Generalized Langevin equation and the linear regression model with memory.

Area of Science:

• Computational Physics
• Fluid Dynamics
• Statistical Mechanics

Background:

• Reduced-order models offer significant computational savings compared to direct numerical simulations for complex turbulence models.
• Developing rigorous guidelines for constructing reduced models from true dynamical systems is crucial for practical applications.

Purpose of the Study:

• To present a novel approach for deriving a linear reduced model with memory from a one-dimensional turbulent wave system.
• To evaluate the performance of this model in uncertainty quantification tasks.

Main Methods:

• Discretization in time of the generalized Langevin equation (GLE) governing the wave profile.
• Formalization using Mori-Zwanzig (MZ) projection theory to obtain an exact reduced-order equation.
• Comparison with linear and nonlinear Markovian models derived from the same framework.

Main Results:

• A linear non-Markovian model with memory was successfully derived for the Majda-McLaughlin-Tabak (MMT) turbulent system.
• The derived model demonstrated performance in prediction and filtering tasks within uncertainty quantification.
• Comparative analysis highlighted the model's effectiveness against existing Markovian approaches.

Conclusions:

• The developed linear reduced model with memory provides a viable and computationally efficient alternative for turbulence simulation.
• This approach facilitates robust uncertainty quantification for turbulent signals.
• Optimal selection of statistical models is key for applying reduced-model approaches effectively.