Mathematics Colloquium

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High Dimensional Sparse Approximation of Elliptic PDEs with Lognormal Coefficients

Speaker: Albert Cohen, Université Pierre et Marie Curie, Paris

Location: Warren Weaver Hall 1302

Date: Monday, January 30, 2017, 3:45 p.m.

Synopsis:

Various mathematical problems are challenged by the fact they involve functions of a very large number of variables. Such problems arise naturally in learning theory, partial differential equations or numerical models depending on parametric or stochastic variables. They typically result in numerical difficulties due to the so-called "curse of dimensionality".

We shall adress the particular example of elliptic partial differential equations with diffusion coefficients of lognormal form, that is, of the form exp(b) where b is a gaussian random field, which is one typical model in groundwater modeling. We discuss numerical strategies that allow us to break the curse of dimensionality under mild assumptions on the random field.

These strategies rely on sparse polynomial approximations by best n-term truncation of tensorized Hermite expansions in stochastic variables which represent the gaussian fields. One interesting conclusion is that in certain relevant cases, the most often used Karhunen-Loeve representation of the random field is not best choice in terms of the resulting sparsity and approximability of Hermite expansion.