Heterogeneous problems that take place at multiple scales are ubiquitous in science and engineering. Examples are wind turbines made from composites or groundwater flow relevant e.g., for the design of flood prevention measures. However, finite element or finite volume methods require an often prohibitively large amount of computational time for such tasks. Multiscale methods that are based on ansatz functions which incorporate the local behavior of the (numerical) solution of the partial differential equations (PDEs) have been developed to tackle these heterogeneous problems. Localizable multiscale methods that allow controlling the error due to localization and the (global) approximation error at a (quasi-optimal) rate and do not rely on structural assumptions such as scale separation or periodicity have only been developed within the last decade. Here, localizable multiscale methods allow the efficient construction of the basis functions by solving the PDE (in parallel) on several small subdomains at low cost.
While there has been a significant progress in recent years for these types of multiscale methods for linear PDEs, very few results have been obtained so far for nonlinear PDEs. In this talk, we will show how randomized methods and their probabilistic numerical analysis can be exploited for the construction and numerical analysis of such types of multiscale methods for nonlinear PDEs.