Breaking the Curse of Dimensionality

Abstract

In light of worsening climate change and an increased interest in adapting infrastructure to cope with its effects, model-based decision support has become an essential tool for policy makers. In conditions of deep uncertainty, models may be used to explore a large space of possible system behaviours and so encourage a wider consideration of the possible futures. Such methods, where the focus is intentionally broad, fall under the remit of exploratory modelling and are a potential antidote to traditional predictive modelling methods where only a marginal treatment of uncertainty is attempted. One serious issue limiting the full exploitation of exploratory modelling is its computational intensity, the many computational experiments requiring large amounts on computing power which makes some analyses too expensive to attempt. In order to fully exploit the promise of exploratory modelling new methods of reducing computational intensity are needed. Polynomial chaos expansions (PCEs) are one class of methods which may fulfill this role. Our results conclusively demonstrate that PCEs are capable of accurately reproducing statistical moments and determining Sobol sensitivity indices significantly faster than direct-sampling methods, often requiring orders of magnitude fewer function evaluations. However, we found that the curse of dimensionality rendered conventional PCEs too costly for use with higher-dimensional models. We found that conventional sparse grids were effective at reducing the computational cost associated with fitting PCEs with high-dimensional models, as long as the model output was sufficiently smooth. For models where sparse grids were able to converge with reasonable accuracy, supplementing the PCCs with a modified Gersnter's dimension-adaptive algorithm further improved convergence times. The anisotropic refinement strategy employed by the algorithm allows for accurate determination of Sobol sensitivity indices with a minimum of computational effort.