Regularized Inversion of the Two-Dimensional Born Operator for Potential Field Reconstruction

• Ansh Desai, Applied Mathematics, University of Delaware

• Peter Monk, Department of Mathematical Sciences, University of Delaware

Time harmonic inverse scattering concerns the ill-posed, nonlinear problem of determining the potential field, or properties, of an inaccessible, penetrable scatterer based on distant measurements of scattered electromagnetic or acoustic waves from the medium. In electromagnetic applications, the potential field represents the relative permittivity of the medium, whereas for acoustic applications, it denotes the square of the refractive index.

Knowledge of the potential field is very useful for medical, seismic, and radar imaging. In model cases involving weak scatterers, the far field pattern—i.e., the asymptotic behavior of the scattered field far from the scatterer’s boundary—can be well approximated by the Born integral operator, which acts as a linearization of the true far field.

In order to test the Born approximation we make the assumption that a two-dimensional scatterer’s approximate location is known a priori. To generate far field data measurements, we chose various scattering shapes and employed a finite element method with a radial Perfectly Matched Layer to truncate the computational domain and compute accurate far field data for various incident fields. This data is the input data for the inversion scheme.

To reconstruct the potential field, we utilized Tikhonov regularization to apply the inverted Born operator to the far field dataset. This approach yielded accurate approximations of the potential fields–particularly for simple, weak scatterers. When the weak scattering approximation does not hold, the quality of the reconstruction is degraded, and we plan to develop a neural network corrector for the Born approximation as a future project.