Propagation of Stochasticity in Heterogeneous Media and Applications to Uncertainty Quantification

Abstract

This chapter reviews several results on the derivation of asymptotic models for solutions to partial differential equations (PDE) with highly oscillatory random coefficients. We primarily consider elliptic models with random diffusion or random potential terms. In the regime of small correlation length of the random coefficients, the solution may be described either as the sum of a leading, deterministic, term, and random fluctuations, whose macroscopic law is described, or as a random solution of a stochastic partial differential equation (SPDE). Several models for such random fluctuations or SPDEs are reviewed here.

The second part of the chapter focuses on potential applications of such macroscopic models to uncertainty quantification and effective medium models. The main advantage of macroscopic models, when they are available, is that the small correlation length parameter no longer appears. Highly oscillatory coefficients are typically replaced by (additive or multiplicative) white noise or fractional white noise forcing. Quantification of uncertainties, such as, for instance, the probability that a certain function of the PDE solution exceeds a certain threshold, then sometimes takes an explicit, solvable, form. In most cases, however, the propagation of stochasticity from the random forcing to the PDE solution needs to be estimated numerically. Since (fractional) white noise oscillates at all scales (as a fractal object), its numerical discretization involves a large number of random degrees of freedom, which renders accurate computations extremely expensive. Some remarks on a coupled polynomial chaos – Monte Carlo framework and related concentration (Efron-Stein) inequalities to numerically solve problems with small-scale randomness conclude this chapter.