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.
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机译：在这项工作中，我们考虑多项式空间（此后称为RDP）上的随机离散L ^ 2投影，用于与带有随机输入参数的偏微分方程模型的解相关的标量（QOI）的近似。在RDP技术中，首先像标准的蒙特卡洛方法那样，为随机输入参数的独立样本计算QOI，然后使用离散最小二乘法通过输入参数的多元多项式函数对QOI进行近似。我们考虑几个示例，包括具有随机渗透率的Darcy方程，具有随机弹性系数的线性弹性方程，以及具有随机几何形状和随机流体粘度的Navier-Stokes方程。我们表明，RDP技术非常适合平稳地依赖于中等数量的随机参数的QOI。我们的数值测试证实了[G. Migliorati，F。Nobile，E。von Schwerin和R. Tempone，多项式空间的离散L ^ 2 $投影的随机评估分析，MOX报告46-2011，米兰理工大学，意大利米兰，提交]，结果表明，在单个均匀分布的随机参数的情况下，如果采样点的数量与多项式空间维数的平方成正比，则RDP技术是稳定且最优收敛的。在此，最优性意味着RDP误差的加权$ L ^ 2 $范数在给定的多项式空间内，可以达到最佳对数因子，从上方限制了最佳的$ L ^ infty $误差。在有几个随机输入参数的情况下，数值证据表明，采样点数量的二次增长条件可以放宽为线性增长，并且仍然可以实现稳定和最佳的收敛。这使得RDP技术非常适合中等高维不确定性量化。
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