Seminar in Numerical Analysis: Fabio Nobile (EPF Lausanne)

Perturbation methods and low rank approximations for the Darcy equation with lognormal permeability

We consider the Darcy equation to describe the flow in a saturated porous medium. The permeability of the medium is described as a log-normal random field, eventually conditioned to available direct measurements, to account for its relatively large uncertainty and heterogeneity.

We consider perturbation methods based on Taylor expansion of the solution of the PDE around the nominal permeability value. Successive higher order corrections to the statistical moments such as pointwise mean and covariance of the solution can be obtained recursively from the computation of high order correlation functions which, on their turn, solve high dimensional problems. To overcome the curse of dimensionality in computing and storing such high order correlations, we adopt a low-rank format, namely the so called tensor-train (TT) format.

We show that, on the one hand, the Taylor series does not converge globally, so that it only makes sense to compute corrections up to a maximum critical order, beyon which the accuracy of the solution deteriorates insetad of improving. On the other hand, we show on some numerical test cases, the effectiveness of the proposed approach in case of a moderately small variance of the log-normal permeability field.

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