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：提出了一种非线性优化方法，用于求解具有空间变化特性的逆向介质问题。为避免标准网格表示导致的大量未知控制变量，该模型在由特定椭圆算子的前几个特征函数张成的低维子空间内最小化失配函数，而该椭圆算子本身依赖于前一次迭代。通过反复调整搜索空间的维度和基函数，正则化在每次迭代中自然融入，无需额外的吉洪诺夫惩罚项。在角度条件下证明了收敛性，并将其纳入所提出的自适应谱反演算法。将该方法应用于椭圆逆向问题时，与采用$L^2$-吉洪诺夫正则化的标准网格反演相比具有显著优势。通过时间依赖逆散射问题的数值实验，进一步验证了新增角度条件带来的精度提升。