Jornet, M.; Calatayud, J.; Le Maître, OP.; Cortés, J. (2020). Variance reduction methods and multilevel Monte Carlo strategy for the density estimation of random second order linear differential equations solutions. International Journal for Uncertainty Quantification. 10(5):467-497. https://doi.org/10.1615/Int.J.UncertaintyQuantification.2020032659
Por favor, use este identificador para citar o enlazar este ítem: http://hdl.handle.net/10251/161701
Título:
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Variance reduction methods and multilevel Monte Carlo strategy for the density estimation of random second order linear differential equations solutions
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Autor:
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Jornet, Marc
Calatayud, Julia
Le Maître, Olivier P.
Cortés, J.-C.
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Entidad UPV:
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Universitat Politècnica de València. Departamento de Matemática Aplicada - Departament de Matemàtica Aplicada
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Fecha difusión:
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Resumen:
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[EN] This paper concerns the estimation of the density function of the solution to a random nonautonomous second-order linear differential equation with analytic data processes. In a recent contribution, we proposed to ...[+]
[EN] This paper concerns the estimation of the density function of the solution to a random nonautonomous second-order linear differential equation with analytic data processes. In a recent contribution, we proposed to express the density function as an expectation, and we used a standard Monte Carlo algorithm to approximate the expectation. Although the algorithms worked satisfactorily for most test problems, some numerical challenges emerged for others, due to large statistical errors. In these situations, the convergence of the Monte Carlo simulation slows down severely, and noisy features plague the estimates. In this paper, we focus on computational aspects and propose several variance reduction methods to remedy these issues and speed up the convergence. First, we introduce a pathwise selection of the approximating processes which aims at controlling the variance of the estimator. Second, we propose a hybrid method, combining Monte Carlo and deterministic quadrature rules, to estimate the expectation. Third, we exploit the series expansions of the solutions to design a multilevel Monte Carlo estimator. The proposed methods are implemented and tested on several numerical examples to highlight the theoretical discussions and demonstrate the significant improvements achieved.
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Palabras clave:
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Random linear differential equation
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Probability density function
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Standard and multilevel Monte Carlo simulation
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Analysis of algorithms
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Derechos de uso:
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Reserva de todos los derechos
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Fuente:
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International Journal for Uncertainty Quantification. (issn:
2152-5080
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DOI:
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10.1615/Int.J.UncertaintyQuantification.2020032659
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Editorial:
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Begell House
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Versión del editor:
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https://doi.org/10.1615/Int.J.UncertaintyQuantification.2020032659
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Código del Proyecto:
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info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-89664-P/ES/PROBLEMAS DINAMICOS CON INCERTIDUMBRE SIMULABLE: MODELIZACION MATEMATICA, ANALISIS, COMPUTACION Y APLICACIONES/
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Agradecimientos:
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This work is supported by Spanish "Ministerio de Economia y Competitividad" Grant No. MTM2017-89664P.M. Jornet acknowledges the doctorate scholarship granted by PAID, as well as "Ayudas movilidad de estudiantes de doctorado ...[+]
This work is supported by Spanish "Ministerio de Economia y Competitividad" Grant No. MTM2017-89664P.M. Jornet acknowledges the doctorate scholarship granted by PAID, as well as "Ayudas movilidad de estudiantes de doctorado de la Universitat Polit`ecnica de Valencia para estancias en 2019," for financing his research stay at CMAP. J. Calatayud acknowledges "Fundacio Ferran Sunyer i Balaguer," "Institut d'Estudis Catalans," and the award from "Borses Ferran Sunyer i Balaguer 2019" for funding her research stay at CMAP. All authors are grateful to Inria (Centre de Saclay, DeFi Team), which hosted M. Jornet and J. Calatayud during their research stays at Ecole Polytechnique. The authors thank the reviewers for the valuable comments and suggestions, which have greatly enriched the quality of the paper.
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Tipo:
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Artículo
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