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

Abstract

ABSTRACTAn 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)], that can evolve into a useful approximation to a hypothetical target result [u∗(.,Tmax),v∗(.,Tmax),w∗(Tmax)], at some realistic 0 $ ]]>Tmax>0. Highly non smooth target data are considered, that may not correspond to actual solutions at time Tmax. 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 512×512 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 (−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 Tmax, and/or the nonlinearity, are not sufficiently small.