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.
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)