# Approximation of Quantities of Interest in Stochastic PDEs by the Random Discrete L^2 Projection on Polynomial Spaces

Handle URI:
http://hdl.handle.net/10754/555667
Title:
Approximation of Quantities of Interest in Stochastic PDEs by the Random Discrete L^2 Projection on Polynomial Spaces
Authors:
Migliorati, G.; Nobile, F.; von Schwerin, E.; Tempone, Raul ( 0000-0003-1967-4446 )
Abstract:
In this work we consider the random discrete L^2 projection on polynomial spaces (hereafter RDP) for the approximation of scalar quantities of interest (QOIs) related to the solution of a partial differential equation model with random input parameters. In the RDP technique the QOI is first computed for independent samples of the random input parameters, as in a standard Monte Carlo approach, and then the QOI is approximated by a multivariate polynomial function of the input parameters using a discrete least squares approach. We consider several examples including the Darcy equations with random permeability, the linear elasticity equations with random elastic coefficient, and the Navier--Stokes equations in random geometries and with random fluid viscosity. We show that the RDP technique is well suited to QOIs that depend smoothly on a moderate number of random parameters. Our numerical tests confirm the theoretical findings in [G. Migliorati, F. Nobile, E. von Schwerin, and R. Tempone, Analysis of the Discrete $L^2$ Projection on Polynomial Spaces with Random Evaluations, MOX report 46-2011, Politecnico di Milano, Milano, Italy, submitted], which have shown that, in the case of a single uniformly distributed random parameter, the RDP technique is stable and optimally convergent if the number of sampling points is proportional to the square of the dimension of the polynomial space. Here optimality means that the weighted $L^2$ norm of the RDP error is bounded from above by the best $L^\infty$ error achievable in the given polynomial space, up to logarithmic factors. In the case of several random input parameters, the numerical evidence indicates that the condition on quadratic growth of the number of sampling points could be relaxed to a linear growth and still achieve stable and optimal convergence. This makes the RDP technique very promising for moderately high dimensional uncertainty quantification.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Center for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ)
Citation:
Approximation of Quantities of Interest in Stochastic PDEs by the Random Discrete L^2 Projection on Polynomial Spaces 2013, 35 (3):A1440 SIAM Journal on Scientific Computing
Journal:
SIAM Journal on Scientific Computing
Issue Date:
30-May-2013
DOI:
10.1137/120897109
Type:
Article
ISSN:
1064-8275; 1095-7197
http://epubs.siam.org/doi/abs/10.1137/120897109
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

DC FieldValue Language
dc.contributor.authorMigliorati, G.en
dc.contributor.authorNobile, F.en
dc.contributor.authorvon Schwerin, E.en
dc.contributor.authorTempone, Raulen
dc.date.accessioned2015-05-25T08:31:18Zen
dc.date.available2015-05-25T08:31:18Zen
dc.date.issued2013-05-30en
dc.identifier.citationApproximation of Quantities of Interest in Stochastic PDEs by the Random Discrete L^2 Projection on Polynomial Spaces 2013, 35 (3):A1440 SIAM Journal on Scientific Computingen
dc.identifier.issn1064-8275en
dc.identifier.issn1095-7197en
dc.identifier.doi10.1137/120897109en
dc.identifier.urihttp://hdl.handle.net/10754/555667en
dc.description.abstractIn this work we consider the random discrete L^2 projection on polynomial spaces (hereafter RDP) for the approximation of scalar quantities of interest (QOIs) related to the solution of a partial differential equation model with random input parameters. In the RDP technique the QOI is first computed for independent samples of the random input parameters, as in a standard Monte Carlo approach, and then the QOI is approximated by a multivariate polynomial function of the input parameters using a discrete least squares approach. We consider several examples including the Darcy equations with random permeability, the linear elasticity equations with random elastic coefficient, and the Navier--Stokes equations in random geometries and with random fluid viscosity. We show that the RDP technique is well suited to QOIs that depend smoothly on a moderate number of random parameters. Our numerical tests confirm the theoretical findings in [G. Migliorati, F. Nobile, E. von Schwerin, and R. Tempone, Analysis of the Discrete $L^2$ Projection on Polynomial Spaces with Random Evaluations, MOX report 46-2011, Politecnico di Milano, Milano, Italy, submitted], which have shown that, in the case of a single uniformly distributed random parameter, the RDP technique is stable and optimally convergent if the number of sampling points is proportional to the square of the dimension of the polynomial space. Here optimality means that the weighted $L^2$ norm of the RDP error is bounded from above by the best $L^\infty$ error achievable in the given polynomial space, up to logarithmic factors. In the case of several random input parameters, the numerical evidence indicates that the condition on quadratic growth of the number of sampling points could be relaxed to a linear growth and still achieve stable and optimal convergence. This makes the RDP technique very promising for moderately high dimensional uncertainty quantification.en
dc.relation.urlhttp://epubs.siam.org/doi/abs/10.1137/120897109en
dc.rightsArchived with thanks to SIAM Journal on Scientific Computingen
dc.subjectPDE stochastic dataen
dc.subjectdiscrete least squaresen
dc.subjectpolynomial approximationen
dc.titleApproximation of Quantities of Interest in Stochastic PDEs by the Random Discrete L^2 Projection on Polynomial Spacesen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentCenter for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ)en
dc.identifier.journalSIAM Journal on Scientific Computingen
dc.eprint.versionPublisher's Version/PDFen
dc.contributor.institutionMATHICSE-CSQI,École Polytechnique Fédérale de Lausanne, Lausanne CH-1015, Switzerland, and MOX-Dipartimento di Matematica “Francesco Brioschi,” Politecnico di Milano, Milano 20133, Italyen
dc.contributor.institutionMOX-Dipartimento di Matematica “Francesco Brioschi,” Politecnico di Milano, Milano 20133, Italyen
kaust.authorvon Schwerin, Eriken
kaust.authorTempone, Raulen