Many mathematical models of physical processes contain uncertainties due to incomplete models or measurement errors and lack of knowledge associated with the model inputs. We consider processes which are formulated in terms of classical partial differential equations (PDEs). The challenge and novelty is that the PDEs contain random coefficient functions, e.g., some transformations of Gaussian random fields. Random PDEs are much more flexible and can model more complex situations compared to classical PDEs with deterministic coefficients. However, each sample of a PDE-based model is extremely expensive. To alleviate the high costs the numerical analysis community has developed so-called multilevel estimators which work with a hierarchy of PDE models with different resolution and cost. We review the basic idea of multilevel estimators and discuss our own recent contributions:

i) a multilevel best linear unbiased estimator to approximate the expectation of a scalar output quantity of interest associated with a random PDE [1, 2],

ii) a multilevel sequential Monte Carlo method for Bayesian inverse problems [3],

iii) a multilevel sequential importance method to estimate the probability of rare events [4].

[1] D. Schaden, E. Ullmann: On multilevel best linear unbiased estimators. SIAM/ASA J. Uncert. Quantif. 8(2), pp. 601-635, 2020

[2] D. Schaden, E. Ullmann: Asymptotic analysis of multilevel best linear unbiased estimators, arXiv:2012.03658

[3] J. Latz, I. Papaioannou, E. Ullmann: Multilevel Sequential² Monte Carlo for Bayesian Inverse Problems. J. Comput. Phys., 368, pp. 154-178, 2018

[4] F. Wagner, J. Latz, I. Papaioannou, E. Ullmann: Multilevel sequential importance sampling for rare event estimation. SIAM J. Sci. Comput. 42(4), pp. A2062–A2087, 2020