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



Alfred S. Carasso


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.
NIST Interagency/Internal Report (NISTIR) - 7967
Report Number


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], (Accessed April 14, 2024)
Created November 26, 2013, Updated June 2, 2021