Identifying sources of ground water pollution, and deblurring nanoscale imagery, as well as astronomical galaxy images, often involves the numerical computation of parabolic equations backward in time. However, very little computational experience has so far been accumulated on backward continuation in nonlinear parabolic equations. In this paper, an iterative procedure originating in Spectroscopy in the 1930's, is adapted into a useful tool for solving a wide class of 2D nonlinear backward parabolic equations. In addition, previously unsuspected difficulties are uncovered, that may preclude useful backward continuation in parabolic equations deviating too strongly from the linear, autonomous, self adjoint, canonical model. This paper explores backward continuation in selected nonlinear equations by creating fictitious blurred images, obtained by using given sharp images as initial data in these equations, and selecting the corresponding solutions at some positive time $T$. Successful backward continuation from $t=T$ to $t=0$, would recover the original sharp image. Visual recognition provides meaningful evaluation of the degree of success or failure in the reconstructed solutions. New and instructive examples are developed, illustrating the unexpected influence of certain types of nonlinearities. Visually indistinguishable blurred images are presented, with vastly different deblurring results. The equations studied represent nonlinear generalizations of Brownian motion, and the blurred images may be interpreted as visually expressing the results of novel stochastic processes.
Citation: Journal of Research (NIST JRES) - 118.010Report Number:
NIST Pub Series: Journal of Research (NIST JRES)
Pub Type: NIST Pubs
advection dispersion equation, backward parabolic equations, hydrologic inversion, image deblurring, ill-posed continuation, non uniqueness, Van Cittert iteration