Mori-Zwanzig reduced models for uncertainty quantification
This work addresses the computational bottleneck of uncertainty quantification in time-dependent problems by providing a reduced modeling framework that balances accuracy and efficiency.
The authors apply the Mori-Zwanzig formalism to construct reduced models for polynomial chaos expansion coefficients in uncertainty quantification, and propose a Markovian reformulation to reduce computational cost while retaining accuracy. Results show that including memory effects is essential for successful reduced models.
In many time-dependent problems of practical interest the parameters and/or initial conditions entering the equations describing the evolution of the various quantities exhibit uncertainty. One way to address the problem of how this uncertainty impacts the solution is to expand the solution using polynomial chaos expansions and obtain a system of differential equations for the evolution of the expansion coefficients. We present an application of the Mori-Zwanzig (MZ) formalism to the problem of constructing reduced models of such systems of differential equations. In particular, we construct reduced models for a subset of the polynomial chaos expansion coefficients that are needed for a full description of the uncertainty caused by uncertain parameters or initial conditions. Even though the MZ formalism is exact, its straightforward application to the problem of constructing reduced models for estimating uncertainty involves the computation of memory terms whose cost can become prohibitively expensive. For those cases, we present a Markovian reformulation of the MZ formalism which can lead to approximations that can alleviate some of the computational expense while retaining an accuracy advantage over reduced models that discard the memory altogether. Our results support the conclusion that successful reduced models need to include memory effects.