Abstract: In uncertainty quantification (UQ) problems where a large number of uncertain parameters are involved, the Monte Carlo (MC) method is often the method of last resort. This is due to the large amount of expensive model evaluations required by the standard MC method. Fortunately, so-called multilevel sampling methods, such as the Multilevel Monte Carlo (MLMC) method, can heavily reduce this computational burden. Instead of performing many expensive model evaluations with high accuracy, these methods start from many cheap model evaluations with low accuracy, and subsequently add fewer and fewer evaluations of a correction term with increasing accuracy, but also increasing cost.

In a first part of this talk, we present several improved multilevel sampling methods, including Multilevel Quasi-Monte Carlo (MLQMC), Adaptive Multi-Index Monte Carlo (AMIMC) and Multiple Semicoarsened Multigrid Multi-Index Monte Carlo (MSG-MIMC). Next, we focus on the recent progress in applying these state-of-the-art methods to real life engineering problems.