论文标题
$ H^1 $正则化的参数识别问题的有效解决方案
Efficient solution of parameter identification problems with $H^1$ regularization
论文作者
论文摘要
我们考虑在$ h^1 $正则化下对空间分布的参数的识别。通过高斯 - 纽顿迭代来解决相关的最小化问题，导致在每个步骤中要解决的线性问题，可以将其作为边界价值问题施放，涉及涉及拉普拉斯式的低级别修饰。使用代数多机作为快速拉普拉斯求解器，可以使用Sherman-Morrison-Woodbury公式来构建这些线性问题的预处理，该线性问题表现出极好的缩放W.R.T.相关问题参数。我们首先在功能设置中开发这种方法，从而获得了选择$ h^1 $正则化的边界条件的一致方法。然后，我们构建了一种解决离散线性系统的方法，该系统基于将任何快速泊松求解器与伍德伯里公式相结合的方法。然后，通过缩放实验证明了该方法的功效。这些是针对在电阻率断层扫描中产生的常见非线性参数鉴定问题进行的。
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.
