Material derivatives of boundary integral operators in electromagnetism and application to inverse scattering problems

Abstract

This paper deals with the material derivative analysis of the boundary integral operators arising from the scattering theory of time-harmonic electromagnetic waves and its application to inverse problems. We present new results using the Piola transform of the boundary parametrisation to transport the integral operators on a fixed reference boundary. The transported integral operators are infinitely differentiable with respect to the parametrisations and simplified expressions of the material derivatives are obtained. Using these results, we extend a nonlinear integral equations approach developed for solving acoustic inverse obstacle scattering problems to electromagnetism. The inverse problem is formulated as a pair of nonlinear and ill-posed integral equations for the unknown boundary representing the boundary condition and the measurements, for which the iteratively regularized GaussNewton method can be applied. The algorithm has the interesting feature that it avoids the numerous numerical solution of boundary value problems at each iteration step. Numerical experiments are presented in the special case of star-shaped obstacles.

A spectrally accurate method for the direct and inverse scattering problems by multiple 3D dielectric obstacles

“…The latter issue can be handled by combining the iterative algorithm to qualitative methods such as topological gradient based method [6] or T-matrix based method [48]. The domain-derivative based formulation in the Gauss-Newton iterations can also be replaced by the material-derivative based formulation [31] that avoids the solution of the forward problem in the nonlinear least square. The analysis and comparison of the resulting hybrid methods for more complicated configurations (multiple-layered media) is the subject of future work.…”

Application of the second domain derivative in inverse electromagnetic scattering

“…Furthermore, in case of acoustic scattering problems several numerical implementations are documented (see [6,11] for three dimensional examples). Presumably due to the computational effort, there are only a few results for the full vector valued electromagnetic inverse scattering problem (see [9,15,21]). These approaches are based on boundary integral equations for the electromagnetic scattering problem and the first domain derivative.…”

Shape optimization of Stokesian peristaltic pumps using boundary integral methods

“…Boundary integral equation (BIE) methods, on the other hand, avoid volume discretization altogether for linear partial differential equations (PDEs) and are highly scalable even for moving geometry problems [22]. While BIE methods have been used widely for shape optimization problems, including in linear elasticity, acoustics, electrostatics, electromagnetics and heat flow (e.g., [17,33,2,31,10]), we are not aware of their application to optimization of peristaltic pumps transporting simple (or complex) fluids.…”