Reconstruction of shapes and refractive indices from backscattering experimental data using the adaptivity

Abstract

We consider the inverse problem of the reconstruction of the spatially\ndistributed dielectric constant $\\varepsilon_{r}\\left(\\mathbf{x}\\right), \\\n\\mathbf{x}\\in \\mathbb{R}^{3}$, which is an unknown coefficient in the Maxwell's\nequations, from time-dependent backscattering experimental radar data\nassociated with a single source of electric pulses. The refractive index is\n$n\\left(\\mathbf{x}\\right) =\\sqrt{\\varepsilon_{r}\\left(\\mathbf{x}\\right)}.$ The\ncoefficient $\\varepsilon_{r}\\left(\\mathbf{x}\\right) $ is reconstructed using a\ntwo-stage reconstruction procedure. In the first stage an approximately\nglobally convergent method proposed is applied to get a good first\napproximation of the exact solution. In the second stage a locally convergent\nadaptive finite element method is applied, taking the solution of the first\nstage as the starting point of the minimization of the Tikhonov functional.\nThis functional is minimized on a sequence of locally refined meshes. It is\nshown here that all three components of interest of targets can be\nsimultaneously accurately imaged: refractive indices, shapes and locations.\n