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

G. Migliorati, F. Nobile, E. von Schwerin, Raul Tempone

Research output: Contribution to journalArticlepeer-review

57 Scopus citations


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.
Original languageEnglish (US)
Pages (from-to)A1440-A1460
Number of pages1
JournalSIAM Journal on Scientific Computing
Issue number3
StatePublished - May 30 2013

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01


Dive into the research topics of 'Approximation of Quantities of Interest in Stochastic PDEs by the Random Discrete L^2 Projection on Polynomial Spaces'. Together they form a unique fingerprint.

Cite this