# Approximation of Quantities of Interest in stochastic PDEs by the random discrete L2 projection on polynomial spaces

**Code:**

49/2012

**Title:**

Approximation of Quantities of Interest in stochastic PDEs by the random discrete L2 projection on polynomial spaces

**Date:**

Tuesday 27th November 2012

**Author(s):**

Migliorati, G.; Nobile, F.; von Schwerin, E.; Tempone, R.

**Abstract:**

In this work we consider the random discrete L2 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. The RDP technique consists of randomly sampling the input parameters and computing the corresponding values of the QOI, as in a standard Monte Carlo approach. Then, the QOI is approximated as a multivariate polynomial
function of the input parameters by a discrete least squares approach.
We consider several examples including the Darcy equations with random permeability; the linear elasticity equations with random elastic coefficient; the Navier-Stokes equations in random geometries and with random fluid viscosity.
We show that the RDP technique is well suited for QOIs that depend smoothly on a moderate number of random parameters. Our numerical tests confirm the
theoretical findings in [14], which have shown that, in the case of a single random parameter uniformly distributed, the RDP technique is stable and optimally convergent if the number of sampling points scales quadratically with the dimension of the polynomial space. However, in the case of several random input parameters, numerical evidence shows that this condition could be relaxed and a linear scaling seems enough to achieve stable and optimal convergence, making the RDP technique very promising for high dimensional uncertainty quantification.