Numerical Approximation of Flows in Random Porous Media

The objective of this thesis is to develop efficient numerical schemes to successfully tackle problems arising from the study of groundwater flows in a porous saturated medium; we deal therefore with partial differential equations(PDE) having random coefficients and we are interested in computing statistics related to specific quantities of interest (QoI), e.g. a linear functional of the solution of the PDE or the solution itself. We mainly consider the approximation of the pressure in the medium through a stochastic Darcy problem with random lognormally distributed permeability relying on Matérn-type covariance functions to take into account a wide range of possible smoothness of the permeability field. Once the problem has been reformulated in terms of a countable number of random variables, we analyze sparse grid polynomial approximations of the QoI. We propose different strategies to exploit the anisotropicity of the QoI with respect to the different random entries; to this end we consider a priori'' and a posteriori'' strategies to drive the exploration of the multi-index set that defines the sparse grid, associating a profit to each multi-index either by using explicit theoretical estimates or by actually solving the PDE and computing on the fly the corresponding sparse grid interpolant. We show on several numerical examples the effectiveness of this strategy in treating the case of smooth permeability fields. In order to cover also the case of rough input permeabilities we consider, instead, Multi Level Monte Carlo techniques based on the use of a suitable control variate. Such a control variate is obtained from the solution of an auxiliary Darcy problem with a regularized input permeability which leads to pressure distributions that are smoother and less oscillatory than the original ones, but still highly correlated with them. We use then a sparse grid approximation to compute effectively the mean of the control variate and provide explicit bounds for the corresponding estimator as well as a complexity result. We also consider groundwater transport problems and focus, in particular, on arrival times properly defined starting from particle trajectories driven by the stochastic Darcy velocity and subject to molecular diffusion taking place at porous level. In this case, by using suitable PDEs whose solution can be linked to specific expectations (with respect to all Brownian motions) thanks to the famous Feynman-Kac formula, we compute statistics of such arrival times, e.g. their expected value or the probability of exiting the physical domain in a given time horizon. We discuss several scenarios and readapt the methodologies previously developed involving adaptive sparse grid stochastic collocation and Monte Carlo type schemes to this case.

A posteriori error estimation for partial differential equations with random input data

Diane Sylvie Guignard

This thesis is devoted to the derivation of error estimates for partial differential equations with random input data, with a focus on a posteriori error estimates which are the basis for adaptive strategies. Such procedures aim at obtaining an approximation of the solution with a given precision while minimizing the computational costs. If several sources of error come into play, it is then necessary to balance them to avoid unnecessary work. We are first interested in problems that contain small uncertainties approximated by finite elements. The use of perturbation techniques is appropriate in this setting since only few terms in the power series expansion of the exact random solution with respect to a parameter characterizing the amount of randomness in the problem are required to obtain an accurate approximation. The goal is then to perform an error analysis for the finite element approximation of the expansion up to a certain order. First, an elliptic model problem with random diffusion coefficient with affine dependence on a vector of independent random variables is studied. We give both a priori and a posteriori error estimates for the first term in the expansion for various norms of the error. The results are then extended to higher order approximations and to other sources of uncertainty, such as boundary conditions or forcing term. Next, the analysis of nonlinear problems in random domains is proposed, considering the one-dimensional viscous Burgers' equation and the more involved incompressible steady-state Navier-Stokes equations. The domain mapping method is used to transform the equations in random domains into equations in a fixed reference domain with random coefficients. We give conditions on the mapping and the input data under which we can prove the well-posedness of the problems and give a posteriori error estimates for the finite element approximation of the first term in the expansion. Finally, we consider the heat equation with random Robin boundary conditions. For this parabolic problem, the time discretization brings an additional source of error that is accounted for in the error analysis. The second part of this work consists in the analysis of a random elliptic diffusion problem that is approximated in the physical space by the finite element method and in the stochastic space by the stochastic collocation method on a sparse grid. Considering a random diffusion coefficient with affine dependence on a vector of independent random variables, we derive a residual-based a posteriori error estimate that controls the two sources of error. The stochastic error estimator is then used to drive an adaptive sparse grid algorithm which aims at alleviating the so-called curse of dimensionality inherent to tensor grids. Several numerical examples are given to illustrate the performance of the adaptive procedure.

EPFL2016

An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data

Fabio Nobile

This work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms ( input data of the model). The method consists of a Galerkin approximation in the space variables and a collocation, in probability space, on sparse tensor product grids utilizing either Clenshaw-Curtis or Gaussian knots. Even in the presence of nonlinearities, the collocation approach leads to the solution of uncoupled deterministic problems, just as in the Monte Carlo method. This work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem. These procedures seem to be very effective for the problems under study. The proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation: the first approach is effective for problems depending on random variables which weigh approximately equally in the solution, while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables, as in the case where input random variables are Karhunen-Loeve truncations of "smooth" random fields. This work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates ( sub) exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime, with respect to the total number of collocation points. It also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems. Numerical examples illustrate the theoretical results and are used to compare this approach with several others, including the standard Monte Carlo. In particular, for moderately large-dimensional problems, the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods.

Society for Industrial and Applied Mathematics2008

Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients

Fabio Nobile

We consider the problem of numerically approximating statistical moments of the solution of a time-dependent linear parabolic partial differential equation (PDE), whose coefficients and/or forcing terms are spatially correlated random fields. The stochastic coefficients of the PDE are approximated by truncated Karhunen–Loève expansions driven by a finite number of uncorrelated random variables. After approximating the stochastic coefficients, the original stochastic PDE turns into a new deterministic parametric PDE of the same type, the dimension of the parameter set being equal to the number of random variables introduced. After proving that the solution of the parametric PDE problem is analytic with respect to the parameters, we consider global polynomial approximations based on tensor product, total degree or sparse polynomial spaces and constructed by either a Stochastic Galerkin or a Stochastic Collocation approach. We derive convergence rates for the different cases and present numerical results that show how these approaches are a valid alternative to the more traditional Monte Carlo Method for this class of problems. Copyright © 2009 John Wiley & Sons, Ltd.

2009

An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data

Fabio Nobile

Society for Industrial and Applied Mathematics2008

A posteriori error estimation for partial differential equations with random input data

Diane Sylvie Guignard

EPFL2016