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Controlling error in multi-level approximations of stochastic PDEs.

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Raphaël Bulle
(University of Luxembourg, Université de Bourgogne Franche-Comté.)

Stochastic PDEs have became a very important tool in the modelling of real-
world systems which involve a large lack of knowledge on the data. The first
example we have in mind is economics but the applications field of SPDEs
is becoming wider and wider, including hydrology, geology, meteorology, me-
chanical engineering and so on.
Here, we consider a problem in hydrology of groundwater flow modelling
in porous media. Such a problem can be described by a linear elliptic SPDE
called Darcy problem which consists mainly in a Poisson problem involving
a random field as parameter. This parameter stands for the conductive per-
meability of the ground and is based on a set of measurements of the real
structure of the ground. Since a limited number of these measures is feasible,
the model is subject to a large amount of uncertainties which propagate to
the numerical solution of our SPDE.
In this talk we will combine deterministic and stochastic methods in order
to derive a multi-fidelity scheme estimating some quantity of interest on the
exact solution of the Darcy problem. The deterministic part consists namely
in finite element methods, a posteriori error estimation and adaptive mesh
refinement. The stochastic part is an improved Monte-Carlo method called
multi-level Monte-Carlo allowing us, among others, to reduce the variance of
the classical Monte-Carlo method.