Structured covariance completion via proximal algorithms

This paper studies the interplay between dynamics and statistics of a stochastically driven dynamical system. Motivation is provided by applications in fluid flow modeling and control. In this context, second-order statistics around the mean velocity profile can be obtained, for a subset of variables, from experiments or numerical simulations. The basic idea is to determine a parsimonious perturbation of the generator of the linearized Navier-Stokes equations, together with directions of excitation sources, that can account for the observed statistics. This covariance completion problem is to determine minimum energy and low-rank perturbation of the linearized dynamics to reconcile them with the partially available second-order statistics - such models are valuable as tools for analysis and control purposes. The resulting optimization problem can be cast as a convex semidefinite program (SDP). However, general-purpose SDP solvers cannot handle typical problem-sizes that are of interest in fluid flows. We develop customized algorithms that allow handling such covariance completion problems for substantially larger scales. These algorithms exploit the structure of the problem and utilize the method of multipliers and the proximal augmented Lagrangian method.