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因果扩散及其反向扩散问题

Causal diffusion and its backwards diffusion problem

University of Innsbruck（因斯布鲁克大学）

Richard Kowar

备注：
In the replacement I have rewritten the abstract and the introduction. Moreover, I have added Remark 1 and simplified a little bit the proof of Theorem 4. The reference 25 is updated, since the paper is now published
发表信息：
J. Math. Anal. Appl. 400, pp. 222-238, 2013
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AI总结：

本文通过用因果扩散模型替代非因果扩散方程，研究反向扩散问题，推导了格林函数的解析表示，并证明了反向扩散问题在特定条件下的不适定性，同时进行了数值模拟。

摘要：

本文通过将直接问题中的非因果扩散方程替换为文献[Kow11]中针对恒定扩散速度开发的因果扩散模型，重新审视了反向扩散问题。为此，我们推导了在波矢-时间空间中任意波矢维度$N$下因果扩散格林函数的解析表示。我们证明，对于空间维度$N\geq 3$（$N=2$），如果数据采集时间大于特征时间周期$\tau$（$2\tau$），则相应的反向扩散问题是不适定的，但并非指数不适定。与非因果情况相反，对于$N=1$，该反问题是适定的。此外，我们在时空域和波矢-时间域中对因果扩散和非因果扩散进行了理论和数值比较。最后，通过Landweber方法对反向扩散问题进行了数值模拟。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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