Multilevel Solvers for Stochastic Fluid Flows

Uncertainty is ubiquitous in many areas of science and engineering. It may result from the inadequacy of mathematical models to represent the reality or from unknown physical parameters that are required as inputs for these models. Uncertainty may also arise due to the inherent randomness of the system being analyzed. For many problems of practical interest, uncertainty quantification (UQ) can involve computations that are intractable even for the modern supercomputers, if conventional mathematical techniques are utilized. The reason is typically a product of complexity factors associated with many samples needed to compute the statistics, and for each sample, complexity associated with the spatio-temporal scales characteristics to the system. The main objective of this research is to obtain multilevel solvers for stochastic fluid flow problems with high-dimensional uncertainties. In our approach, the complexity arising due to sampling is overcome by the multilevel Monte Carlo (MLMC) method and complexity due to spatio-temporal scales is eliminated via the multigrid solver. Historically, Monte Carlo (MC) type methods have been proven to be the methods of choice for problems with a large uncertainty dimension as they do not suffer from the curse of dimensionality. A well-known computational bottleneck associated with the plain MC method is the slow convergence of the sampling error. For problems involving a wide range of space and time scales, ensuring a low mean square error will require a large number of MC samples on a very fine computational mesh making the estimator very expensive. Inspired by the multigrid ideas, the MLMC method generalizes the standard MC to multiple grids, exhibiting an exceptional improvement. The efficiency of the MLMC method comes from solving the problem of interest on a coarse grid and subsequently adding corrections based on fewer mesh resolutions. On the coarsest grid, a large number of samples can be computed inexpensively. The corrections computed on fewer grids, have smaller variances and can be estimated accurately using only fewer samples. The estimates at different levels are then combined using a telescopic sum...