We present a time-domain inverse scattering approach using functional analysis and variational method

which is applied to reconstruct dispersive dielectric properties of stratified lossy media.This approach firstly formulates a cost functional

according to least squares criterion

to turn the inverse problem into a constrained minimization problem.The resulting constrained minimization problem is then transformed into an unconstrained minimization problem by a penalty function technique.And then

the closed Fréchet derivatives of the Lagrange function with respect to the properties are derived based on the calculus of variations.Thus

the unconstrained minimization problem can be solved by using any gradient-based algorithm and the finite-difference time-domain (FDTD) method.Also

the first-order Tikhonov's regularization is adopted to cope with noise and the ill-posedness of the problem.In numerical examples

the presented algorithm is applied to a one-dimensional (1-D) breast model in several cases with the help of a set of nonlinear conjugate gradient (CG) methods

and the simulated results demonstrate its robustness and validity in the lesions identification.