DFG Individual Project: "Direct and inverse interaction problems with unbounded interfaces between acoustic, electromagnetic and elastic waves"


July 16, 2012 - July 15, 2014

Project Head:

G. Hu (Nonlinear Optimization and Inverse Problems)

Project Description:

Direct and inverse interaction problems between acoustic, electromagnetic and elastic waves occur in many applications in natural sciences and engineering. The project is devoted to the investigation of scattering of time harmonic acoustic and electromagnetic waves by an unbounded elastic body in the case of periodic structures (diffraction gratings) as well as in the non-periodic case (rough surfaces). This leads to direct and inverse transmission problems between the Helmholtz (or Maxwell) equations and the Navier equation in unbounded domains, the analytical and numerical treatment of which is challenging. One objective of the project is to develop a new solvability theory (existence and uniqueness of solutions, Fredholm property) for the direct scattering problems using variational methods. In the more general and difficult case of rough interfaces, this requires the derivation of novel a priori estimates in weighted Sobolev spaces. The second goal of the project is the development and theoretical justification of efficient numerical methods for the solution of the direct and inverse interaction problems. The approximate solution of the direct problems will be based on finite element and boundary element methods, whereas for the solution of the inverse problem of reconstructing the interface from near and far field measurements of the scattered acoustic or electromagnetic field, optimization and factorization methods will be used. For both tasks, inspiration should be taken from recent results on electromagnetic and elastic diffraction gratings and rough surfaces and on interaction problems with bounded elastic obstacles.