一种直接线性反演：仅从内部位移信息恢复不连续弹性参数
A direct linear inversion for discontinuous elastic parameters recovery from internal displacement information only
AI总结：
本文提出并分析了一种新的直接方法，用于解决线性弹性逆问题。通过内部介质中的位移场测量，能够稳定地恢复弹性参数，即使参数不连续且无需边界信息。方法基于刚度到力算子的弱定义，将问题转化为线性系统，并证明了在最小光滑性假设下，剪切模量恢复具有L²稳定性。
摘要：
本文旨在提出并分析一种新的直接方法，用于解决线性弹性逆问题。给定介质内部某些位移场的测量，我们表明即使参数不连续且没有边界信息，也能实现弹性参数的稳定重建。我们提供了一种基于刚度到力算子弱定义的一般方法，将问题转化为线性系统。我们证明，在剪切模量恢复的情况下，仅需一个测量即可在最小光滑性假设下获得L²稳定性。该稳定性结果通过证明待求解的线性算子具有闭合范围而获得。我们随后描述了一种直接离散化方法，能够稳定地重建各向同性和各向异性刚度张量。
英文摘要：
The aim of this paper is to present and analyze a new direct method for solving the linear elasticity inverse problem. Given measurements of some displacement fields inside a medium, we show that a stable reconstruction of elastic parameters is possible, even for discontinuous parameters and without boundary information. We provide a general approach based on the weak definition of the stiffness-to-force operator which conduces to see the problem as a linear system. We prove that in the case of shear modulus reconstruction, we have an $L^2$-stability with only one measurement under minimal smoothness assumptions. This stability result is obtained though the proof that the linear operator to invert has closed range. We then describe a direct discretization which provides stable reconstructions of both isotropic and anisotropic stiffness tensors.
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