论文标题
用于椭圆形库奇问题的Mann迭代正则化方法
A Mann iterative regularization method for elliptic Cauchy problems
论文作者
论文摘要
我们在常规套装$ω\子集r^2 $上调查了带有$ c^\ infty $ cefficients的线性椭圆运算符的Cauchy问题，这是一个经典的问题的经典示例。 Cauchy数据是在歧管$γ\ subset \partialΩ$上给出的，我们的目标是重建椭圆方程的$ h^1（ω）$解决方案，该椭圆方程$ \ partialω/γ$。 此处提出的方法与与椭圆形的Cauchy问题相关的固定点方程组成了分段Mann迭代。我们的算法概括了Maz'ya等人开发的迭代方法，Maz'ya等人提出了一种基于解决连续良好的混合边界价值问题的方法。我们在理论上和数值上分析了正则化和收敛属性。
We investigate the Cauchy problem for linear elliptic operators with $C^\infty$-coefficients at a regular set $Ω\subset R^2$, which is a classical example of an ill-posed problem. The Cauchy data are given at the manifold $Γ\subset \partialΩ$ and our goal is to reconstruct the trace of the $H^1(Ω)$ solution of an elliptic equation at $\partial Ω/ Γ$. The method proposed here composes the segmenting Mann iteration with a fixed point equation associated with the elliptic Cauchy problem. Our algorithm generalizes the iterative method developed by Maz'ya et al., who proposed a method based on solving successive well-posed mixed boundary value problems. We analyze the regularizing and convergence properties both theoretically and numerically.
