Causal diffusion and its backwards diffusion problem

Abstract

This article starts over the backwards diffusion problem by replacing the \emph{noncausal} diffusion equation, the direct problem, by the \emph{causal} diffusion model developed in \cite{Kow11} for the case of constant diffusion speed. For this purpose we derive an analytic representation of the Green function of causal diffusion in the wave vector-time space for arbitrary (wave vector) dimension $N$. We prove that the respective backwards diffusion problem is ill-posed, but not exponentially ill-posed, if the data acquisition time is larger than a characteristic time period $τ$ ($2\,τ$) for space dimension $N\geq 3$ (N=2). In contrast to the noncausal case, the inverse problem is well-posed for N=1. Moreover, we perform a theoretical and numerical comparison between causal and noncausal diffusion in the \emph{space-time domain} and the \emph{wave vector-time domain}. The paper is concluded with numerical simulations of the backwards diffusion problem via the Landweber method.