Compensating operators and stable backward in time marching in nonlinear parabolic equations.

Published

November 26, 2013

Author(s)

Alfred S. Carasso

Abstract

Step by step time-marching schemes are fundamental tools in the numerical exploration of well-posed nonlinear evolutionary partial differential equations. However, when the initial value problem is ill-posed, such stepwise time-marching numerical schemes are necessary unconditionally unstable and result in explosive noise amplification. This paper outlines a novel step by step stabilized time-marching procedure for computing nonlinear parabolic equations on 2D rectangular regions, backward in time. Very little is known either analytically, or computationally, about this class of exponentially ill-posed problems. The procedure uses easily synthesized FFT-based compensating operators at every time step to quench the instability. A fictitious nonlinear image deblurring problem is used to evaluate the effectiveness of this computational approach. The method is compared with a previously introduced global in time nonlinear Van Cittert iterative procedure that is significantly more time consuming and impractical on large problems.

Citation

NIST Interagency/Internal Report (NISTIR) - 7967

Report Number

7967

NIST Pub Series

NIST Interagency/Internal Report (NISTIR)

Pub Type

NIST Pubs

Download Paper

https://doi.org/10.6028/NIST.IR.7967

Citation

Carasso, A. (2013), Compensating operators and stable backward in time marching in nonlinear parabolic equations., NIST Interagency/Internal Report (NISTIR), National Institute of Standards and Technology, Gaithersburg, MD, [online], https://doi.org/10.6028/NIST.IR.7967 (Accessed December 13, 2024)