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Publication# MATHICSE Technical Report : Analysis of the discrete $L^2$ projection on polynomial spaces with random evaluations

Giovanni Migliorati, Fabio Nobile, Erik Gustaf Bogislaw Von Schwerin

*MATHICSE, *2011

Report or working paper

Report or working paper

Abstract

We analyse the problem of approximating a multivariate function by discrete least-squares projection on a polynomial space starting from random, noise-free observations. An area of possible application of such technique is Uncertainty Quantification (UQ) for computational models. We prove an optimal convergence estimate, up to a logarithmic factor, in the monovariate case, when the observation points are sampled in a bounded domain from a probability density function bounded away from zero, provided the number of samples scales quadratically with the dimension of the polynomial space. Several numerical tests are presented both in the monovariate and multivariate case, confirming our theoretical estimates. The numerical tests also clarify how the convergence rate depends on the number of sampling points, on the polynomial degree, and on the smoothness of the target function.

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Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathema

Least squares

The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by

Polynomial

In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and po

We address adaptive multivariate polynomial approximation by means of the discrete least-squares method with random evaluations, to approximate in the L2 probability sense a smooth function depending on a random variable distributed according to a given probability density. The polynomial least-squares approximation is computed using random noiseless pointwise evaluations of the target function. Here noiseless means that the pointwise evaluation of the function is not polluted by the presence of noise. Recent works Migliorati et al. (Found Comput Math 14:419–456, 2014), Cohen et al. (Found Comput Math 13:819–834, 2013), and Chkifa et al. (Discrete least squares polynomial approximation with random evaluations – application to parametric and stochastic elliptic PDEs, EPFL MATHICSE report 35/2013, submitted) have analyzed the univariate and multivariate cases, providing error estimates for (a priori) given sequences of polynomial spaces. In the present work, we apply the results developed in the aforementioned analyses to devise adaptive least-squares polynomial approximations. We build a sequence of quasi-optimal best n-term sets to approximate multivariate functions that feature strong anisotropy in moderately high dimensions. The adaptive approximation relies on a greedy selection of basis functions, which preserves the downward closedness property of the polynomial approximation space. Numerical results show that the adaptive approximation is able to catch effectively the anisotropy in the function.

, ,

We analyse the problem of approximating a multivariate function by discrete least-squares projection on a polynomial space starting from random, noise-free observations. An area of possible application of such technique is Uncertainty Quantification (UQ) for computational models. We prove an optimal convergence estimate, up to a logarithmic factor, in the monovariate case, when the observation points are sampled in a bounded domain from a probability density function bounded away from zero, provided the number of samples scales quadratically with the dimension of the polynomial space. Several numerical tests are presented both in the monovariate and multivariate case, confirming our theoretical estimates. The numerical tests also clarify how the convergence rate depends on the number of sampling points, on the polynomial degree, and on the smoothness of the target function

2014, ,

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.