论文标题
用于椭圆pdes源识别的正规化操作员
A regularization operator for source identification for elliptic PDEs
论文作者
论文摘要
我们研究了Dirichlet边界数据的原型椭圆PDE的源识别问题。这个问题是不适合的，并且参与的远期操作员有明显的零空间。标准的Tikhonov正则化产生的解决方案接近最小$ l^2 $ -NORM最小二乘解决方案，因为正则化参数趋于零。我们表明，这种方法“始终”表明未知的局部来源非常接近PDE域的边界，而不论真实局部来源的位置如何。 我们提出了一种替代正规化程序，该程序是根据新型的正则操作员实现的，该程序更适合识别位于PDE域中任何地方的局部来源。我们的方法是由Tikhonov正则化的经典理论激励的，并产生了标准的二次优化问题。由于新方法是为抽象操作员方程式得出的，因此可以应用于许多其他源识别问题。本文包含几个数值实验和新方法的分析。
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.
