Stabilized backward in time explicit marching schemes in the numerical computation of ill-posed time-reversed hyperbolic/parabolic systems

This paper develops stabilized explicit marching difference schemes that can successfully solve a significant but limited class of multidimensional, ill-posed, backward in time problems for coupled hyperbolic/parabolic systems associated with vibrating thermoelastic plates and coupled sound and heat flow. Stabilization is achieved by applying compensating smoothing operators at each time step, to quench the instability. Analysis of convergence is restricted to the transparent case of linear, autonomous, selfadjoint spatial differential operators, and almost best-possible error bounds are obtained for backward in time reconstruction in that class of problems. However, the actual computational schemes can be applied to more general problems, including examples with variable time dependent coefficients, as well as nonlinearities. The stabilized explicit schemes are unconditionally stable, marching forward or backward in time, but the smoothing operation at each step leads to a distortion away from the true solution. This is the stabilization penalty. It is shown that in many problems of interest, that distortion is small enough to allow for useful results. Backward in time continuation is illustrated using $512\times 512$ pixel images. Such images are associated with highly irregular non smooth intensity data that severely challenge ill-posed reconstruction procedures. Several computational experiments show that efficient FFT-synthesized smoothing operators, based on ${(-\mathrm{\Delta })}^p$ with real $p>2$, can be successfully applied in a broad range of problems.