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

Giovanni Migliorati, Fabio Nobile, Erik Gustaf Bogislaw Von Schwerin

*Siam Publications, *2013

Journal paper

Journal paper

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.

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Partial differential equation

In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function.
The function is often thought of as

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

Giovanni Migliorati, Fabio Nobile, Erik Gustaf Bogislaw Von Schwerin

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.

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.

The multiquery solution of parametric partial differential equations (PDEs), that is, PDEs depending on a vector of parameters, is computationally challenging and appears in several engineering contexts, such as PDE-constrained optimization, uncertainty quantification or sensitivity analysis. When using the finite element (FE) method as approximation technique, an algebraic system must be solved for each instance of the parameter, leading to a critical bottleneck when we are in a multiquery context, a problem which is even more emphasized when dealing with nonlinear or time dependent PDEs. Several techniques have been proposed to deal with sequences of linear systems, such as truncated Krylov subspace recycling methods, deflated restarting techniques and approximate inverse preconditioners; however, these techniques do not satisfactorily exploit the parameter dependence. More recently, the reduced basis (RB) method, together with other reduced order modeling (ROM) techniques, emerged as an efficient tool to tackle parametrized PDEs.
In this thesis, we investigate a novel preconditioning strategy for parametrized systems which arise from the FE discretization of parametrized PDEs. Our preconditioner combines multiplicatively a RB coarse component, which is built upon the RB method, and a nonsingular fine grid preconditioner. The proposed technique hinges upon the construction of a new Multi Space Reduced Basis (MSRB) method, where a RB solver is built at each step of the chosen iterative method and trained to accurately solve the error equation.
The resulting preconditioner directly exploits the parameter dependence, since it is tailored to the class of problems at hand, and significantly speeds up the solution of the parametrized linear system.
We analyze the proposed preconditioner from a theoretical standpoint, providing assumptions which lead to its well-posedness and efficiency.
We apply our strategy to a broad range of problems described by parametrized PDEs:
(i) elliptic problems such as advection-diffusion-reaction equations, (ii) evolution problems such as time-dependent advection-diffusion-reaction equations or linear elastodynamics equations (iii) saddle-point problems such as Stokes equations, and, finally, (iv) Navier-Stokes equations.
Even though the structure of the preconditioner is similar for all these classes of problems, its fine and coarse components must be accurately chosen in order to provide the best possible results.
Several comparisons are made with respect to the current state-of-the-art preconditioning and ROM techniques.
Finally, we employ the proposed technique to speed up the solution of problems in the field of cardiovascular modeling.