Reduced Order Nonlinear Filters for Multi-Scale Systems with Correlated Sensor Noise

This paper provides theoretical results and numerical demonstration for nonlinear filtering of systems with multiple timescales and correlated signal-sensor noise. The motivation of this work is to provide the necessary theoretical bedrock upon which computationally efficient algorithms may be further developed to handle the problem of data assimilation in ever-increasingly higher dimensional complex systems; specifically with a focus on Dynamic Data-Driven Application Systems. As a main result, we provide details of the convergence of the filter equation to a homogenized (reduced order) filter in the correlated case. We present a particle filtering method that makes use of the reduced order filtering equation to efficiently solve high-dimensional multi-scale models. We numerically demonstrate an implementation of the particle method on a two-dimensional multi-scale problem with correlated noise, and a scalable testbed atmospheric model that is chaotic and has multiple timescales.