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, [Formula: see text], that can evolve into a useful approximation to a hypothetical target result [Formula: see text], at some realistic [Formula: see text]. Highly non smooth target data are considered, that may not correspond to actual solutions at time [Formula: see text]. 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 [Formula: see text] 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 [Formula: see text] 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 [Formula: see text], and/or the nonlinearity, are not sufficiently small.

Keywords

• Time reversed hyperbolic/parabolic systems
• non smooth data assimilation
• stabilized explicit backward marching scheme
• nonlinear problems
• numerical experiments
• 35M31