We consider the inverse conductivity problem with discontinuous conductivities. We show in a rigorous way, by a convergence analysis, that one can construct a completely discrete minimization problem whose solution is a good approximation of a solution to the inverse problem. The minimization problem contains a regularization term which is given by a total variation penalization and is characterized by a regularization parameter. The discretization involves at the same time the boundary measurements, by the use of the complete electrode model, the unknown conductivity and the solution to the direct problem. The electrodes are characterized by a parameter related to their size, which in turn controls the number of electrodes to be used. The discretization of the unknown and of the solution to the direct problem is characterized by another parameter related to the size of the mesh involved. In our analysis we show how to precisely choose the regularization, electrodes size and mesh size parameters with respect to the noise level in such a way that the solution to the discrete regularized problem is meaningful. In particular we obtain that the electrodes and mesh size parameters should decay polynomially with respect to the noise level.
翻译:我们从不连续的导体中考虑相反的传导问题。 我们通过趋同分析, 严格地表明, 电极可以构建一个完全独立的最小化问题, 其解决办法是完全近似于对反向问题的解决方案。 最小化问题包含一个正规化的术语, 其特点是完全的变异处罚, 并具有正规化参数的特征。 离异化同时涉及边界测量, 使用完整的电极模型, 未知的传导性和直接问题的解决方案。 电极的特征是与其大小有关的参数, 后者反过来控制要使用的电极数量。 未知的离散化和直接问题的解决方案的特征是另一个与所涉网状大小有关的参数。 在我们的分析中, 我们展示了如何精确地选择与噪音水平有关的正规化、 电极大小和网状大小参数, 从而使得离散的标准化问题的解决方案有意义。 我们特别了解到, 电极和网状尺寸参数应该与噪声水平发生多相衰变。