Reconstructing the past from imprecise knowledge of the present: Effective non-uniqueness in solving parabolic equations backward in time

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 existence of distinct solutions wred(x,t) and wgreen(x,t), on 0 ≤ t ≤ 1. These solutions have the property that the traces wred(x,1) and wgreen(x,1) at time t = 1 are close enough to be visually indistinguishable, while the corresponding initial values wred(x,0) and wgreen(x,0) are vastly different, well‐behaved, physically plausible functions, with comparable L2 norms. This implies effective non‐uniqueness in the recovery of wred(x,0) from approximate data for wred(x,1). In all these examples, the Van Cittert iterative procedure is used as a tool to discover unsuspected, valid, additional solutions wgreen(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. Published 2012. This article is a US Government work and is in the public domain in the USA.