Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

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



Alfred S. Carasso


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 analyzes 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 of the negative Laplacian are helpful, and $(-\Δ)^p$ 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 {\em 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 {\em best-possible} estimate, only by the stabilization penalty. The method is a valuable complement to the well-known quasi-reversibility method. As illustrative examples, the paper uses fictitiously blurred $512 \times 512$ pixel images, obtained by using sharp images as initial values in linear or nonlinear well-posed, forward viscous wave equations. Deblurring these images uses the stabilized explicit scheme on the corresponding ill-posed, time-reversed equation. These images are associated with highly irregular underlyi
Inverse Problems in Science and Engineering


FFT Laplacian stabilization, forward or backward time marching, image deblurring, irreversible systems, non-integer power Laplacian, quasi-reversibility method, stabilized explicit scheme, viscous wave equation.


Carasso, A. (2015), Stable explicit stepwise marching scheme in ill-posed time-reversed viscous wave equations, Inverse Problems in Science and Engineering, [online], (Accessed June 23, 2024)


If you have any questions about this publication or are having problems accessing it, please contact

Created August 12, 2015, Updated June 2, 2021