An observation-driven time-dependent basis for a reduced description of transient stochastic systems

We present a variational principle for the extraction of a time-dependent orthonormal basis from random realizations of transient systems. The optimality condition of the variational principle leads to a closed-form evolution equation for the orthonormal basis and its coefficients. The extracted modes are associated with the most transient subspace of the system, and they provide a reduced description of the transient dynamics that may be used for reduced-order modelling, filtering and prediction. The presented method is matrix free; it relies only on the observables of the system and ignores any information about the underlying system. In that sense, the presented reduction is purely observation driven and may be applied to systems whose models are not known. The presented method has linear computational complexity and memory storage requirement with respect to the number of observables and the number of random realizations. Therefore, it may be used for a large number of observations and samples. The effectiveness of the proposed method is tested on four examples: (i) stochastic advection equation, (ii) stochastic Burgers equation, (iii) a reduced description of transient instability of Kuramoto-Sivashinsky, and (iv) a transient vertical jet governed by the incompressible Navier-Stokes equation. In these examples, we contrast the performance of the time-dependent basis versus static basis such as proper orthogonal decomposition, dynamic mode decomposition and polynomial chaos expansion.