MATHICSE Technical Report : A continuation multi level Monte Carlo (C-MLMC) method for uncertainty quantification in compressible aerodynamics

In this work we apply the Continuation Multi-Level Monte Carlo (C-MLMC) algorithm proposed by [Collier et al, BIT 2014] to efficiently propagate operational and geometrical uncertainties in compressible aerodynamics numerical simulations. The key idea of MLMC is that one can draw MC samples simultaneously and independently on several approximations of the problem under investigations on a hierarchy of nested computational grids (levels). The expectation of an output quantity is computed as a sample average using the coarsest solutions and corrected by averages of the differences of the solutions of two consecutive grids in the hierarchy. By this way, most of the computational effort is transported from the finest level (as in a standard Monte Carlo approach) to the coarsest one. In the continuation algorithm (C-MLMC) the parameters that control the number of levels and realizations per level are computed on the y to further reduce the overall computational cost. The C-MLMC is applied to the quasi 1D convergent-divergent Laval nozzle and the 2D transonic RAE-2822 airfoil.