--- OA_place: publisher OA_type: hybrid _id: '19027' 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.' acknowledgement: The work of the authors was supported by the Austrian Science Fund (FWF) projectF65. article_processing_charge: Yes (in subscription journal) article_type: original arxiv: 1 author: - first_name: Federico full_name: Cornalba, Federico id: 2CEB641C-A400-11E9-A717-D712E6697425 last_name: Cornalba orcid: 0000-0002-6269-5149 - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X citation: ama: Cornalba F, Fischer JL. Multilevel Monte Carlo methods for the Dean–Kawasaki equation from fluctuating hydrodynamics. SIAM Journal on Numerical Analysis. 2025;63(1):262-287. doi:10.1137/23M1617345 apa: Cornalba, F., & Fischer, J. L. (2025). Multilevel Monte Carlo methods for the Dean–Kawasaki equation from fluctuating hydrodynamics. SIAM Journal on Numerical Analysis. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/23M1617345 chicago: Cornalba, Federico, and Julian L Fischer. “Multilevel Monte Carlo Methods for the Dean–Kawasaki Equation from Fluctuating Hydrodynamics.” SIAM Journal on Numerical Analysis. Society for Industrial and Applied Mathematics, 2025. https://doi.org/10.1137/23M1617345. ieee: F. Cornalba and J. L. Fischer, “Multilevel Monte Carlo methods for the Dean–Kawasaki equation from fluctuating hydrodynamics,” SIAM Journal on Numerical Analysis, vol. 63, no. 1. Society for Industrial and Applied Mathematics, pp. 262–287, 2025. 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. mla: Cornalba, Federico, and Julian L. Fischer. “Multilevel Monte Carlo Methods for the Dean–Kawasaki Equation from Fluctuating Hydrodynamics.” SIAM Journal on Numerical Analysis, vol. 63, no. 1, Society for Industrial and Applied Mathematics, 2025, pp. 262–87, doi:10.1137/23M1617345. short: F. Cornalba, J.L. Fischer, SIAM Journal on Numerical Analysis 63 (2025) 262–287. corr_author: '1' date_created: 2025-02-16T23:02:34Z date_published: 2025-02-01T00:00:00Z date_updated: 2025-09-30T10:30:31Z day: '01' ddc: - '510' department: - _id: JuFi doi: 10.1137/23M1617345 external_id: arxiv: - '2311.08872' isi: - '001447583400011' file: - access_level: open_access checksum: 53505647e848ed50f7e0d00c369b14e7 content_type: application/pdf creator: dernst date_created: 2025-02-17T08:32:23Z date_updated: 2025-02-17T08:32:23Z file_id: '19029' file_name: 2025_SIAMNumerAnaly_Cornalba.pdf file_size: 2435019 relation: main_file success: 1 file_date_updated: 2025-02-17T08:32:23Z has_accepted_license: '1' intvolume: ' 63' isi: 1 issue: '1' language: - iso: eng month: '02' oa: 1 oa_version: Published Version page: 262-287 project: - _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2 grant_number: F6504 name: Taming Complexity in Partial Differential Systems publication: SIAM Journal on Numerical Analysis publication_identifier: eissn: - 1095-7170 issn: - 0036-1429 publication_status: published publisher: Society for Industrial and Applied Mathematics quality_controlled: '1' scopus_import: '1' status: public title: Multilevel Monte Carlo methods for the Dean–Kawasaki equation from fluctuating hydrodynamics tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345 volume: 63 year: '2025' ... --- _id: '9335' abstract: - lang: eng text: 'Various degenerate diffusion equations exhibit a waiting time phenomenon: depending on the “flatness” of the compactly supported initial datum at the boundary of the support, the support of the solution may not expand for a certain amount of time. We show that this phenomenon is captured by particular Lagrangian discretizations of the porous medium and the thin film equations, and we obtain sufficient criteria for the occurrence of waiting times that are consistent with the known ones for the original PDEs. For the spatially discrete solution, the waiting time phenomenon refers to a deviation of the edge of support from its original position by a quantity comparable to the mesh width, over a mesh-independent time interval. Our proof is based on estimates on the fluid velocity in Lagrangian coordinates. Combining weighted entropy estimates with an iteration technique à la Stampacchia leads to upper bounds on free boundary propagation. Numerical simulations show that the phenomenon is already clearly visible for relatively coarse discretizations.' acknowledgement: This research was supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics”. article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X - first_name: Daniel full_name: Matthes, Daniel last_name: Matthes citation: ama: Fischer JL, Matthes D. The waiting time phenomenon in spatially discretized porous medium and thin film equations. SIAM Journal on Numerical Analysis. 2021;59(1):60-87. doi:10.1137/19M1300017 apa: Fischer, J. L., & Matthes, D. (2021). The waiting time phenomenon in spatially discretized porous medium and thin film equations. SIAM Journal on Numerical Analysis. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/19M1300017 chicago: Fischer, Julian L, and Daniel Matthes. “The Waiting Time Phenomenon in Spatially Discretized Porous Medium and Thin Film Equations.” SIAM Journal on Numerical Analysis. Society for Industrial and Applied Mathematics, 2021. https://doi.org/10.1137/19M1300017. ieee: J. L. Fischer and D. Matthes, “The waiting time phenomenon in spatially discretized porous medium and thin film equations,” SIAM Journal on Numerical Analysis, vol. 59, no. 1. Society for Industrial and Applied Mathematics, pp. 60–87, 2021. ista: Fischer JL, Matthes D. 2021. The waiting time phenomenon in spatially discretized porous medium and thin film equations. SIAM Journal on Numerical Analysis. 59(1), 60–87. mla: Fischer, Julian L., and Daniel Matthes. “The Waiting Time Phenomenon in Spatially Discretized Porous Medium and Thin Film Equations.” SIAM Journal on Numerical Analysis, vol. 59, no. 1, Society for Industrial and Applied Mathematics, 2021, pp. 60–87, doi:10.1137/19M1300017. short: J.L. Fischer, D. Matthes, SIAM Journal on Numerical Analysis 59 (2021) 60–87. date_created: 2021-04-18T22:01:42Z date_published: 2021-01-01T00:00:00Z date_updated: 2023-08-08T13:10:40Z day: '01' department: - _id: JuFi doi: 10.1137/19M1300017 external_id: arxiv: - '1911.04185' isi: - '000625044600003' intvolume: ' 59' isi: 1 issue: '1' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1911.04185 month: '01' oa: 1 oa_version: Preprint page: 60-87 publication: SIAM Journal on Numerical Analysis publication_identifier: issn: - 0036-1429 publication_status: published publisher: Society for Industrial and Applied Mathematics quality_controlled: '1' scopus_import: '1' status: public title: The waiting time phenomenon in spatially discretized porous medium and thin film equations type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 59 year: '2021' ... --- _id: '9352' abstract: - lang: eng text: This paper provides an a priori error analysis of a localized orthogonal decomposition method for the numerical stochastic homogenization of a model random diffusion problem. If the uniformly elliptic and bounded random coefficient field of the model problem is stationary and satisfies a quantitative decorrelation assumption in the form of the spectral gap inequality, then the expected $L^2$ error of the method can be estimated, up to logarithmic factors, by $H+(\varepsilon/H)^{d/2}$, $\varepsilon$ being the small correlation length of the random coefficient and $H$ the width of the coarse finite element mesh that determines the spatial resolution. The proof bridges recent results of numerical homogenization and quantitative stochastic homogenization. acknowledgement: 'This work was initiated while the authors enjoyed the kind hospitality of the Hausdorff Institute for Mathematics in Bonn during the trimester program Multiscale Problems: Algorithms, Numerical Analysis, and Computation. D. Peterseim would like to acknowledge the kind hospitality of the Erwin Schrödinger International Institute for Mathematics and Physics (ESI), where parts of this research were developed under the frame of the thematic program Numerical Analysis of Complex PDE Models in the Sciences.' article_processing_charge: No article_type: original arxiv: 1 author: - first_name: Julian L full_name: Fischer, Julian L id: 2C12A0B0-F248-11E8-B48F-1D18A9856A87 last_name: Fischer orcid: 0000-0002-0479-558X - first_name: Dietmar full_name: Gallistl, Dietmar last_name: Gallistl - first_name: Dietmar full_name: Peterseim, Dietmar last_name: Peterseim citation: ama: Fischer JL, Gallistl D, Peterseim D. A priori error analysis of a numerical stochastic homogenization method. SIAM Journal on Numerical Analysis. 2021;59(2):660-674. doi:10.1137/19M1308992 apa: Fischer, J. L., Gallistl, D., & Peterseim, D. (2021). A priori error analysis of a numerical stochastic homogenization method. SIAM Journal on Numerical Analysis. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/19M1308992 chicago: Fischer, Julian L, Dietmar Gallistl, and Dietmar Peterseim. “A Priori Error Analysis of a Numerical Stochastic Homogenization Method.” SIAM Journal on Numerical Analysis. Society for Industrial and Applied Mathematics, 2021. https://doi.org/10.1137/19M1308992. ieee: J. L. Fischer, D. Gallistl, and D. Peterseim, “A priori error analysis of a numerical stochastic homogenization method,” SIAM Journal on Numerical Analysis, vol. 59, no. 2. Society for Industrial and Applied Mathematics, pp. 660–674, 2021. ista: Fischer JL, Gallistl D, Peterseim D. 2021. A priori error analysis of a numerical stochastic homogenization method. SIAM Journal on Numerical Analysis. 59(2), 660–674. mla: Fischer, Julian L., et al. “A Priori Error Analysis of a Numerical Stochastic Homogenization Method.” SIAM Journal on Numerical Analysis, vol. 59, no. 2, Society for Industrial and Applied Mathematics, 2021, pp. 660–74, doi:10.1137/19M1308992. short: J.L. Fischer, D. Gallistl, D. Peterseim, SIAM Journal on Numerical Analysis 59 (2021) 660–674. date_created: 2021-04-25T22:01:31Z date_published: 2021-03-09T00:00:00Z date_updated: 2023-08-08T13:13:37Z day: '09' department: - _id: JuFi doi: 10.1137/19M1308992 external_id: arxiv: - '1912.11646' isi: - '000646030400003' intvolume: ' 59' isi: 1 issue: '2' language: - iso: eng main_file_link: - open_access: '1' url: https://arxiv.org/abs/1912.11646 month: '03' oa: 1 oa_version: Preprint page: 660-674 publication: SIAM Journal on Numerical Analysis publication_identifier: issn: - 0036-1429 publication_status: published publisher: Society for Industrial and Applied Mathematics quality_controlled: '1' scopus_import: '1' status: public title: A priori error analysis of a numerical stochastic homogenization method type: journal_article user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8 volume: 59 year: '2021' ...