We present iDARR, a scalable iterative Data-Adaptive RKHS Regularization method, for solving ill-posed linear inverse problems. The method searches for solutions in subspaces where the true solution can be identified, with the data-adaptive RKHS penalizing the spaces of small singular values. At the core of the method is a new generalized Golub-Kahan bidiagonalization procedure that recursively constructs orthonormal bases for a sequence of RKHS-restricted Krylov subspaces. The method is scalable with a complexity of $O(kmn)$ for $m$-by-$n$ matrices with $k$ denoting the iteration numbers. Numerical tests on the Fredholm integral equation and 2D image deblurring show that it outperforms the widely used $L^2$ and $l^2$ norms, producing stable accurate solutions consistently converging when the noise level decays.

翻译：我们提出iDARR，一种可扩展的迭代自适应再生核希尔伯特空间（RKHS）正则化方法，用于求解病态线性逆问题。该方法在可辨识真实解的子空间中搜索解，利用数据自适应RKHS惩罚小奇异值空间。其核心是一种新的广义Golub-Kahan双对角化过程，可递归构建一系列RKHS约束Krylov子空间的标准正交基。该方法复杂度为$O(kmn)$（其中$m\times n$矩阵的迭代次数为$k$），具有良好的可扩展性。在Fredholm积分方程和二维图像去模糊的数值实验中，该方法优于广泛使用的$L^2$和$l^2$范数，在噪声水平衰减时始终产生稳定精准的收敛解。