Adaptive Finite Element method for reconstructing dielectric properties of malign melanoma in 3D

Journal article, 2026

The paper presents the performance of an adaptive finite element/finite difference domain decomposition method for reconstructing dielectric permittivity and conductivity functions in a 3D malignant melanoma model using measurements of the backscattered electric field at the boundary of the domain. The inverse problem is formulated as the minimization of a regularized Tikhonov functional, which is solved by constructing the corresponding Lagrangian function. The optimality conditions are obtained from the Fréchet derivative of the Lagrangian. The finite element/finite difference domain decomposition method is formulated, and explicit schemes for solving the forward and adjoint problems are also developed. We present both conjugate gradient and adaptive conjugate gradient reconstruction algorithms to compute a stationary point of the Lagrangian. Numerical experiments are conducted in both homogeneous and non-homogeneous settings with adaptive refinement of the finite element mesh. The computational results demonstrate accurate qualitative and quantitative 3D reconstructions of the weighted dielectric properties of malignant melanoma at 6 GHz.