Collocation methods for uncertainty quanti cation in PDE models with random data

Handle URI:
http://hdl.handle.net/10754/624026
Title:
Collocation methods for uncertainty quanti cation in PDE models with random data
Authors:
Nobile, Fabio
Abstract:
In this talk we consider Partial Differential Equations (PDEs) whose input data are modeled as random fields to account for their intrinsic variability or our lack of knowledge. After parametrizing the input random fields by finitely many independent random variables, we exploit the high regularity of the solution of the PDE as a function of the input random variables and consider sparse polynomial approximations in probability (Polynomial Chaos expansion) by collocation methods. We first address interpolatory approximations where the PDE is solved on a sparse grid of Gauss points in the probability space and the solutions thus obtained interpolated by multivariate polynomials. We present recent results on optimized sparse grids in which the selection of points is based on a knapsack approach and relies on sharp estimates of the decay of the coefficients of the polynomial chaos expansion of the solution. Secondly, we consider regression approaches where the PDE is evaluated on randomly chosen points in the probability space and a polynomial approximation constructed by the least square method. We present recent theoretical results on the stability and optimality of the approximation under suitable conditions between the number of sampling points and the dimension of the polynomial space. In particular, we show that for uniform random variables, the number of sampling point has to scale quadratically with the dimension of the polynomial space to maintain the stability and optimality of the approximation. Numerical results show that such condition is sharp in the monovariate case but seems to be over-constraining in higher dimensions. The regression technique seems therefore to be attractive in higher dimensions.
Conference/Event name:
Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2014)
Issue Date:
6-Jan-2014
Type:
Presentation
Additional Links:
http://mediasite.kaust.edu.sa/Mediasite/Play/5820fc8fa9ab4a2580ba50d25328dba61d?catalog=ca65101c-a4eb-4057-9444-45f799bd9c52
Appears in Collections:
Conference on Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2014)

Full metadata record

DC FieldValue Language
dc.contributor.authorNobile, Fabioen
dc.date.accessioned2017-06-01T10:20:44Z-
dc.date.available2017-06-01T10:20:44Z-
dc.date.issued2014-01-06-
dc.identifier.urihttp://hdl.handle.net/10754/624026-
dc.description.abstractIn this talk we consider Partial Differential Equations (PDEs) whose input data are modeled as random fields to account for their intrinsic variability or our lack of knowledge. After parametrizing the input random fields by finitely many independent random variables, we exploit the high regularity of the solution of the PDE as a function of the input random variables and consider sparse polynomial approximations in probability (Polynomial Chaos expansion) by collocation methods. We first address interpolatory approximations where the PDE is solved on a sparse grid of Gauss points in the probability space and the solutions thus obtained interpolated by multivariate polynomials. We present recent results on optimized sparse grids in which the selection of points is based on a knapsack approach and relies on sharp estimates of the decay of the coefficients of the polynomial chaos expansion of the solution. Secondly, we consider regression approaches where the PDE is evaluated on randomly chosen points in the probability space and a polynomial approximation constructed by the least square method. We present recent theoretical results on the stability and optimality of the approximation under suitable conditions between the number of sampling points and the dimension of the polynomial space. In particular, we show that for uniform random variables, the number of sampling point has to scale quadratically with the dimension of the polynomial space to maintain the stability and optimality of the approximation. Numerical results show that such condition is sharp in the monovariate case but seems to be over-constraining in higher dimensions. The regression technique seems therefore to be attractive in higher dimensions.en
dc.relation.urlhttp://mediasite.kaust.edu.sa/Mediasite/Play/5820fc8fa9ab4a2580ba50d25328dba61d?catalog=ca65101c-a4eb-4057-9444-45f799bd9c52en
dc.titleCollocation methods for uncertainty quanti cation in PDE models with random dataen
dc.typePresentationen
dc.conference.dateJanuary 6-10, 2014en
dc.conference.nameAdvances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2014)en
dc.conference.locationKAUSTen
dc.contributor.institutionÉcole Polytechnique Fédérale de Lausanneen
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