Funding Opportunity W911NF-12-R-0012-02 (Mathematics)
Division of Applied Mathematics, Army Research Office (ARO)

We address a rather neglected but very important research area in stochastic multiscale modeling, namely the propagation of uncertainty across domains characterized by partially correlated processes but with vastly different correlation lengths in space/time. This class of problems includes stochastic/stochastic domain interaction but also stochastic/deterministic coupling. Both the deterministic and stochastic processes may be described by either partial differential equations (PDEs or SPDES) or particle systems, e.g. smooth particle hydrodynamics (SPH) or dissipative particle dynamics (DPD). The domains we consider may be embedded or adjacent but partially overlapping. The fundamental open question we address is the construction of proper transmission boundary conditions at the interface that preserve global statistical properties as it is computationally prohibitive to preserve continuity of trajectories across domains. No rigorous theory or even effective empirical algorithms have yet been developed for interfaces defined in terms of functionals of the stochastic field, e.g. mean, variance or multi-point correlations. In addition to the high-dimensionality challenge, the stochastic processes across domains are correlated and hence new theoretical developments are required to represent such stochastic correlated systems.

To overcome these difficulties we propose to develop a new general framework for multi-scale propagation of uncertainty in heterogeneous stochastic systems based on multi-level domain decomposition methods. The key idea relies on new types of interface conditions and new time-integrators combined with reduced-order representations and generalized Schwarz methods. This is a timely quest, motivated in particular by recent advances in stochastic modeling techniques, domain decomposition methods and parallel computational algorithms. Specifically, we propose to develop two different approaches based on (1) state-based coupling, where moments of the state variable are employed for information exchange at the interfaces, and (2) PDF-based coupling, where the probability density function is used. The former is based on a theory we developed recently for multi-correlated processes to determine series expansions in terms of a reduced set of random variables and to represent random functions with characteristic components at different spatial scales, e.g. superposition of two (or more) processes with different correlation length. The latter is based also on a new theory that extends the Fokker-Planck-Kolmogorv equation to processes with correlated noise of arbitrary correlation length. The three-year research plan consists of theoretical and numerical developments as well as a general software framework that will implement the proposed algorithms in a seamless way for parallel environments. Demonstration examples will be drawn from multi-scale problems in solid and fluid mechanics as well as general advection-diffusion-reaction systems.

The proposed work will have a significant and broad impact as it will set the foundations of a new stochastic multi-level approach to uncertainty propagation, which is useful to multiscale modeling in many applications of interest to Army Research Labs. For instance, multi-scale stochastic modeling of complex materials and devices, dynamic data-driven applications, complex networks and multi-fidelity optimization under uncertainty.