Seminar in Numerical Analysis: Andrea Barth (University of Stuttgart)
Multilevel Monte Carlo methods were introduced to lower the computational complexity for the calculation of, for instance, the expectation of a random quantity. More precisely, in comparison to standard Monte Carlo methods the computational complexity is (asymptotically) equal to the calculation of one sample of the problem on the finest grid used. The price to pay for this increase in efficiency is that the problem needs to be solved not only on one (fine) grid, but on a hierarchy of discretizations. This implies first that the solution has to be represented on all grids and second, that the variance of the detail (the difference of approximate solutions on two consecutive grids) converges with the refinement of the grid.
In this talk, I will give an introduction to multilevel Monte Carlo methods in the case when the variance of the detail does not converge uniformly. The idea is illustrated by the calculation of the expectation for an elliptic problem with a random multiscale coefficient and then extended to approximations of statistical solutions to the Navier-Stokes equations.
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