Preconditioned Adjoint Data-Assimilation for Two-dimensional Decaying Isotropic Turbulence

The inverse problem of reconstructing the history of flow from sparse measurements using the Navier-Stokes equations is formulated as a constrained cost minimization, where the cost function is defined as the mismatch between the true and modeled measurements. The adjoint fields assist in efficiently evaluating the gradient of the cost function with respect to the initial condition of the flow evolution. Our previous studies indicate that the adjoint fields grow exponentially in backward time and favor small-scale structures in the initial condition, complicating adjoint-based data assimilation. To achieve favorable reconstruction quality across scales, the optimization process is preconditioned by modifying the inner product definition for the forward-adjoint duality relation with a weighting kernel in Fourier space. We demonstrate that this approach of preconditioning resembles the adjoint of large-eddy simulations, with the flexibility to adjust the filter as needed. We focus on two-dimensional decaying isotropic turbulence within a periodic domain and perform data assimilation using a discrete adjoint solver. The results with different preconditioned adjoint solvers are discussed and compared, highlighting the potential of preconditioned adjoint in data assimilation techniques.