In our group, we have successfully used and improved adaptive sparse grids for a range of higher-dimensional applications. They range from exascale simulations of hot fusion plasmas via the mining of vast datasets to topology optimization and financial applications.
In this talk, we consider non-intrusive stochastic collocation for uncertainty quantification, as our applications require us to treat the underlying simulation code as a black box. We propose spatially adaptive sparse grids for both the estimation of the stochastic densities and the stochastic collocation.
With sparse grids, the numerical discretization is still possible in higher-dimensional settings, and the integrated sparse grid approach leads to fast and efficient algorithms and implementations. This allows us to start with data that is provided by measurements and to combine the estimated densities with the model function's surrogate without introducing additional sampling or approximation errors. Bayesian updating allows us to incorporate observations and to adaptively refine the surrogate based on the posterior. Efficient and scalable algorithms for the evaluation of the surrogate function are available, which can achieve close-to-peak performance even on hybrid hardware.