Mori-Zwanzig Approach to Uncertainty Quantification

Determining the statistical properties of nonlinear random systems is a problem of major interest in many areas of physics and engineering. Even with recent theoretical and computational advancements, no broadly applicable technique has yet been developed for dealing with the challenging problems of high dimensionality, low regularity and random frequencies often exhibited by the system. The Mori-Zwanzig and the effective propagator approaches discussed in this chapter have the potential of overcoming some of these limitations, in particular the curse of dimensionality and the lack of regularity. The key idea stems from techniques of irreversible statistical mechanics, and it relies on developing exact evolution equations and corresponding numerical methods for quantities of interest, e.g., functionals of the solution to stochastic ordinary and partial differential equations. Such quantities of interest could be low-dimensional objects in infinite-dimensional phase spaces, e.g., the lift of an airfoil in a turbulent flow, the local displacement of a structure subject to random loads (e.g., ocean waves loading on an offshore platform), or the macroscopic properties of materials with random microstructure (e.g., modeled atomistically in terms of particles). We develop the goal-oriented framework in two different, although related, mathematical settings: the first one is based on the Mori-Zwanzig projection operator method, and it yields exact reduced-order equations for the quantity of interest. The second approach relies on effective propagators, i.e., integrals of exponential operators with respect to suitable distributions. Both methods can be applied to nonlinear systems of stochastic ordinary and partial differential equations subject to random forcing terms, random boundary conditions, or random initial conditions.