AI 中文总结
研究从稀疏边界测量中唯一确定测度值源的逆源问题，结合正则性等分析工具及数值研究，将相关文献从点源或\(L^2\)源扩展到一般拉东测度，实现点源和曲线上源的重建。
AI 中文摘要
本文致力于从稀疏边界测量中唯一确定测度值源的逆源问题。所考虑的测量包括在域边界上两个不同点的时间间隔内的通量观测。这项工作的主要目标是将现有的关于稀疏边界测量的逆源问题的文献扩展到识别一类一般的拉东测度，目前该文献仅限于点源或\(L^2\)源。我们的方法结合了多种分析工具，包括正则性性质、边界表示以及具有奇异源的扩散方程解的时间解析性。我们的理论分析通过对该问题的数值研究得到补充。特别是，我们研究了点源和支撑在曲线上的源的重建，并给出数值实验来说明从稀疏边界通量测量中恢复此类源的情况。
英文摘要
This article is devoted to the inverse source problem of uniquely determining a measure-valued source from sparse boundary measurements. The measurements considered consist of flux observations over a time interval at two distinct points on the boundary of the domain. The main objective of this work is to extend the existing literature on inverse source problems from sparse boundary measurements, which has so far been limited to point sources or L2 sources, to the identification of a general class of Radon measures. Our approach combines several analytical tools, including regularity properties, boundary representations, and the time analyticity of solutions to the diffusion equation with singular sources. Our theoretical analysis is complemented by a numerical study of the problem. In particular, we investigate the reconstruction of point sources and of a source supported on a curve, and present numerical experiments illustrating the recovery of such sources from sparse boundary flux measurements.
