Multilevel Quasi-Monte Carlo Methods for Lognormal Diffusion

Robert Scheichl
Department of Mathematical Sciences
University of Bath
Bath, UK


Abstract:

Large-scale PDE-constrained uncertainty propagation and Bayesian inference are inherently difficult and computationally intensive problems. Sampling methods are among the most accurate and promising methods for these problems. Their cost is dimension independent, thus rendering them very suitable for the infinite-dimensional PDE setting. However, classical sampling approaches, such as Monte Carlo methods or Metropolis-Hastings MCMC, are very slow to converge and hence infeasible for realistic applications. The problem becomes even more involved when the differential operator depends in a non-affine way on the random parameters and when the quantities of interest depend nonlinearly on the PDE solution. A popular model problem for this situation is the lognormal diffusion problem which is of actual practical interest in hydrology. Quasi-Monte Carlo methods and their multilevel extensions provide alternatives with potentially vastly improved computational efficiencies, in terms of cost to accuracy. We provide a dimension-independent (tractable) convergence analysis for the lognormal problem for the forward uncertainty propagation case as well as for the inverse Bayesian inference setting supported by numerical experiments. The analysis in the lognormal case and for nonlinear quantities of interest require several non-trivial extensions of the convergence theory in the uniform, affine case.

This is joint work with F.Y. Kuo, I.G. Graham, J.A. Nicholls, Ch. Schwab, I.H Sloan, A. Stuart, A.L. Teckentrup, E. Ullmann.