基于稀疏数据的heat equation源项识别
On the identification of source term in the heat equation from sparse data
摘要：
本文考虑从$\partialΩ\times(0,\infty)$的稀疏子集上的叠加横向数据恢复$f(x,t)=p(x)q(t)$中的非齐次热方程的源项$f(x,t)=p(x)q(t)$，其中$Ω$是$\mathbb{R}^2$中的有界区域，具有光滑边界$\partialΩ$.具体来说，我们将需要一个小的有限数量的$N$测量点上$\partialΩ$和证明的唯一性结果，即恢复对$(p,q)$在一个给定的类，明智的选择$N=2$点。当然，由于重叠数据的缺乏，这个问题是严重不适定的。然而，我们将表明，提供的数据噪声水平是低的，有效的数值重建可能会得到。
英文摘要：
We consider the recovery of a source term $f(x,t)=p(x)q(t)$ for the nonhomogeneous heat equation in $Ω\times (0,\infty)$ where $Ω$ is a bounded domain in $\mathbb{R}^2$ with smooth boundary $\partialΩ$ from overposed lateral data on a sparse subset of $\partialΩ\times(0,\infty)$. Specifically, we shall require a small finite number $N$ of measurement points on $\partialΩ$ and prove a uniqueness result; namely the recovery of the pair $(p,q)$ within a given class, by a judicious choice of $N=2$ points. Naturally, with this paucity of overposed data, the problem is severely ill-posed. Nevertheless we shall show that provided the data noise level is low, effective numerical reconstructions may be obtained.
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