The Mori-Zwanzig Approach to Dimension Reduction and Uncertainty Quantification
Technical Report,01 Dec 2015,30 Sep 2019
University of California Santa Cruz United States
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The Mori-Zwanzig MZ formulation is a technique originally developed in statistical mechanics to formally integrate out phase variables in nonlinear dynamical systems by means of a projection operator. One of the main features of such formulation is that it allows us to systematically derive exact equations of motion for quantities of interest macroscopic observables, based on microscopic equations of motion. Such equations can be found in a variety of applications, including molecular dynamics, fluid dynamics, and solid-state physics. Computing the solution to the MZ equation is a challenging task. One of the main difficulties is the approximation of the memory integral convolution term, and the fluctuation term noise, which encode the interaction between the so-called orthogonal dynamics and the dynamics of the quantity of interest. The orthogonal dynamics is essentially a high-dimensional nonlinear flow that satisfies a hard-to-solve integro-differential equation. Such flow has, in general, the same order of magnitude and dynamical properties as the quantity of interest, i.e., there is no general scale separation between the so-called resolved and the unresolved variables of the system. As a consequence, approximating the MZ memory integral and the fluctuation term in these cases is a daunting task, because of the strong coupling between the orthogonal dynamics and the dynamics of the macroscopic observables. In this project, we developed an in-depth mathematical analysis of the MZ formulation for both deterministic and stochastic dynamical systems, and established an effective computational framework that allows us to perform numerical simulations of the MZ equation.
- Theoretical Mathematics