# Publication Citation: Reconstructing the past from imprecise knowledge of the present: Some examples of non uniqueness in solving parabolic equations backward in time.

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Author(s): Alfred S. Carasso; Reconstructing the past from imprecise knowledge of the present: Some examples of non uniqueness in solving parabolic equations backward in time. May 24, 2012 Identifying sources of ground water pollution, and deblurring astronomical galaxy images, are two important applications generating growing interest in the numerical computation of parabolic equations backward in time. However, while backward uniqueness typically prevails in parabolic equations, the precise data needed for the existence of a particular backward solution is seldom available. This paper discusses previously unexplored non uniqueness issues, originating from trying to reconstruct a particular solution from imprecise data. Explicit 1D examples of linear and nonlinear parabolic equations are presented, in which there is strong computational evidence for the {\em existence} of distinct solutions $w^{red}(x,t)$ and $w^{green}(x,t)$, on $~0 \leq t \leq 1$. These solutions have the property that the traces $w^{red}(x,1)$ and $w^{green}(x,1)$ at time $t=1$, are close enough to be visually indistinguishable, while the corresponding initial values $w^{red}(x,0)$ and $w^{green}(x,0)$, are vastly different, well-behaved, physically plausible functions, with comparable $L^2$ norms. This implies effective non uniqueness in the recovery of $w^{red}(x,0)$ from approximate data for $w^{red}(x,1)$. In all these examples, the Van Cittert iterative procedure is used as a tool to discover unsuspected, valid, additional solutions $w^{green}(x,0)$. This methodology can generate numerous other examples, and indicates that multidimensional problems are likely to be a rich source of striking non uniqueness phenomena. Siam Journal on Numerical Analysis 36 3 pp. 249 - 261 advection dispersion equation; backward parabolic equations; hydrologic inversion; image deblurring; ill-posed continuation; non uniqueness; Van Cittert iteration. Math Click here to retrieve PDF version of paper (729KB)