Quasi optimal and adaptive sparse grids with control variates for PDEs with random diffusion coefficient

Handle URI:
http://hdl.handle.net/10754/624842
Title:
Quasi optimal and adaptive sparse grids with control variates for PDEs with random diffusion coefficient
Authors:
Tamellini, Lorenzo
Abstract:
In this talk we discuss possible strategies to minimize the impact of the curse of dimensionality effect when building sparse-grid approximations of a multivariate function u = u(y1, ..., yN ). More precisely, we present a knapsack approach , in which we estimate the cost and the error reduction contribution of each possible component of the sparse grid, and then we choose the components with the highest error reduction /cost ratio. The estimates of the error reduction are obtained by either a mixed a-priori / a-posteriori approach, in which we first derive a theoretical bound and then tune it with some inexpensive auxiliary computations (resulting in the so-called quasi-optimal sparse grids ), or by a fully a-posteriori approach (obtaining the so-called adaptive sparse grids ). This framework is very general and can be used to build quasi-optimal/adaptive sparse grids on bounded and unbounded domains (e.g. u depending on uniform and normal random distributions for yn), using both nested and non-nested families of univariate collocation points. We present some theoretical convergence results as well as numerical results showing the efficiency of the proposed approach for the approximation of the solution of elliptic PDEs with random diffusion coefficients. In this context, to treat the case of rough permeability fields in which a sparse grid approach may not be suitable, we propose to use the sparse grids as a control variate in a Monte Carlo simulation.
Conference/Event name:
Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2016)
Issue Date:
5-Jan-2016
Type:
Presentation
Additional Links:
http://mediasite.kaust.edu.sa/Mediasite/Play/0b9b0fbdc1cb451c94007ea73abd44151d?catalog=ca65101c-a4eb-4057-9444-45f799bd9c52
Appears in Collections:
Conference on Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2016)

Full metadata record

DC FieldValue Language
dc.contributor.authorTamellini, Lorenzoen
dc.date.accessioned2017-06-08T06:32:29Z-
dc.date.available2017-06-08T06:32:29Z-
dc.date.issued2016-01-05-
dc.identifier.urihttp://hdl.handle.net/10754/624842-
dc.description.abstractIn this talk we discuss possible strategies to minimize the impact of the curse of dimensionality effect when building sparse-grid approximations of a multivariate function u = u(y1, ..., yN ). More precisely, we present a knapsack approach , in which we estimate the cost and the error reduction contribution of each possible component of the sparse grid, and then we choose the components with the highest error reduction /cost ratio. The estimates of the error reduction are obtained by either a mixed a-priori / a-posteriori approach, in which we first derive a theoretical bound and then tune it with some inexpensive auxiliary computations (resulting in the so-called quasi-optimal sparse grids ), or by a fully a-posteriori approach (obtaining the so-called adaptive sparse grids ). This framework is very general and can be used to build quasi-optimal/adaptive sparse grids on bounded and unbounded domains (e.g. u depending on uniform and normal random distributions for yn), using both nested and non-nested families of univariate collocation points. We present some theoretical convergence results as well as numerical results showing the efficiency of the proposed approach for the approximation of the solution of elliptic PDEs with random diffusion coefficients. In this context, to treat the case of rough permeability fields in which a sparse grid approach may not be suitable, we propose to use the sparse grids as a control variate in a Monte Carlo simulation.en
dc.relation.urlhttp://mediasite.kaust.edu.sa/Mediasite/Play/0b9b0fbdc1cb451c94007ea73abd44151d?catalog=ca65101c-a4eb-4057-9444-45f799bd9c52en
dc.titleQuasi optimal and adaptive sparse grids with control variates for PDEs with random diffusion coefficienten
dc.typePresentationen
dc.conference.dateJanuary 5-10, 2016en
dc.conference.nameAdvances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2016)en
dc.conference.locationKAUSTen
dc.contributor.institutionUniversità di Paviaen
All Items in KAUST are protected by copyright, with all rights reserved, unless otherwise indicated.