A continuation multilevel Monte Carlo algorithm

Handle URI:
http://hdl.handle.net/10754/563752
Title:
A continuation multilevel Monte Carlo algorithm
Authors:
Collier, Nathan; Haji Ali, Abdul Lateef ( 0000-0002-6243-0335 ) ; Nobile, Fabio; von Schwerin, Erik; Tempone, Raul ( 0000-0003-1967-4446 )
Abstract:
We propose a novel Continuation Multi Level Monte Carlo (CMLMC) algorithm for weak approximation of stochastic models. The CMLMC algorithm solves the given approximation problem for a sequence of decreasing tolerances, ending when the required error tolerance is satisfied. CMLMC assumes discretization hierarchies that are defined a priori for each level and are geometrically refined across levels. The actual choice of computational work across levels is based on parametric models for the average cost per sample and the corresponding variance and weak error. These parameters are calibrated using Bayesian estimation, taking particular notice of the deepest levels of the discretization hierarchy, where only few realizations are available to produce the estimates. The resulting CMLMC estimator exhibits a non-trivial splitting between bias and statistical contributions. We also show the asymptotic normality of the statistical error in the MLMC estimator and justify in this way our error estimate that allows prescribing both required accuracy and confidence in the final result. Numerical results substantiate the above results and illustrate the corresponding computational savings in examples that are described in terms of differential equations either driven by random measures or with random coefficients. © 2014, Springer Science+Business Media Dordrecht.
KAUST Department:
Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Stochastic Numerics Research Group
Publisher:
Springer Science + Business Media
Journal:
BIT Numerical Mathematics
Issue Date:
5-Sep-2014
DOI:
10.1007/s10543-014-0511-3
Type:
Article
ISSN:
00063835
Sponsors:
Raul Tempone is a member of the Strategic Research Initiative on Uncertainty Quantification in Computational Science and Engineering at KAUST (SRI-UQ). The authors would like to recognize the support of King Abdullah University of Science and Technology (KAUST) AEA project "Predictability and Uncertainty Quantification for Models of Porous Media" and University of Texas at Austin AEA Round 3 "Uncertainty quantification for predictive modeling of the dissolution of porous and fractured media". We would also like to acknowledge the use of the following open source software packages: PETSc [4], PetIGA [8], NumPy, matplotlib [21].
Appears in Collections:
Articles; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorCollier, Nathanen
dc.contributor.authorHaji Ali, Abdul Lateefen
dc.contributor.authorNobile, Fabioen
dc.contributor.authorvon Schwerin, Eriken
dc.contributor.authorTempone, Raulen
dc.date.accessioned2015-08-03T12:08:55Zen
dc.date.available2015-08-03T12:08:55Zen
dc.date.issued2014-09-05en
dc.identifier.issn00063835en
dc.identifier.doi10.1007/s10543-014-0511-3en
dc.identifier.urihttp://hdl.handle.net/10754/563752en
dc.description.abstractWe propose a novel Continuation Multi Level Monte Carlo (CMLMC) algorithm for weak approximation of stochastic models. The CMLMC algorithm solves the given approximation problem for a sequence of decreasing tolerances, ending when the required error tolerance is satisfied. CMLMC assumes discretization hierarchies that are defined a priori for each level and are geometrically refined across levels. The actual choice of computational work across levels is based on parametric models for the average cost per sample and the corresponding variance and weak error. These parameters are calibrated using Bayesian estimation, taking particular notice of the deepest levels of the discretization hierarchy, where only few realizations are available to produce the estimates. The resulting CMLMC estimator exhibits a non-trivial splitting between bias and statistical contributions. We also show the asymptotic normality of the statistical error in the MLMC estimator and justify in this way our error estimate that allows prescribing both required accuracy and confidence in the final result. Numerical results substantiate the above results and illustrate the corresponding computational savings in examples that are described in terms of differential equations either driven by random measures or with random coefficients. © 2014, Springer Science+Business Media Dordrecht.en
dc.description.sponsorshipRaul Tempone is a member of the Strategic Research Initiative on Uncertainty Quantification in Computational Science and Engineering at KAUST (SRI-UQ). The authors would like to recognize the support of King Abdullah University of Science and Technology (KAUST) AEA project "Predictability and Uncertainty Quantification for Models of Porous Media" and University of Texas at Austin AEA Round 3 "Uncertainty quantification for predictive modeling of the dissolution of porous and fractured media". We would also like to acknowledge the use of the following open source software packages: PETSc [4], PetIGA [8], NumPy, matplotlib [21].en
dc.publisherSpringer Science + Business Mediaen
dc.subjectBayesian inferenceen
dc.subjectMonte Carloen
dc.subjectMultilevel Monte Carloen
dc.subjectPartial differential equations with random dataen
dc.subjectStochastic differential equationsen
dc.titleA continuation multilevel Monte Carlo algorithmen
dc.typeArticleen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentStochastic Numerics Research Groupen
dc.identifier.journalBIT Numerical Mathematicsen
dc.contributor.institutionEnvironmental Sciences Division, Oak Ridge National Lab, Climate Change Science Institute (CCSI), Oak Ridge, United Statesen
dc.contributor.institutionMATHICSE-CSQI, EPF de Lausanne, Lausanne, Switzerlanden
dc.contributor.institutionDepartment of Mathematics, Kungliga Tekniska Högskolan, Stockholm, Swedenen
kaust.authorTempone, Raulen
kaust.authorHaji Ali, Abdul Lateefen
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