Reconstruction of wide spectrum forcing in transport-diffusion and Navier-Stokes equations
Jochen Bröcker, Giulia Carigi, Tobias Kuna, Vincent R. Martinez
TL;DR
The paper tackles the challenge of reconstructing unknown external forcings in infinite-dimensional dissipative systems, specifically transport–diffusion and the 2D Navier–Stokes equations, from incomplete observations. It develops two data-assimilation strategies: (i) the Sieve Algorithm, an iterative scheme that alternates state estimation and forcing reconstruction using low/high-mode projections, and (ii) the Nudging Algorithm, a continuous-time method that synchronizes observed and modeled states while recovering the forcing. A key contribution is allowing forcing terms of quasi-finite rank that can inject energy across all scales, extending prior work limited to band-limited spectra, and providing rigorous convergence analyses under explicit conditions on observational resolution, mode cutoff N, and nudging/relaxation parameters μ. The results demonstrate exponential convergence of both the reconstructed forcing and the state to the true quantities in 2D NSE and transport–diffusion, with the Nudging approach offering a conceptually simpler and more robust framework for time-independent forcings. These findings have potential impact for geophysical data assimilation and environmental monitoring where forcings are uncertain and multi-scale, energy-containing, and only partially observed.
Abstract
This article considers the problem of reconstructing unknown driving forces based on incomplete knowledge of the system and its state. This is studied in both a linear and nonlinear setting that is paradigmatic in geophysical fluid dynamics and various applications. Two algorithms are proposed to address this problem: one that iteratively reconstructs forcing and another that provides a continuous-time reconstruction. Convergence is shown to be guaranteed provided that observational resolution is sufficiently high and algorithmic parameters are properly tuned according to the prior information; these conditions are quantified precisely. The class of reconstructable forces identified here include those which are time-dependent and potentially inject energy at all length scales. This significantly expands upon the class of forces in previous studies, which could only accommodate those with band-limited spectra. The second algorithm moreover provides a conceptually streamlined approach that allows for a more straightforward analysis and simplified practical implementation.