Arbitrary multi-resolution multi-wavelet-based polynomial chaos expansion for data-driven uncertainty quantification

Various real world problems deal with data-driven uncertainty. In particular, in geophysical applications the amount of available data is often limited, posing a challenge in the construction of an appropriate stochastic discretization. Arbitrary polynomial chaos is an alternative to tackle this challenge. Approximating the dependence of model output on the uncertain model parameters by expansion in an orthogonal polynomial basis using data-driven principles. This type of global polynomial representation suffers often from Gibbs’ phenomena, especially if applied in non-linear convection dominated problems that require to deal with discontinuities. The multi-resolution or multi-element framework has been successfully used for reducing Gibbs’ phenomena in intrusive stochastic discretizations. In the present work, we introduce a multi-resolution extension of the arbitrary polynomial chaos expansion which is based on the construction of piecewise polynomial. Gaussian quadrature nodes and weights that are computed using only stochastic (localized) moments provided by the underlying raw data. We enhance our approach by a multi-wavelet based stochastic adaptivity that assures a significant reduction of the computational costs. Numerical experiments of increasing complexity demonstrate the performance of the non-intrusive implementation of the introduced methods in relevant scenarios. The use of a carbon dioxide storage benchmark scenario allows one to compare the presented methodology with other stochastic discretization techniques applied to this benchmark.