Identifying the discontinuous diffusion coefficient in an elliptic equation with observation data of the gradient of the solution is an important nonlinear and ill-posed inverse problem. Models with total variational (TV) regularization have been widely studied for this problem, while the theoretically required nonsmoothness property of the TV regularization and the hidden convexity of the models are usually sacrificed when numerical schemes are considered in the literature. In this paper, we show that the favorable nonsmoothness and convexity properties can be entirely kept if the well-known alternating direction method of multipliers (ADMM) is applied to the TV-regularized models, hence it is meaningful to consider designing numerical schemes based on the ADMM. Moreover, we show that one of the ADMM subproblems can be well solved by the active-set Newton method along with the Schur complement reduction method, and the other one can be efficiently solved by the deep convolutional neural network (CNN). The resulting ADMM-Newton-CNN approach is demonstrated to be easily implementable and very efficient even for higher-dimensional spaces with fine mesh discretization.
翻译:在椭圆方程式中确定不连续扩散系数并观察该溶液梯度的数据是一个重要的非线性和错误的反向问题。对这个问题已广泛研究了完全变异(TV)正规化的模式,同时在理论上需要的电视正规化的非移动特性和模型的隐藏共性通常在文献中考虑到数字方法时被牺牲。在本文中,我们表明,如果在电视正规化模式中应用众所周知的倍数交替方向方法(ADMM),那么可以完全保持有利的非移动和共性特性,因此,考虑设计基于ADMM的数值方案是有意义的。 此外,我们表明,一个ADMMM的子质可以通过主动设定的牛顿方法以及Schur补充减少法来很好地解决,而另一个则可以通过深层的电磁神经网络(CNN)来有效解决。 由此形成的ADMM-Newton-CN方法证明,即使在具有精细的离心的高层空间上,也很容易执行,而且非常高效。