Causal diffusion and its backwards diffusion problem

TL;DR

This paper investigates the backwards diffusion problem using a causal diffusion model, deriving an analytic Green function, analyzing ill-posedness, and comparing causal and noncausal diffusion through theoretical and numerical methods.

Contribution

It introduces a causal diffusion model for backwards diffusion, derives its Green function, and analyzes the problem's well-posedness and differences from noncausal diffusion.

Findings

Backwards causal diffusion is ill-posed but not exponentially ill-posed for certain data acquisition times.

The inverse problem is well-posed in one dimension.

Numerical simulations demonstrate differences between causal and noncausal diffusion.

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 $\tau$ ($2\,\tau$) 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.