{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T05:37:10Z","timestamp":1775540230619,"version":"3.50.1"},"reference-count":57,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2024,8,15]],"date-time":"2024-08-15T00:00:00Z","timestamp":1723680000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,8,15]],"date-time":"2024-08-15T00:00:00Z","timestamp":1723680000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Found Comput Math"],"published-print":{"date-parts":[[2024,12]]},"DOI":"10.1007\/s10208-023-09625-8","type":"journal-article","created":{"date-parts":[[2024,8,15]],"date-time":"2024-08-15T16:01:50Z","timestamp":1723737710000},"page":"1819-1869","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Approximations of Dispersive PDEs in the Presence of Low-Regularity Randomness"],"prefix":"10.1007","volume":"24","author":[{"given":"Yvonne","family":"Alama Bronsard","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yvain","family":"Bruned","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Katharina","family":"Schratz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,8,15]]},"reference":[{"key":"9625_CR1","unstructured":"Y. Alama Bronsard, Y. Bruned, K. Schratz, Low regularity integrators via decorated trees. arXiv:2202.01171."},{"key":"9625_CR2","doi-asserted-by":"publisher","unstructured":"Y. Alama Bronsard, Error analysis of a class of semi-discrete schemes for solving the Gross-Pitaevskii equation at low regularity. J. Comput. Appl. Math., 418, (2023), p. 114632. https:\/\/doi.org\/10.1016\/j.cam.2022.114632.","DOI":"10.1016\/j.cam.2022.114632"},{"key":"9625_CR3","unstructured":"Y. Alama Bronsard, A symmetric low-regularity integrator for the nonlinear Schr\u00f6dinger equation. arXiv:2301.13109."},{"key":"9625_CR4","unstructured":"I. Ampatzoglou, C. Collot, P. Germain, Derivation of the kinetic wave equation for quadratic dispersive problems in the inhomogeneous setting . arXiv:2107.11819."},{"key":"9625_CR5","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1016\/j.apnum.2023.01.002","volume":"186","author":"C-E Br\u00e9hier","year":"2023","unstructured":"C-E. Br\u00e9hier, D. Cohen, Analysis of a splitting scheme for a class of nonlinear stochastic Schr\u00f6dinger equations. Applied Numerical Mathematics, 186, (2023), p.57-83. https:\/\/doi.org\/10.1016\/j.apnum.2023.01.002.","journal-title":"Applied Numerical Mathematics"},{"key":"9625_CR6","unstructured":"E. Bronasco, Exotic B-series and S-series: algebraic structures and order conditions for invariant measure sampling, arXiv:2209.11046."},{"key":"9625_CR7","doi-asserted-by":"publisher","unstructured":"Y. Bruned, A. Chandra, I. Chevyrev, M. Hairer, Renormalising SPDEs in regularity structures. J. Eur. Math. Soc. (JEMS), 23, no.\u00a03, (2021), 869-947. https:\/\/doi.org\/10.4171\/JEMS\/1025.","DOI":"10.4171\/JEMS\/1025"},{"key":"9625_CR8","unstructured":"Y. Bruned, K. Ebrahimi-Fard, Bogoliubov type recursions for renormalisation in regularity structures. To appear in Annales de l\u2019Institut Henri Poincar\u00e9 (D) Combinatorics, Physics and their Interactions. arXiv:2006.05284."},{"key":"9625_CR9","doi-asserted-by":"publisher","unstructured":"Y. Bruned, M. Hairer, L. Zambotti. Algebraic renormalisation of regularity structures. Invent. Math. 215, no.\u00a03, (2019), 1039\u20131156. https:\/\/doi.org\/10.1007\/s00222-018-0841-x.","DOI":"10.1007\/s00222-018-0841-x"},{"key":"9625_CR10","doi-asserted-by":"publisher","unstructured":"Y. Bruned, M. Hairer, L. Zambotti. Renormalisation of Stochastic Partial Differential Equations. EMS Newsletter 115, no.\u00a03, (2020), 7\u201311. https:\/\/doi.org\/10.4171\/NEWS\/115\/3.","DOI":"10.4171\/NEWS\/115\/3"},{"key":"9625_CR11","doi-asserted-by":"publisher","unstructured":"Y. Bruned, D. Manchon. Algebraic deformation for (S)PDEs. J. Math. Soc. Japan, 75, no.\u00a02, (2023), 485\u2013526. https:\/\/doi.org\/10.2969\/jmsj\/88028802.","DOI":"10.2969\/jmsj\/88028802"},{"key":"9625_CR12","unstructured":"Y.\u00a0Bruned, U.\u00a0Nadeem, Convergence of space-discretised gKPZ via Regularity Structures. arXiv:2207.09946."},{"key":"9625_CR13","doi-asserted-by":"publisher","unstructured":"Y.\u00a0Bruned, K.\u00a0Schratz. Resonance based schemes for dispersive equations via decorated trees. Forum of Mathematics, Pi, 10, E2. https:\/\/doi.org\/10.1017\/fmp.2021.13.","DOI":"10.1017\/fmp.2021.13"},{"key":"9625_CR14","doi-asserted-by":"publisher","unstructured":"N. Burq, N. Tzvetkov. Random data Cauchy theory for supercritical wave equations. I. Local theory. Invent. Math. 173, no.\u00a03, (2008), 449\u2013475. https:\/\/doi.org\/10.1007\/s00222-008-0124-z.","DOI":"10.1007\/s00222-008-0124-z"},{"key":"9625_CR15","doi-asserted-by":"publisher","unstructured":"N. Burq, N. Tzvetkov. Probabilistic well-posedness for the cubic wave equation. J. Eur. Math. Soc. (JEMS) 16, no.\u00a01, (2014), 1\u201330. https:\/\/doi.org\/10.4171\/JEMS\/426","DOI":"10.4171\/JEMS\/426"},{"key":"9625_CR16","doi-asserted-by":"publisher","first-page":"79","DOI":"10.2307\/2004720","volume":"26","author":"JC Butcher","year":"1972","unstructured":"J. C. Butcher, An algebraic theory of integration methods. Math. Comp. 26, (1972), 79\u2013106. https:\/\/doi.org\/10.2307\/2004720.","journal-title":"Math. Comp."},{"key":"9625_CR17","unstructured":"N. Camps, L. Gassot, S. Ibrahim, Refined probabilistic local well-posedness for a cubic Schr\u00f6dinger half-wave equation. arXiv:2209.14116."},{"key":"9625_CR18","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1007\/s10208-007-9016-7","volume":"8","author":"E Celledoni","year":"2008","unstructured":"E. Celledoni, D. Cohen, B. Owren, Symmetric exponential integrators with an application to the cubic Schr\u00f6dinger equation. Found. Comput. Math. 8, (2008), 303\u2013317. https:\/\/doi.org\/10.1007\/s10208-007-9016-7.","journal-title":"Found. Comput. Math."},{"key":"9625_CR19","unstructured":"A.\u00a0Chandra, M.\u00a0Hairer. An analytic BPHZ theorem for regularity structures. arXiv:1612.08138."},{"key":"9625_CR20","doi-asserted-by":"publisher","first-page":"877","DOI":"10.1007\/s10543-012-0385-1","volume":"52","author":"D Cohen","year":"2012","unstructured":"D. Cohen, L. Gauckler, One-stage exponential integrators for nonlinear Schr\u00f6dinger equations over long times. BIT 52, (2012), 877\u2013903. https:\/\/doi.org\/10.1007\/s10543-012-0385-1.","journal-title":"BIT"},{"key":"9625_CR21","unstructured":"C. Collot, P. Germain On the derivation of the homogeneous kinetic wave equation. To appear in Comm. Pure Appl. Math. arXiv:1912.10368."},{"key":"9625_CR22","doi-asserted-by":"publisher","unstructured":"A. Connes, D. Kreimer, Hopf algebras, renormalization and noncommutative geometry. Comm. Math. Phys. 199, no.\u00a01, (1998), 203\u2013242. https:\/\/doi.org\/10.1007\/s002200050499.","DOI":"10.1007\/s002200050499"},{"key":"9625_CR23","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1007\/s002200050779","volume":"210","author":"A Connes","year":"2000","unstructured":"A. Connes, D. Kreimer, Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem. Commun. Math. Phys. 210, (2000), 249\u201373. https:\/\/doi.org\/10.1007\/s002200050779.","journal-title":"Commun. Math. Phys."},{"key":"9625_CR24","doi-asserted-by":"publisher","unstructured":"J. Colliander, O. Tadahiro, Almost sure well-posedness of the cubic nonlinear Schr\u00f6dinger equation below $$L^{2}({\\textbf{T}})$$, Duke Math. J. 161, no.\u00a03, (2012), 367\u2013414. https:\/\/doi.org\/10.1215\/00127094-1507400.","DOI":"10.1215\/00127094-1507400"},{"key":"9625_CR25","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1090\/S0025-5718-2010-02395-6","volume":"80","author":"A Debussche","year":"2011","unstructured":"A. Debussche, Weak approximation of stochastic partial differential equations: the nonlinear case. Math. Comput. 80 (2011): 89\u2013117.","journal-title":"Math. Comput."},{"key":"9625_CR26","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1090\/S0025-5718-2010-02395-6","volume":"80","author":"A Debussche","year":"2011","unstructured":"M. Christ, Power series solution of a nonlinear Schr\u00f6dinger equation. In Mathematical aspects of nonlinear dispersive equations, volume 163 of Ann. of Math. Stud., pages 131\u2013155. Princeton Univ. Press, Princeton, NJ, 2007.","journal-title":"Math. Comput."},{"key":"9625_CR27","unstructured":"Y.\u00a0Deng, Z.\u00a0Hani, Full derivation of the wave kinetic equation. To appear in Invent. Math. arXiv:2104.11204."},{"key":"9625_CR28","doi-asserted-by":"publisher","DOI":"10.1017\/fmp.2021.6","volume":"9","author":"Y Deng","year":"2021","unstructured":"Y.\u00a0Deng, Z.\u00a0Hani, On the derivation of the wave kinetic equation for NLS. Forum of Mathematics, Pi, 9, (2021), e6. https:\/\/doi.org\/10.1017\/fmp.2021.6.","journal-title":"Forum of Mathematics, Pi"},{"key":"9625_CR29","doi-asserted-by":"publisher","unstructured":"Y.\u00a0Deng, A.\u00a0R. Nahmod, H.\u00a0Yue, Random tensors, propagation of randomness, and nonlinear dispersive equations. Invent. Math. 228, no.\u00a02, (2022), 539\u2013686. https:\/\/doi.org\/10.1007\/s00222-021-01084-8.","DOI":"10.1007\/s00222-021-01084-8"},{"key":"9625_CR30","unstructured":"D.\u00a0Erhard, M.\u00a0Hairer. A scaling limit of the parabolic Anderson model with exclusion interaction. arXiv:2103.13479."},{"key":"9625_CR31","unstructured":"Y. Feng, G. Maierhofer, K. Schratz, Long-time error bounds of low-regularity integrators for nonlinear Schr\u00f6dinger equations. arXiv:2302.00383."},{"key":"9625_CR32","doi-asserted-by":"publisher","DOI":"10.4171\/100","volume-title":"Geometric Numerical Integration and Schr\u00f6dinger Equations","author":"E Faou","year":"2012","unstructured":"E. Faou, Geometric Numerical Integration and Schr\u00f6dinger Equations. European Math. Soc. Publishing House, Z\u00fcrich 2012."},{"key":"9625_CR33","doi-asserted-by":"publisher","unstructured":"Z. Guo, S. Kwon, T. Oh, Poincar\u00e9-Dulac normal form reduction for unconditional well-posedness of the periodic cubic NLS, Comm. Math. Phys. 322, no.\u00a01, (2013), 19-48. https:\/\/doi.org\/10.1007\/s00220-013-1755-5.","DOI":"10.1007\/s00220-013-1755-5"},{"key":"9625_CR34","doi-asserted-by":"publisher","unstructured":"M. Gubinelli, Rough solutions for the periodic Korteweg-de Vries equation. Comm. Pure Appl. Anal. 11, no.\u00a04, (2012), 709\u2013733. https:\/\/doi.org\/10.3934\/cpaa.2012.11.709.","DOI":"10.3934\/cpaa.2012.11.709"},{"key":"9625_CR35","doi-asserted-by":"publisher","unstructured":"M. Hairer, A theory of regularity structures. Invent. Math. 198, no.\u00a02, (2014), 269\u2013504. https:\/\/doi.org\/10.1007\/s00222-014-0505-4.","DOI":"10.1007\/s00222-014-0505-4"},{"key":"9625_CR36","doi-asserted-by":"publisher","DOI":"10.4171\/078","volume-title":"Splitting for Partial Differential Equations with Rough Solutions","author":"H Holden","year":"2010","unstructured":"H. Holden, K. H. Karlsen, K.-A. Lie, N. H. Risebro, Splitting for Partial Differential Equations with Rough Solutions. European Math. Soc. Publishing House, Z\u00fcrich 2010."},{"key":"9625_CR37","volume-title":"Geometric Numerical Integration","author":"E Hairer","year":"2006","unstructured":"E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Second edition, Springer, Berlin 2006."},{"key":"9625_CR38","doi-asserted-by":"publisher","unstructured":"M. Hochbruck, A. Ostermann, Exponential integrators. Acta Numer. 19, (2010), 209\u2013286. https:\/\/doi.org\/10.1017\/S0962492910000048https:\/\/www.cambridge.org\/core\/journals\/acta-numerica\/article\/abs\/exponential-integrators\/8ED12FD70C2491C4F3FB7A0ACF922FCD.","DOI":"10.1017\/S0962492910000048"},{"key":"9625_CR39","doi-asserted-by":"publisher","unstructured":"L. Ignat, E. Zuazua, Numerical dispersive schemes for the nonlinear Schr\u00f6dinger equation. SIAM J. Numer. Anal. 47, no.\u00a02, (2009), 1366\u20131390. https:\/\/doi.org\/10.1137\/070683787.","DOI":"10.1137\/070683787"},{"key":"9625_CR40","unstructured":"L. Ji, A. Ostermann, F. Rousset, K. Schratz, Low regularity error estimates for the time integration of 2D NLS. arXiv:2301.10639."},{"key":"9625_CR41","doi-asserted-by":"publisher","unstructured":"A. Laurent, G. Vilmart, Exotic aromatic B-series for the study of long time integrators for a class of ergodic SDEs. Math. Comput. 89, no.\u00a0321, (2020) , 169\u2013202 . https:\/\/doi.org\/10.1090\/mcom\/3455.","DOI":"10.1090\/mcom\/3455"},{"key":"9625_CR42","doi-asserted-by":"publisher","first-page":"372","DOI":"10.1137\/0704033","volume":"4","author":"JD Lawson","year":"1967","unstructured":"J. D. Lawson, Generalized Runge\u2013Kutta processes for stable systems with large Lipschitz constants. SIAM J. Numer. Anal. 4:372\u2013380 (1967).","journal-title":"SIAM J. Numer. Anal."},{"key":"9625_CR43","doi-asserted-by":"publisher","unstructured":"C. Lubich, On splitting methods for Schr\u00f6dinger-Poisson and cubic nonlinear Schr\u00f6dinger equations. Math. Comp. 77, no.\u00a04, (2008), 2141\u20132153. https:\/\/doi.org\/10.1090\/S0025-5718-08-02101-7.","DOI":"10.1090\/S0025-5718-08-02101-7"},{"key":"9625_CR44","doi-asserted-by":"crossref","unstructured":"B. Leimkuhler, S. Reich, Simulating Hamiltonian dynamics. Cambridge Monographs on Applied and Computational Mathematics 14. Cambridge University Press, Cambridge, 2004.","DOI":"10.1017\/CBO9780511614118"},{"key":"9625_CR45","doi-asserted-by":"crossref","unstructured":"G. Lord, C.E. Powell, T. Shardlow, An Introduction to Computational Stochastic PDEs. Cambridge Texts in Applied Mathematics, 2014.","DOI":"10.1017\/CBO9781139017329"},{"key":"9625_CR46","unstructured":"G. Maierhofer, K. Schratz, Bridging the gap: symplecticity and low regularity on the example of the KdV equation. arXiv:2205.05024."},{"key":"9625_CR47","doi-asserted-by":"publisher","unstructured":"R.I. McLachlan, G.R.W. Quispel, Splitting methods. Acta Numer. 11, (2002), 341\u2013434. https:\/\/doi.org\/10.1017\/S0962492902000053https:\/\/www.cambridge.org\/core\/journals\/acta-numerica\/article\/abs\/splitting-methods\/122F5736DAF3D88598989E68FE4D2EF2.","DOI":"10.1017\/S0962492902000053"},{"key":"9625_CR48","doi-asserted-by":"publisher","first-page":"583","DOI":"10.1007\/s10208-013-9167-7","volume":"13","author":"HZ Munthe-Kaas","year":"2013","unstructured":"H. Z Munthe-Kaas, A. Lundervold, On post-Lie algebras, Lie-Butcher series and moving frames. Found. Comput. Math. 13, (2013), 583-613. https:\/\/doi.org\/10.1007\/s10208-013-9167-7","journal-title":"Found. Comput. Math."},{"key":"9625_CR49","doi-asserted-by":"publisher","unstructured":"A. R. Nahmod, G. Staffilani, Almost sure well-posedness for the periodic 3D quintic nonlinear Schr\u00f6dinger equation below the energy space. J. Eur. Math. Soc. (JEMS) 17, no.\u00a07, (2015), 1687-1759. https:\/\/doi.org\/10.4171\/JEMS\/543.","DOI":"10.4171\/JEMS\/543"},{"key":"9625_CR50","doi-asserted-by":"publisher","first-page":"725","DOI":"10.1007\/s10208-020-09468-7","volume":"21","author":"A Ostermann","year":"2021","unstructured":"A. Ostermann, F. Rousset, K. Schratz, Error estimates of a Fourier integrator for the cubic Schr\u00f6dinger equation at low regularity. Found. Comput. Math. 21, (2021), 725-765. https:\/\/doi.org\/10.1007\/s10208-020-09468-7.","journal-title":"Found. Comput. Math."},{"key":"9625_CR51","doi-asserted-by":"publisher","unstructured":"A. Ostermann, F. Rousset, K. Schratz, Fourier integrator for periodic NLS: low regularity estimates via discrete Bourgain spaces. J. Eur. Math. Soc. (JEMS) 25, no. 10, (2023), 3913\u20133952. https:\/\/doi.org\/10.4171\/JEMS\/1275.","DOI":"10.4171\/JEMS\/1275"},{"key":"9625_CR52","doi-asserted-by":"publisher","first-page":"731","DOI":"10.1007\/s10208-017-9352-1","volume":"18","author":"A Ostermann","year":"2018","unstructured":"A.\u00a0Ostermann, K.\u00a0Schratz, Low regularity exponential-type integrators for semilinear Schr\u00f6dinger equations. Found. Comput. Math. 18, (2018), 731\u2013755. https:\/\/doi.org\/10.1007\/s10208-017-9352-1.","journal-title":"Found. Comput. Math."},{"key":"9625_CR53","doi-asserted-by":"crossref","unstructured":"F. Rousset, K. Schratz, A general framework of low regularity integrators, SIAM J. Numer. Anal. 59, (2021) 1735\u20131768. arXiv:2010.01640.","DOI":"10.1137\/20M1371506"},{"key":"9625_CR54","doi-asserted-by":"publisher","unstructured":"F. Rousset, K. Schratz, Convergence error estimates at low regularity for time discretizations of KdV. Pure Appl. Anal. 4, no.\u00a01, (2022), 127-152. https:\/\/doi.org\/10.2140\/paa.2022.4.127.","DOI":"10.2140\/paa.2022.4.127"},{"key":"9625_CR55","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-3093-4","volume-title":"Numerical Hamiltonian Problems","author":"JM Sanz-Serna","year":"1994","unstructured":"J.M. Sanz-Serna, M.P. Calvo, Numerical Hamiltonian Problems. Chapman and Hall, London, 1994."},{"key":"9625_CR56","unstructured":"G.\u00a0Staffilani, M.-B. Tran. On the wave turbulence theory for stochastic and random multidimensional kdv type equations. arXiv:2106.09819."},{"key":"9625_CR57","doi-asserted-by":"publisher","unstructured":"C.\u00a0 Sun, B.\u00a0 Xia Probabilistic well-posedness for supercritical wave equations with periodic boundary condition on dimension three. Illinois J. Math. 60.2 (2016), pp. 481\u2013503. https:\/\/doi.org\/10.1215\/ijm\/1499760018.","DOI":"10.1215\/ijm\/1499760018"}],"container-title":["Foundations of Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-023-09625-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10208-023-09625-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-023-09625-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,12,4]],"date-time":"2024-12-04T17:05:46Z","timestamp":1733331946000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10208-023-09625-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,15]]},"references-count":57,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2024,12]]}},"alternative-id":["9625"],"URL":"https:\/\/doi.org\/10.1007\/s10208-023-09625-8","relation":{},"ISSN":["1615-3375","1615-3383"],"issn-type":[{"value":"1615-3375","type":"print"},{"value":"1615-3383","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,8,15]]},"assertion":[{"value":"3 June 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 April 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 July 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 August 2024","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}