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Mori-Zwanzig reduced models for uncertainty quantification I: Parametric uncertainty

In many time-dependent problems of practical interest the parameters 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 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 the uncertain parameters. We also present a Markovian reformulation of the Mori-Zwanzig reduced equation which replaces the solution of the orthogonal dynamics equation with an infinite hierarchy of ordinary differential equations. The viscous Burgers equation with uncertain viscosity parameter is used to illustrate the construction. For this example we provide a way to estimate the necessary parameters that appear in the reduced model without having to solve the full system.