论文标题
修改的Tikhonov正则化，用于识别多个来源
Modified Tikhonov regularization for identifying several sources
论文作者
论文摘要
我们研究是否可以使用修改版的Tikhonov正则化版本来识别原型椭圆pde的Dirichlet边界数据中的几个局部来源。本文扩展了[5]中介绍的结果。事实证明，区分两个或更多源的可能性取决于第二或第四阶PDE的平滑属性。因此，涉及域的几何形状以及来源相对于该域的边界的位置决定了可识别性。 我们还为识别单个局部来源提供了唯一性结果。该结果是根据抽象运营商框架得出的，因此不仅适用于本文研究的模型问题。我们的方案产生二次优化问题，因此可以使用标准软件工具来解决。除了理论研究外，本文还包含了几个数值实验。
We study whether a modified version of Tikhonov regularization can be used to identify several local sources from Dirichlet boundary data for a prototypical elliptic PDE. This paper extends the results presented in [5]. It turns out that the possibility of distinguishing between two, or more, sources depends on the smoothing properties of a second or fourth order PDE. Consequently, the geometry of the involved domain, as well as the position of the sources relative to the boundary of this domain, determines the identifiability. We also present a uniqueness result for the identification of a single local source. This result is derived in terms of an abstract operator framework and is therefore not only applicable to the model problem studied in this paper. Our schemes yield quadratic optimization problems and can thus be solved with standard software tools. In addition to a theoretical investigation, this paper also contains several numerical experiments.
