Multilevel Monte Carlo on a high-dimensional parameter space for transmission problems with geometric uncertainties
For practitioners in uncertainty quantification, this provides a validated method to handle high-dimensional geometric uncertainties with non-smooth responses, though the analysis is specific to transmission problems.
The paper shows that multilevel Monte Carlo can efficiently compute expectations of quantities of interest that depend non-smoothly on high-dimensional parameters, as in transmission problems with uncertain interfaces, and provides convergence results and optimal sample distribution robust to parameter dimension.
In the framework of uncertainty quantification, we consider a quantity of interest which depends non-smoothly on the high-dimensional parameter representing the uncertainty. We show that, in this situation, the multilevel Monte Carlo algorithm is a valid option to compute moments of the quantity of interest (here we focus on the expectation), as it allows to bypass the precise location of discontinuities in the parameter space. We illustrate how such lack of smoothness occurs for the point evaluation of the solution to a (Helmholtz) transmission problem with uncertain interface, if the point can be crossed by the interface for some realizations. For this case, we provide a space regularity analysis for the solution, in order to state converge results in the L1-norm for the finite element discretization. The latter are then used to determine the optimal distribution of samples among the Monte Carlo levels. Particular emphasis is given on the robustness of our estimates with respect to the dimension of the parameter space.