Inverse medium problems involve the reconstruction of a spatially varying unknown medium from available observations by exploring a restricted search space of possible solutions. Standard grid-based representations are very general but all too often computationally prohibitive due to the high dimension of the search space. Adaptive spectral (AS) decompositions instead expand the unknown medium in a basis of eigenfunctions of a judicious elliptic operator, which depends itself on the medium. Here the AS decomposition is combined with a standard inexact Newton-type method for the solution of time-harmonic scattering problems governed by the Helmholtz equation. By repeatedly adapting both the eigenfunction basis and its dimension, the resulting adaptive spectral inversion (ASI) method substantially reduces the dimension of the search space during the nonlinear optimization. Rigorous estimates of the AS decomposition are proved for a general piecewise constant medium. Numerical results illustrate the accuracy and efficiency of the ASI method for time-harmonic inverse scattering problems, including a salt dome model from geophysics.

中文翻译：

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逆介质问题的自适应谱分解

逆介质问题涉及通过探索可能解决方案的受限搜索空间，从可用观察中重建空间变化的未知介质。标准的基于网格的表示非常通用，但由于搜索空间的高维度，计算量常常令人望而却步。自适应谱 (AS) 分解以明智的椭圆算子的本征函数为基础，而是扩展未知介质，该算子本身取决于介质。在这里，AS 分解与标准的非精确牛顿型方法相结合，用于解决由亥姆霍兹方程控制的时谐散射问题。通过反复调整本征函数基及其维数，由此产生的自适应谱反演 (ASI) 方法在非线性优化过程中大大降低了搜索空间的维数。对于一般分段常数介质，证明了对 AS 分解的严格估计。数值结果说明了 ASI 方法对时谐逆散射问题的准确性和效率，包括来自地球物理学的盐丘模型。