The aim of this article is to present a hybrid finite element/finite difference method which is used for reconstructions of electromagnetic properties within a realistic breast phantom. This is done by studying the mentioned properties' (electric permittivity and conductivity in this case) representing coefficients in a constellation of Maxwell's equations. This information is valuable since these coefficient can reveal types of tissues within the breast, and in applications could be used to detect shapes and locations of tumours. Because of the ill-posed nature of this coefficient inverse problem, we approach it as an optimization problem by introducing the corresponding Tikhonov functional and in turn Lagrangian. These are then minimized by utilizing an interplay between finite element and finite difference methods for solutions of direct and adjoint problems, and thereafter by applying a conjugate gradient method to an adaptively refined mesh.

翻译：本文旨在提出一种混合有限元/有限差分方法，用于在真实乳腺体模内重建电磁特性。这是通过研究表征麦克斯韦方程组中系数的相关特性（此处为介电常数和电导率）来实现的。这些信息具有重要价值，因为此类系数能够揭示乳腺内的组织类型，并可在实际应用中用于检测肿瘤的形状与位置。鉴于该系数反问题的不适定性，我们将其视为优化问题进行处理，引入了相应的吉洪诺夫泛函及对应的拉格朗日函数。随后通过有限元法与有限差分法在正问题与伴随问题求解中的协同作用实现泛函最小化，并在自适应加密网格上应用共轭梯度法完成优化过程。