We consider the identification of spatially distributed parameters under $H^1$ regularization. Solving the associated minimization problem by Gauss-Newton iteration results in linearized problems to be solved in each step that can be cast as boundary value problems involving a low-rank modification of the Laplacian. Using algebraic multigrid as a fast Laplace solver, the Sherman-Morrison-Woodbury formula can be employed to construct a preconditioner for these linear problems which exhibits excellent scaling w.r.t. the relevant problem parameters. We first develop this approach in the functional setting, thus obtaining a consistent methodology for selecting boundary conditions that arise from the $H^1$ regularization. We then construct a method for solving the discrete linear systems based on combining any fast Poisson solver with the Woodbury formula. The efficacy of this method is then demonstrated with scaling experiments. These are carried out for a common nonlinear parameter identification problem arising in electrical resistivity tomography.

翻译：我们考虑在$H^1$正则化框架下识别空间分布参数的问题。通过高斯-牛顿迭代求解相关极小化问题，每一步需要求解的线性化问题可转化为涉及拉普拉斯算子低秩修正的边值问题。利用代数多重网格作为快速拉普拉斯求解器，可采用谢尔曼-莫里森-伍德伯里公式为这些线性问题构建预条件子，该预条件子在相关问题参数方面展现出优异的可扩展性。我们首先在泛函框架下发展该方法，从而建立一套一致性的边界条件选取准则（这些边界条件源于$H^1$正则化）。随后构建离散线性系统的求解方法，其核心思想是将任意快速泊松求解器与伍德伯里公式相结合。最终通过扩展性实验验证该方法的有效性，实验针对电阻率层析成像中典型的非线性参数辨识问题展开。