A nonlinear optimization method is proposed for the solution of inverse medium problems with spatially varying properties. To avoid the prohibitively large number of unknown control variables resulting from standard grid-based representations, the misfit is instead minimized in a small subspace spanned by the first few eigenfunctions of a judicious elliptic operator, which itself depends on the previous iteration. By repeatedly adapting both the dimension and the basis of the search space, regularization is inherently incorporated at each iteration without the need for extra Tikhonov penalization. Convergence is proved under an angle condition, which is included into the resulting \emph{Adaptive Spectral Inversion} (ASI) algorithm. The ASI approach compares favorably to standard grid-based inversion using $L^2$-Tikhonov regularization when applied to an elliptic inverse problem. The improved accuracy resulting from the newly included angle condition is further demonstrated via numerical experiments from time-dependent inverse scattering problems.

翻译：提出了一种适用于空间变化特性的非线性反演方法。为避免标准网格离散导致待反演变量数目过大的问题，该方法在一个由特定椭圆算子前若干特征函数张成的小规模子空间中对拟合残差进行极小化，而该算子本身依赖于前一次迭代结果。通过反复自适应调整搜索空间的维度和基函数，正则化在每次迭代中被内嵌执行，无需额外添加Tikhonov惩罚项。在角度条件成立的前提下验证了算法的收敛性，该条件被纳入最终形成的自适应谱反演（ASI）算法。与采用$L^2$-Tikhonov正则化的标准网格反演方法相比，ASI方法在处理椭圆反问题时展现出更优性能。时域散射反问题数值实验进一步证实了新增角度条件对反演精度的提升效果。