We study a source identification problem for a prototypical elliptic PDE from Dirichlet boundary data. This problem is ill-posed, and the involved forward operator has a significant nullspace. Standard Tikhonov regularization yields solutions which approach the minimum $ L^2 $-norm least-squares solution as the regularization parameter tends to zero. We show that this approach 'always' suggests that the unknown local source is very close to the boundary of the domain of the PDE, regardless of the position of the true local source.We propose an alternative regularization procedure, realized in terms of a novel regularization operator, which is better suited for identifying local sources positioned anywhere in the domain of the PDE. Our approach is motivated by the classical theory for Tikhonov regularization and yields a standard quadratic optimization problem. Since the new methodology is derived for an abstract operator equation, it can be applied to many other source identification problems. This paper contains several numerical experiments and an analysis of the new methodology.

中文翻译：

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椭圆形PDE来源识别的正则化运算符

我们从Dirichlet边界数据研究原型椭圆PDE的源识别问题。这个问题是不恰当的，并且所涉及的前向运算符具有很大的空空间。标准Tikhonov正则化产生的解接近正则化参数趋于零的最小L ^ 2 $-范数最小二乘解。我们表明，这种方法``总是''表明未知的本地源非常靠近PDE的域边界，而不管真实本地源的位置如何。正则化运算符，它更适合于标识位于PDE域中任意位置的本地源。我们的方法是由Tikhonov正则化的经典理论所激发的，并产生了标准的二次优化问题。由于新方法是针对抽象算子方程式得出的，因此可以将其应用于许多其他源识别问题。本文包含一些数值实验和对新方法的分析。