Efficient adaptive stochastic Galerkin methods for parametric operator equations

Paper: Efficient adaptive stochastic Galerkin methods forparametric operator equations

Authors: A. Bespalov (School of Mathematics) and D. Silvester
Link: http://dx.doi.org/10.1137/15M1027048

In this paper, we design and implement an efficient algorithm for solving elliptic PDE problems with correlated random data. Such problems are typical when modelling phenomena with inherent uncertainties (e.g., groundwater flow). Since the underlying differential operators depend on a large, possibly infinite, number of random parameters, naive application of numerical methods often results in the ‘curse of dimensionality’ (exponential growth of the algorithm’s complexity).

fe-solution-paper

The algorithm designed in this paper avoids the ‘curse of dimensionality’ by adaptively ‘building’ a polynomial space over a low-dimensional manifold in the infinitely-dimensional parameter space so that the discretisation error is reduced most effectively.