Stable explicit stepwise marching scheme in ill-posed time-reversed viscous wave equations

Abstract

The numerical computation of ill-posed, nonlinear, multidimensional initial value problems presents considerable difficulties. Conventional stepwise marching schemes for such problems, whether explicit or implicit, are necessarily unconditionally unstable and result in explosive noise amplification. Following previous work on backward parabolic equations, this paper develops and analyses a stabilized explicit marching scheme for ill-posed time-reversed viscous wave equations. The method uses easily synthesized linear smoothing operators at each time step to quench the instability. Smoothing operators based on positive real powers p of the negative Laplacian are helpful, and can be realized efficiently on rectangular domains using FFT algorithms. The stabilized explicit scheme is unconditionally stable, marching forward or backward in time, and can be applied to nonlinear viscous wave equations by simply lagging the nonlinearity at the previous time step. However, 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 often small enough to allow for useful results. In the canonical case of linear autonomous selfadjoint time-reversed viscous wave equations, with solutions satisfying prescribed bounds, it is proved that the stabilized explicit scheme leads to an error estimate differing from the best-possible estimate, only by the stabilization penalty. The procedure is a valuable complement to the well-known quasi-reversibility method. As illustrative examples, the paper uses fictitiously blurred pixel images, obtained using sharp images as initial values in linear or nonlinear well-posed, forward viscous wave equations. Such images are associated with highly irregular underlying data intensity surfaces that can severely challenge reconstruction procedures. Deblurring these images proceeds by applying the stabilized explicit scheme on the corresponding ill-posed, time-reversed equation. Instructive computational experiments demonstrate the capabilities of the method on 2D rectangular regions.