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We perform a general optimization of the parameters in the Multilevel MonteCarlo (MLMC) discretization hierarchy based on uniform discretization methodswith general approximation orders and computational costs. We optimizehierarchies with geometric and non-geometric sequences of mesh sizes and showthat geometric hierarchies, when optimized, are nearly optimal and have thesame asymptotic computational complexity as non-geometric optimal hierarchies.We discuss how enforcing constraints on parameters of MLMC hierarchies affectsthe optimality of these hierarchies. These constraints include an upper and alower bound on the mesh size or enforcing that the number of samples and thenumber of discretization elements are integers. We also discuss the optimaltolerance splitting between the bias and the statistical error contributionsand its asymptotic behavior. To provide numerical grounds for our theoreticalresults, we apply these optimized hierarchies together with the ContinuationMLMC Algorithm. The first example considers a three-dimensional ellipticpartial differential equation with random inputs. Its space discretization isbased on continuous piecewise trilinear finite elements and the correspondinglinear system is solved by either a direct or an iterative solver. The secondexample considers a one-dimensional It\^o stochastic differential equationdiscretized by a Milstein scheme.

Optimization of mesh hierarchies in Multilevel Monte Carlo samplers