The inverse conductivity problem aims at determining the unknown conductivity inside a bounded domain from boundary measurements. In practical applications, algorithms based on minimizing a regularized residual functional subject to PDE constraints have been widely used to deal with this problem. However, such approaches typically require repeated iterations and solving the forward problem at each iteration, which leads to a heavy computational cost. To address this issue, we first reformulate the inverse conductivity problem as a minimization problem involving a regularized residual functional. We then transform this minimization problem into a variational problem and establish the equivalence between them. This reformulation enables the employment of the finite element method to reconstruct the shape of the object from finitely many measurements. Notably, the proposed approach allows us to identify the object directly without requiring any iterative procedure. {\it A prior} error estimates are rigorously established to demonstrate the theoretical soundness of the finite element method. Based on these estimates, we provide a criterion for selecting the regularization parameter. Additionally, several numerical examples are presented to verify the feasibility of the proposed approach in shape reconstruction.

翻译：反电导率问题旨在通过边界测量值确定有界域内的未知电导率。在实际应用中，基于最小化受偏微分方程约束的正则化残差泛函的算法已被广泛用于处理该问题。然而，此类方法通常需要重复迭代并在每次迭代中求解正问题，从而导致沉重的计算成本。为解决此问题，我们首先将反电导率问题重新表述为涉及正则化残差泛函的最小化问题。随后，我们将此最小化问题转化为变分问题，并建立两者之间的等价性。这一重新表述使得能够采用有限元方法从有限多次测量中重构物体的形状。值得注意的是，所提出的方法允许我们直接识别物体，而无需任何迭代过程。我们严格建立了先验误差估计，以证明该有限元方法的理论合理性。基于这些估计，我们提供了正则化参数的选择准则。此外，本文给出了若干数值算例，以验证所提方法在形状重构中的可行性。