1 + 1维wave equation反问题的正则化策略
Regularization strategy for inverse problem for 1+1 dimensional wave equation
摘要：
考虑了具有波速$c(x)$的1 + 1维波动方程的边值反问题.我们给出了一个正则化策略，用于反演映射$\mathcal A:c\mapsto Λ,$，其中$Λ$是对应于波速$c$的双曲Neumann-to-Dirichlet映射。更准确地说，我们考虑的情况下，当我们给出了一个扰动的纽曼到狄利克雷映射$\tilde Λ=Λ+\mathcal E $，其中$\mathcal E$对应于测量误差，并重建一个近似的波速$\tilde c$。我们强调，$\tilde Λ$可能不在地图$\mathcal A$的范围内。我们证明了重建的波速$\tilde c$满足$\| \tilde c-c\|_{L^\infty}<C \|E\|^{1/18}$。我们的正则化策略是基于一个新的公式来计算$c$从$Λ$。
英文摘要：
An inverse boundary value problem for a 1+1 dimensional wave equation with wave speed $c(x)$ is considered. We give a regularisation strategy for inverting the map $\mathcal A:c\mapsto Λ,$ where $Λ$ is the hyperbolic Neumann-to-Dirichlet map corresponding to the wave speed $c$. More precisely, we consider the case when we are given a perturbation of the Neumann-to-Dirichlet map $\tilde Λ=Λ+\mathcal E $, where $\mathcal E$ corresponds to the measurement errors, and reconstruct an approximate wave speed $\tilde c$. We emphasize that $\tilde Λ$ may not not be in the range of the map $\mathcal A$. We show that the reconstructed wave speed $\tilde c$ satisfies $\| \tilde c-c\|_{L^\infty}<C \|E\|^{1/18}$. Our regularization strategy is based on a new formula to compute $c$ from $Λ$.
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