Galerkin Methods for Stochastic Elliptic Partial Differential Equations of Flow through Porous Media

The flow through a porous medium is considered in a simple but prototypical setting. Knowledge about the conductivity of the soil, the magnitude of source-terms, or about the in- and out-flow boundary conditions is often very uncertain. These uncertainties inherent in the model result in uncertainties in the results of numerical simulations.

Stochastic methods are one way to model these uncertainties, and in our case we are concerned with spatially varying randomness, and model this by random fields. If the physical system is described by a partial differential equation (PDE), then the combination with the stochastic model results in a stochastic partial differential equation (SPDE). The solution of the SPDE is again a random field, describing both the expected response and quantifying its uncertainty.

SPDEs can be interpreted mathematically in several ways. At the moment we concentrate on randomness in space. If evolution with stochastic input has to be considered, one may combine the techniques described here with the already well established methods in that field.

One may distinguish---as in the case of stochastic ordinary differential equations (SDEs)---between additive and multiplicative noise. As is well known from SDEs, in the case of multiplicative noise one has to be more careful. A similar problem occurs here. Additive noise---particularly for linear problems---is well known and much simpler to deal with, even if the random fields are generalised to stochastic distributions. With multiplicative noise on the other hand the product of a random coefficient field and the solution may have no meaning. As with SDEs, it is a modelling decision how this is resolved.

Additive noise corresponds to the case where the right hand side---the loading or the solution independent source term---is random, whereas when the operator is random, we have multiplicative noise. In the first case it is the external influences which are uncertain, in the latter it is the system under consideration itself. In general, both uncertainties are present.

In an engineering setting, these models have been considered in different fields. Many different kinds of solution procedures have been tried, but mostly Monte Carlo methods have been used. Alternatives to Monte Carlo methods, which first compute the solution and then the required statistic, have been developed in the field of stochastic mechanics, for example perturbation methods, methods based on Neumann-series, or the spectral stochastic finite element-method. The latter expands the input random fields in eigenfunctions of their covariance kernels, and obtains the solution by a Galerkin method in a space of stochastic ansatz functions.

One direction of numerical investigation focuses on computing the moments of the solution. These are very common, but specific response descriptors. Often other functionals of the solution may be desired. Monte Carlo (MC) methods can be used directly for this, but they require a high computational effort. Variance reduction techniques are employed to reduce this somewhat. Quasi Monte Carlo (QMC) methods may reduce the computational effort considerably without requiring much regularity. But often we have high regularity in the stochastic variables, and this is not exploited by QMC methods. The problem of computing such high-dimensional integrals comes up as a subtask also in the stochastic Galerkin method which we pursue. The integrands are often very smooth, and MC and QMC methods do not take much advantage out of this.

For this subtask, we propose sparse grid (Smolyak) quadrature methods as an efficient alternative. These have found increasing attention in recent years.

The stochastic Galerkin methods started from N. Wiener''s polynomial chaos. This has been used extensively in the theoretical white noise analysis in stochastics. This device may of course also be used in the simulation of random fields.

In general, the stochastic Galerkin methods allow a direct representation of the solution. Stochastic Galerkin methods have been applied to various linear problems, using a variety of numerical techniques to accelerate the solution. Recently, nonlinear problems with stochastic loads have been tackled, and some first results for nonlinear stochastic operators have been reached. A convergence theory in has been started, but we are very much at the beginning.

These Galerkin methods allow us to have an explicit functional relationship between the independent random variables and the solution---and this is contrast with usual Monte Carlo approaches, so that subsequent evaluations of functionals---statistics like the mean, covariance, or probabilities of exceedance---are very cheap. This may be seen as a way to systematically calculate more and more accurate "response surfaces".