Data assimilation in 2D hyperbolic/parabolic systems using a stabilized explicit finite difference scheme run backward in time.

An artificial example of a coupled system of three nonlinear partial differential equations generalizing 2D thermoelastic vibrations, is used to demonstrate the effectiveness, as well as the limitations, of a non iterative direct procedure in data assimilation. A stabilized explicit finite difference scheme, run backward in time, is used to find initial values, $ [u(.,0), v(.,0), w(.,0)] $ [ u (. , 0) , v (. , 0) , w (. , 0) ] , that can evolve into a useful approximation to a hypothetical target result $ [u^*(.,T_{\max }), v^*(.,T_{\max }), w^*(T_{\max })] $ [ u ∗ (. , T max) , v ∗ (. , T max) , w ∗ (T max) ] , at some realistic $ T_{\max } > 0 $ T max > 0. Highly non smooth target data are considered, that may not correspond to actual solutions at time $ T_{\max } $ T max . Stabilization is achieved by applying a compensating smoothing operator at each time step. Such smoothing leads to a distortion away from the true solution, but that distortion is small enough to allow for useful results. Data assimilation is illustrated using $ 512 \times 512 $ 512 × 512 pixel images. Such images are associated with highly irregular non smooth intensity data that severely challenge ill-posed reconstruction procedures. Computational experiments show that efficient FFT-synthesized smoothing operators, based on $ (-\Delta)^q $ (− Δ) q with real q>3, can be successfully applied, even in nonlinear problems in non-rectangular domains. However, an example of failure illustrates the limitations of the method in problems where $ T_{\max } $ T max , and/or the nonlinearity, are not sufficiently small. [ABSTRACT FROM AUTHOR]