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|Author(s):||Alfred S. Carasso;|
|Title:||Compensating operators and stable backward in time marching in nonlinear parabolic equations.|
|Published:||November 26, 2013|
|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|
|PDF version:||Click here to retrieve PDF version of paper (912KB)|