Time-parallel data assimilation using a Paraexp-based observer strategy

In this paper, we propose a time-parallel strategy for data assimilation based on the coupling of the Paraexp algorithm with a Luenberger observer. The observer dynamics, being linear, can be reformulated as a linear initial-value problem, making it particularly suitable for the Paraexp approach. To address assimilation over an unbounded time interval, we introduce a windowing strategy together with an adaptive refinement of the time-parallel accuracy across windows. We show that, under a suitable accuracy condition, the exponential convergence rate of the classical observer is preserved. The proposed method is illustrated on several linear systems arising from discretized partial differential equations, including hyperbolic models, for which classical iterative parallel-in-time methods may encounter difficulties. Numerical experiments demonstrate significant efficiency gains and, in some cases, superlinear speedups due to the adaptive accuracy strategy.