Analysis of the Hessian for Inverse Scattering Problems. Part III: Inverse Medium Scattering of Electromagnetic Waves in Three Dimensions-猫眼课题宝

逆散射问题的 Hessian 分析。

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2013

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T. Bui;O. Ghattas

继续我们之前的工作 [6, Inverse Problems, 2012, 28, 055002] 和 [5, Inverse Problems, 2012, 28, 055001]，我们解决了逆散射问题的不适定性通过研究非均匀介质产生的电磁波数据不适合的 Hessian 矩阵。我们推导并分析了 Hessian 矩阵霍尔德空间和索博列夫空间。使用基于积分方程的方法关于牛顿势理论和 Holder 中的紧嵌入 Sobolev 空间，我们证明 Hessian 矩阵可以分解为三个组件，所有这些都被证明是紧凑的运算符。这 Hessian 紧凑性的含义是对于小数据噪声和模型误差，离散 Hessian 矩阵可以近似为低秩矩阵。这反过来又可以快速解决适当正则化的反问题，以及基于高斯的估计不均匀性的不确定性的量化。

Continuing our previous work [6, Inverse Problems, 2012, 28, 055002] and [5, Inverse Problems, 2012, 28, 055001], we address the ill-posedness of the inverse scattering problem of electromagnetic waves due to an inhomogeneous medium by studying the Hessian of the data misfit. We derive and analyze the Hessian in both Holder and Sobolev spaces. Using an integral equation approach based on Newton potential theory and compact embeddings in Holder and Sobolev spaces, we show that the Hessian can be decomposed into three components, all of which are shown to be compact operators. 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.

Analysis of the Hessian for Inverse Scattering Problems. Part III: Inverse Medium Scattering of Electromagnetic Waves in Three Dimensions

逆散射问题的 Hessian 分析。

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