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

Published

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
International Journal on Geomathematics
Volume
5
Issue
1

Keywords

nonlinear backward parabolic equations, ill-posed initial value problem, backward time-marching scheme, FFT compensating operators, non-integer power Laplacian, nonlinear Van Cittert method, nonlinear image deblurring.

Citation

Carasso, A. (2014), Compensating operators and stable backward in time marching in nonlinear parabolic equations., International Journal on Geomathematics, [online], https://doi.org/10.1007/s13137-014-0057-1 (Accessed June 7, 2023)
Created March 1, 2014, Updated June 2, 2021