The simulation and forecast of complex physical processes in science, engineering and industry requires input data which are often subject to considerable uncertainties. This is due to incomplete models, measurement errors or lack of knowledge. Partial differential equations (PDEs) with random coefficients offer the opportunity to incorporate data uncertainties in mathematical models and subsequent computer simulations. However, these PDEs are formulated in a physical domain coupled with a possibly high-dimensional sample space generated by random parameters and can be very computing-intensive.
We outline the key computational challenges by discussing a model elliptic PDE of single phase subsurface flow in a random porous medium. Suggested by real-world data the diffusion coefficient is modeled as a lognormal random field with rough realisations. Hence a large number of random parameters is required. To date, only Monte Carlo based methods are computationally feasible in this case, however the cost of Monte Carlo is often prohibitively large. We employ multilevel Monte Carlo (MLMC), a novel variance reduction technique which can reduce the cost significantly. We explain the basic MLMC idea and combine this technique with mixed finite element discretisations to calculate travel times of particles in groundwater flows (Graham et al., preprint).
For coefficients which can be parameterised by a small number of random variables we employ spectral stochastic Galerkin (SG). A standard SG discretisation of lognormal diffusion problems gives a block-dense Galerkin matrix with sparse blocks. This precludes the development of iterative solvers with optimal complexity from the outset. An alternative problem formulation as convection-diffusion-type problem with random convective velocity has been studied in (Ullmann et al., 2012). We discuss possible mixed formulations of this problem. The saddle point Galerkin matrices are sparse, but have nonsymmetric off-diagonal blocks. We suggest block-diagonal and block-triangular preconditioners for use with GMRES and report on the efficiency of the preconditioners.
- I.G. Graham, R. Scheichl, and E. Ullmann. Mixed Finite Element Analysis of Lognormal Diffusion and Multilevel Monte Carlo Methods. Available from arXiv:1312.6047
- E. Ullmann, H.C. Elman, and O. G. Ernst. Efficient iterative solvers for stochastic Galerkin discretizations of log-transformed random diffusion problems. SIAM J. Sci. Comput., 34:A659-A682, 2012.