An efficient computational framework for data assimilation of fractional-order dynamical system with sparse observations

We introduce an efficient computational framework for data assimilation of fractional dynamical systems, extending traditional data assimilation techniques to fractional models. This framework offers effective computational methods that eliminate the need for complex adjoint model derivations and algorithm redesign. We establish the fundamental problem formulation, develop both the AtD and DtA approaches, and derive adjoint forms and numerical schemes for each method. Additionally, we create a unified fractional-order variational data assimilation framework applicable to both linear and nonlinear models, incorporating both explicit and implicit discrete methods. Specific discretization schemes and gradient formulas are derived for three distinct types of fractional-order models. The method's reliability and convergence are verified, and the effect of observation sparsity and quality is examined through numerical examples.