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3885","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2023,8,24]]}}}