It has become well established that a proper quantification of uncertainty is an important step towards predictive numerical simulations of real-world phenomena. Whether stemming from measurement errors, in- complete knowledge or inherent variabilities, uncertainty is intrinsic to most real-world problems and it needs to be accounted for ab initio. However, quantifying uncertainty in real-world problems is far from trivial, mainly due to the large number of stochastic parameters and significant computational requirements. In this talk, we will discuss two multilevel strategies that aim to overcome these challenges. In particular, we focus on sparse approximation techniques that are constructed adaptively using suitable a posteriori stochastic information. The first approach is designed for uncertainty propagation, while the second is aimed to be used in Bayesian parameter estimation. To illustrate the power and usefulness of these methods, we apply them in several test problems as well as in a real-world application, the analysis of microinstabilities in fusion plasmas, which is one of the key scientific problems in plasma physics and fusion research.
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