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
NIST Pub Series: NIST Interagency/Internal Report (NISTIR)
Pub Type: NIST Pubs