## Section: New Results

### Uncertainty quantification in hydrogeoloy

This work is done in collaboration with A. Debussche, from ENS-Cachan-Rennes and Ipso INRIA team. It is done in the context of the Micas project ( 8.1.2 ).

A PhD thesis was defended this year [11] .

#### Strong and weak error estimates for elliptic partial differential equations with random coefficients

Participant : Julia Charrier.

This work has been presented at a workshop and is published in a journal [23] , [14] .

We consider the problem of numerically approximating the solution of an elliptic partial differential equation with random coefficients and homogeneous Dirichlet boundary conditions. We focus on the case of a lognormal coefficient, we have then to deal with the lack of uniform coercivity and uniform boundedness with respect to the randomness. This model is frequently used in hydrogeology. We approximate this coefficient by a finite dimensional noise using a truncated Karhunen-Loève expansion. We give then estimates of the corresponding error on the solution, both a strong error estimate and a weak error estimate, that is to say an estimate of the error commited on the law of the solution. We obtain a weak rate of convergence which is twice the strong one. Besides this, we give a complete error estimate for the stochastic collocation method in this case, where neither coercivity nor boundedness are stochastically uniform. To conclude, we apply these results of strong and weak convergence to two classical cases of covariance kernel choices: the case of an exponential covariance kernel on a box and the case of an analytic covariance kernel, yielding explicit weak and strong convergence rates.

#### Numerical analysis of a multilevel Monte Carlo method for elliptic PDEs with random coefficients

Participant : Julia Charrier.

This work has been presented at a conference and is submitted in a journal [42] , [22] .

We consider a finite element approximation of elliptic partial differential equations with random coefficients. Such equations arise, for example, in uncertainty quantification in subsurface flow modelling. Models for random coefficients frequently used in these applications, such as log-normal random fields with exponential covariance, have only very limited spatial regularity, and lead to variational problems that lack uniform coercivity and boundedness with respect to the random parameter. In our analysis we overcome these challenges by a careful treatment of the model problem almost surely in the random parameter, which then enables us to prove uniform bounds on the finite element error in standard Bochner spaces. These new bounds can then be used to perform a rigorous analysis of the multilevel Monte Carlo method for these elliptic problems that lack full regularity and uniform coercivity and boundedness. To conclude, we give some numerical results that confirm the new bounds.

#### Numerical analysis of the advection-diffusion of a solute in random media

Participant : Julia Charrier.

This work is submitted in a journal [41] .

We consider the problem of numerically approximating the solution of the coupling of the flow equation in a random porous medium, with the advection-diffusion equation. More precisely, we present and analyse a numerical method to compute the mean value of the spread of a solute introduced at the initial time, and the mean value of the macro-dispersion, defined at the temporal derivative of the spread. We propose a Monte-Carlo method to deal with the uncertainty, i.e. with the randomness of the permeability field. The flow equation is solved using finite element. The advection-diffusion equation is seen as a Fokker-Planck equation, and its solution is approximated thanks to a probabilistic particular method. The spread is indeed the expected value of a function of the solution of the corresponding stochastic differential equation, and is computed using an Euler scheme for the stochastic differential equation and a Monte-Carlo method. Error estimates on the mean spread and on the mean dispersion are established, under various assumptions, in particular on the permeability random field.

#### Model reduction for a 1D stochastic elliptic PDE

Participants : Jocelyne Erhel, Mestapha Oumouni.

This work is done in collaboration with Z. Mghazli, from the university of Kenitra, Morocco, in the context of the Co-Advise and Hydromed projects ( 8.2.1 , 8.3.4 ).

This work has been presented at a conference and published in a journal [15] [34] .

In this paper, we present an efficient method to approximate the expectation of the response of a one-dimensional elliptic problem with stochastic inputs. In conventional methods, the computational effort and cost of the approximation of the response can be dramatic. Our method presented here is based on the Karhunen–Loève (K-L) expansion of the inverse of the diffusion parameter, allowing us to build a base of random variables in reduced numbers, from which we construct a projected solution. We show that the expectation of this projected solution is a good approximation, and give an a priori error estimate. A numerical example is presented to show the efficiency of this approach.

#### Inverse problems in hydrogeology

Participant : Sinda Khalfallah.

This work is done in collaboration with A. ben Abda, from LAMSIN, Tunisia, in the context of the Hydromed and Co-Advise projects ( 8.2.1 , 8.3.4 ). It is also done in collaboration with B. T. Johansson, from University of Birmingham, GB.

This work has been submitted to a journal [40] .

This work is an initial study of a numerical method for identifying multiple leak zones in saturated unsteady flow. Using the conventional saturated groundwater flow equation, the leak identification problem is modelled as a Cauchy problem for the heat equation and the aim is to find the regions on the boundary of the solution domain where the solution vanishes, since leak zones corresponds to null pressure values. To reconstruct the solution to the Cauchy problem in a stable way, we modify and employ an iterative regularizing method proposed recently. In this method, one solves mixed well-posed problems (obtained by changing the boundary conditions) for the heat operator as well as for its adjoint, to get a sequence of approximations to the original Cauchy problem. The mixed problems are solved using a Finite element method (FEM), and the numerical results obtained show that the leak zones can be accurately identified also when there is noise in the data.