On Parallel MLMC for Stationary Single Phase Flow Problem

Many problems that incorporate uncertainty often requires solving a Stochastic Partial Differential Equation. Fast and efficient methods for solving such equations are of particular interest for computational fluid dynamics. Efficient methods for uncertainty quantification in porous media flow simulations are Multilevel Monte Carlo sampling based algorithms. They rely on sample drawing from a probability space. The error is quantified by the root mean square error. Although computationally they are significantly faster than the classical Monte Carlo, parallel implementation is necessity for realistic simulations. The problem of finding optimal processor distribution is considered NP-complete. In this paper, a stationary single-phase flow through a random porous medium is studied as a model problem. Although simple, it is well-established problem in the field, that shows well the computational challenges involving MLMC simulation. For this problem different dynamic scheduling strategies exploiting three-layer parallelism are examined. The considered schedulers consolidate the sample to sample time differences. In this way, more efficient use of computational resources is achieved.