[{"date_created":"2025-02-16T23:02:34Z","date_updated":"2025-09-30T10:30:31Z","quality_controlled":"1","file_date_updated":"2025-02-17T08:32:23Z","publication":"SIAM Journal on Numerical Analysis","type":"journal_article","citation":{"mla":"Cornalba, Federico, and Julian L. Fischer. “Multilevel Monte Carlo Methods for the Dean–Kawasaki Equation from Fluctuating Hydrodynamics.” <i>SIAM Journal on Numerical Analysis</i>, vol. 63, no. 1, Society for Industrial and Applied Mathematics, 2025, pp. 262–87, doi:<a href=\"https://doi.org/10.1137/23M1617345\">10.1137/23M1617345</a>.","apa":"Cornalba, F., &#38; Fischer, J. L. (2025). Multilevel Monte Carlo methods for the Dean–Kawasaki equation from fluctuating hydrodynamics. <i>SIAM Journal on Numerical Analysis</i>. Society for Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/23M1617345\">https://doi.org/10.1137/23M1617345</a>","chicago":"Cornalba, Federico, and Julian L Fischer. “Multilevel Monte Carlo Methods for the Dean–Kawasaki Equation from Fluctuating Hydrodynamics.” <i>SIAM Journal on Numerical Analysis</i>. Society for Industrial and Applied Mathematics, 2025. <a href=\"https://doi.org/10.1137/23M1617345\">https://doi.org/10.1137/23M1617345</a>.","ieee":"F. Cornalba and J. L. Fischer, “Multilevel Monte Carlo methods for the Dean–Kawasaki equation from fluctuating hydrodynamics,” <i>SIAM Journal on Numerical Analysis</i>, vol. 63, no. 1. Society for Industrial and Applied Mathematics, pp. 262–287, 2025.","ama":"Cornalba F, Fischer JL. Multilevel Monte Carlo methods for the Dean–Kawasaki equation from fluctuating hydrodynamics. <i>SIAM Journal on Numerical Analysis</i>. 2025;63(1):262-287. doi:<a href=\"https://doi.org/10.1137/23M1617345\">10.1137/23M1617345</a>","short":"F. Cornalba, J.L. Fischer, SIAM Journal on Numerical Analysis 63 (2025) 262–287.","ista":"Cornalba F, Fischer JL. 2025. Multilevel Monte Carlo methods for the Dean–Kawasaki equation from fluctuating hydrodynamics. SIAM Journal on Numerical Analysis. 63(1), 262–287."},"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","oa":1,"volume":63,"article_type":"original","_id":"19027","isi":1,"department":[{"_id":"JuFi"}],"page":"262-287","has_accepted_license":"1","publication_identifier":{"issn":["0036-1429"],"eissn":["1095-7170"]},"intvolume":"        63","ddc":["510"],"month":"02","language":[{"iso":"eng"}],"tmp":{"image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"abstract":[{"lang":"eng","text":"Stochastic PDEs of fluctuating hydrodynamics are a powerful tool for the description of fluctuations in many-particle systems. In this paper, we develop and analyze a multilevel Monte Carlo (MLMC) scheme for the Dean–Kawasaki equation, a pivotal representative of this class of SPDEs. We prove analytically and demonstrate numerically that our MLMC scheme provides a significant reduction in computational cost (with respect to a standard Monte Carlo method) in the simulation of the Dean–Kawasaki equation. Specifically, we link this reduction in cost to having a sufficiently large average particle density and show that sizeable cost reductions can be obtained even when we have solutions with regions of low density. Numerical simulations are provided in the two-dimensional case, confirming our theoretical predictions. Our results are formulated entirely in terms of the law of distributions rather than in terms of strong spatial norms: this crucially allows for MLMC speed-ups altogether despite the Dean–Kawasaki equation being highly singular."}],"publication_status":"published","OA_place":"publisher","scopus_import":"1","year":"2025","status":"public","corr_author":"1","author":[{"last_name":"Cornalba","id":"2CEB641C-A400-11E9-A717-D712E6697425","orcid":"0000-0002-6269-5149","full_name":"Cornalba, Federico","first_name":"Federico"},{"last_name":"Fischer","first_name":"Julian L","full_name":"Fischer, Julian L","orcid":"0000-0002-0479-558X","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87"}],"publisher":"Society for Industrial and Applied Mathematics","arxiv":1,"date_published":"2025-02-01T00:00:00Z","file":[{"date_created":"2025-02-17T08:32:23Z","content_type":"application/pdf","date_updated":"2025-02-17T08:32:23Z","success":1,"checksum":"53505647e848ed50f7e0d00c369b14e7","access_level":"open_access","file_size":2435019,"file_id":"19029","relation":"main_file","creator":"dernst","file_name":"2025_SIAMNumerAnaly_Cornalba.pdf"}],"article_processing_charge":"Yes (in subscription journal)","day":"01","acknowledgement":"The work of the authors was supported by the Austrian Science Fund (FWF) projectF65.","oa_version":"Published Version","title":"Multilevel Monte Carlo methods for the Dean–Kawasaki equation from fluctuating hydrodynamics","doi":"10.1137/23M1617345","OA_type":"hybrid","project":[{"name":"Taming Complexity in Partial Differential Systems","_id":"fc31cba2-9c52-11eb-aca3-ff467d239cd2","grant_number":"F6504"}],"external_id":{"arxiv":["2311.08872"],"isi":["001447583400011"]},"issue":"1"},{"OA_type":"closed access","title":"Analysis of a modified second-order mixed hybrid BDM1 finite element method for transport problems in divergence form","doi":"10.1137/15M1035379","oa_version":"None","publist_id":"5954","issue":"4","status":"public","year":"2016","publication_status":"published","abstract":[{"text":"We prove optimal second order convergence of a modified lowest-order Brezzi-Douglas-Marini (BDM1) mixed finite element scheme for advection-diffusion problems in divergence form. If advection is present, it is known that the total flux is approximated only with first-order accuracy by the classical BDM1 mixed method, which is suboptimal since the same order of convergence is obtained if the computationally less expensive Raviart-Thomas (RT0) element is used. The modification that was first proposed by Brunner et al. [Adv. Water Res., 35 (2012),pp. 163-171] is based on the hybrid problem formulation and consists in using the Lagrange multipliers for the discretization of the advective term instead of the cellwise constant approximation of the scalar unknown.","lang":"eng"}],"article_processing_charge":"No","day":"02","date_published":"2016-08-02T00:00:00Z","publisher":"Society for Industrial and Applied Mathematics ","author":[{"last_name":"Brunner","first_name":"Fabian","full_name":"Brunner, Fabian"},{"full_name":"Fischer, Julian L","first_name":"Julian L","id":"2C12A0B0-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-0479-558X","last_name":"Fischer"},{"last_name":"Knabner","full_name":"Knabner, Peter","first_name":"Peter"}],"publication_identifier":{"issnl":["0036-1429"],"eissn":["1095-7170"]},"page":"2359 - 2378","_id":"1315","language":[{"iso":"eng"}],"month":"08","intvolume":"        54","citation":{"ieee":"F. Brunner, J. L. Fischer, and P. Knabner, “Analysis of a modified second-order mixed hybrid BDM1 finite element method for transport problems in divergence form,” <i>Journal on Numerical Analysis</i>, vol. 54, no. 4. Society for Industrial and Applied Mathematics , pp. 2359–2378, 2016.","ista":"Brunner F, Fischer JL, Knabner P. 2016. Analysis of a modified second-order mixed hybrid BDM1 finite element method for transport problems in divergence form. Journal on Numerical Analysis. 54(4), 2359–2378.","ama":"Brunner F, Fischer JL, Knabner P. Analysis of a modified second-order mixed hybrid BDM1 finite element method for transport problems in divergence form. <i>Journal on Numerical Analysis</i>. 2016;54(4):2359-2378. doi:<a href=\"https://doi.org/10.1137/15M1035379\">10.1137/15M1035379</a>","short":"F. Brunner, J.L. Fischer, P. Knabner, Journal on Numerical Analysis 54 (2016) 2359–2378.","mla":"Brunner, Fabian, et al. “Analysis of a Modified Second-Order Mixed Hybrid BDM1 Finite Element Method for Transport Problems in Divergence Form.” <i>Journal on Numerical Analysis</i>, vol. 54, no. 4, Society for Industrial and Applied Mathematics , 2016, pp. 2359–78, doi:<a href=\"https://doi.org/10.1137/15M1035379\">10.1137/15M1035379</a>.","chicago":"Brunner, Fabian, Julian L Fischer, and Peter Knabner. “Analysis of a Modified Second-Order Mixed Hybrid BDM1 Finite Element Method for Transport Problems in Divergence Form.” <i>Journal on Numerical Analysis</i>. Society for Industrial and Applied Mathematics , 2016. <a href=\"https://doi.org/10.1137/15M1035379\">https://doi.org/10.1137/15M1035379</a>.","apa":"Brunner, F., Fischer, J. L., &#38; Knabner, P. (2016). Analysis of a modified second-order mixed hybrid BDM1 finite element method for transport problems in divergence form. <i>Journal on Numerical Analysis</i>. Society for Industrial and Applied Mathematics . <a href=\"https://doi.org/10.1137/15M1035379\">https://doi.org/10.1137/15M1035379</a>"},"type":"journal_article","publication":"Journal on Numerical Analysis","date_updated":"2026-05-18T09:48:39Z","quality_controlled":"1","date_created":"2018-12-11T11:51:19Z","extern":"1","article_type":"original","volume":54,"user_id":"ba8df636-2132-11f1-aed0-ed93e2281fdd","keyword":["advection-diffusion problem","mixed finite element methods","suboptimal conver-gence","optimal convergence"]}]