Generalized Langevin equation and the linear regression model with memory.

Abstract

Because the existence of cheap emulators can give rise to potentially drastic computational savings as compared to direct numerical simulations of complex turbulence models, the study of reduced models has good practical relevance, and the guidelines regarding their construction from the true dynamical system model are much in demand. In accordance with this contemporary trend in science and engineering, we provide a new approach for the rigorous derivation of the linear reduced model with memory from the one-dimensional Majda-McLaughlin-Tabak (MMT) prototypical wave turbulence in thermal equilibrium. The basic idea in obtaining the physics-constrained autoregressive model is to perform the discretization in time of the generalized Langevin equation (GLE) governing a single wave profile of the true dynamical system model; the GLE formalized in the Mori-Zwanzig (MZ) projection theory is the exact reduced-order equation and the closed-form rearrangement of the canonical equation of motion for the Hamiltonian system. We study the performance of the resulting linear non-Markovian model in addressing two different manners of uncertainty quantification (UQ) problems: prediction and filtering. In doing so, we perform a comparison analysis against the linear and nonlinear Markovian models describing the MMT system, which are also built upon the GLE under the MZ framework. Finally we discuss an optimal selection of the statistical model in applying the reduced-model approach to the UQ of the turbulent signal.