Adjoint Optimization with Spectral Preconditioning for Sparse Turbulent Flow Reconstruction

Reconstructing flow histories from sparse measurements in two-dimensional decaying isotropic turbulence presents a fundamental inverse problem in fluid dynamics. Adjoint-based data assimilation (DA) efficiently addresses this challenge by leveraging the gradient of a cost function quantifying the mismatch between modeled and observed data. However, conventional adjoint methods often amplify small-scale modes during backward integration, degrading the fidelity of large-scale reconstructions. To address this, we introduce a Fourier-space preconditioning strategy that modifies the inner product in the forward–adjoint duality using a tailored weighting kernel to emphasize low-wavenumber content. This yields filtered adjoint equations reminiscent of large-eddy simulation adjoints, stabilizing computations and enabling scale-selective control. Using a discrete adjoint solver, numerical experiments demonstrate that our method significantly improves reconstruction accuracy and flow coherence compared to unweighted approaches. These results underscore the importance of inner product design in adjoint-based DA, particularly for turbulent flows with sparse observations.