Determining optimal multilevel Monte Carlo parameters with application to fault tolerance

The multilevel Monte Carlo (MLMC) method is characterized by a number of parameters, most notably the number of levels and the number of samples per level. We propose to determine these quantities by solving an integer optimization problem that minimizes the work or the error of the MLMC simulation. A branch-and-bound algorithm to solve these optimization problems is proposed and analyzed.We investigate a fault tolerant MLMC method, in which samples affected by (hard) faults are discarded or replaced, depending on the statistical requirements. As the failure rate increases more and more samples are lost. Thus, the average work to successfully complete a certain number of samples increases. The proposed optimization procedure can react on experienced faults and adapt the number of samples and levels accordingly. Numerical experiments demonstrate the effectiveness of the approach.