We investigate a variational method for ill-posed problems, named $\texttt{graphLa+}\Psi$, which embeds a graph Laplacian operator in the regularization term. The novelty of this method lies in constructing the graph Laplacian based on a preliminary approximation of the solution, which is obtained using any existing reconstruction method $\Psi$ from the literature. As a result, the regularization term is both dependent on and adaptive to the observed data and noise. We demonstrate that $\texttt{graphLa+}\Psi$ is a regularization method and rigorously establish both its convergence and stability properties. We present selected numerical experiments in 2D computerized tomography, wherein we integrate the $\texttt{graphLa+}\Psi$ method with various reconstruction techniques $\Psi$, including Filter Back Projection ($\texttt{graphLa+FBP}$), standard Tikhonov ($\texttt{graphLa+Tik}$), Total Variation ($\texttt{graphLa+TV}$), and a trained deep neural network ($\texttt{graphLa+Net}$). The $\texttt{graphLa+}\Psi$ approach significantly enhances the quality of the approximated solutions for each method $\Psi$. Notably, $\texttt{graphLa+Net}$ is outperforming, offering a robust and stable application of deep neural networks in solving inverse problems.

翻译：我们研究了一种针对不适定问题的变分方法，命名为$\texttt{graphLa+}\Psi$，该方法在正则化项中嵌入了图拉普拉斯算子。该方法的创新之处在于，基于解的先验近似来构建图拉普拉斯，该近似可利用文献中任意现有重建方法$\Psi$获得。由此，正则化项同时依赖于观测数据与噪声并具有自适应性。我们证明了$\texttt{graphLa+}\Psi$是一种正则化方法，并严格建立了其收敛性与稳定性。我们选取了二维计算机断层扫描中的数值实验，将$\texttt{graphLa+}\Psi$方法集成到多种重建技术$\Psi$中，包括滤波反投影（$\texttt{graphLa+FBP}$）、标准Tikhonov方法（$\texttt{graphLa+Tik}$）、全变分方法（$\texttt{graphLa+TV}$）以及训练后的深度神经网络（$\texttt{graphLa+Net}$）。$\texttt{graphLa+}\Psi$方法显著提升了每种$\Psi$方法近似解的质量。值得注意的是，$\texttt{graphLa+Net}$表现优异，为深度神经网络在求解逆问题中的应用提供了稳健且稳定的方案。