Analysis of the Hessian for inverse scattering problems: II. Inverse medium scattering of acoustic waves
Analysis of the Hessian for inverse scattering problems: II. Inverse medium scattering of acoustic waves
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逆散射问题的 Hessian 分析：II．
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
O. Ghattas
中科院分区：
文献类型：
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作者：
T. Bui;O. Ghattas
我们解决了由于不均匀介质而导致的声波的逆问题。 Hessian可以分解为两个组成部分，两者都显示为紧凑型示例，以验证我们的理论结果。 Hessian可以通过低级矩阵近似。
We address the inverse problem for scattering of acoustic waves due to an inhomogeneous medium. We derive and analyze the Hessian in both Hölder and Sobolev spaces. Using an integral equation approach based on Newton potential theory and compact embeddings in Hölder and Sobolev spaces, we show that the Hessian can be decomposed into two components, both of which are shown to be compact operators. Numerical examples are presented to validate our theoretical results. The implication of the compactness of the Hessian is that for small data noise and model error, the discrete Hessian can be approximated by a low-rank matrix. This in turn enables fast solution of an appropriately regularized inverse problem, as well as Gaussian-based quantification of uncertainty in the estimated inhomogeneity.