DFG Project: "Direct and inverse scattering problems for elastic waves"
Duration
July 16, 2009  July 15, 2012Researchers:
Project Head:  J. Elschner (Nonlinear Optimization and Inverse Problems) 

Project Researcher:  G. Hu (Nonlinear Optimization and Inverse Problems) 
Project Description:
Direct and inverse scattering problems for elastic waves occur in many applications in natural sciences and engineering. The project is devoted to the investigation of scattering of time harmonic elastic waves by (in general) unbounded surfaces and interfaces in the case of periodic structures (diffraction gratings) as well as in the nonperiodic case (rough surfaces). This leads to direct and inverse boundary value problems for 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 based on variational formulations. In this respect diffractive structures with nonsmooth interfaces and several elastic materials are of particular interest. A second aim of the project is concerned with uniqueness results and reconstruction methods for the inverse problem of determining the scattering object by near and far field measurements of the scattered elastic field. For both tasks, inspiration should be taken from recent results in the case of acoustic and electromagnetic waves.
Collaborators:
 M. Yamamoto, University of Tokyo
 S.N. ChandlerWilde, University of Reading, UK
 B. Zhang, Academy of Mathematics and Systems Science, Chinese Academy of Sciences
Publications:

G. Hu, F. Qu and B. Zhang,
A linear sampling method for inverse problems of diffraction gratings of mixed type,
Mathematical Methods in the Applied Sciences 35 (2012): pp. 10471066.  J. Elschner, G. Hu.
Elastic scattering by unbounded rough surfaces,
WIAS Preprint No. 1677, 2012.  J. Elschner, G. Hu.
An optimization method in inverse elastic scattering for onedimensional grating profiles,
WIAS Preprint No. 1622, 2011. Communications in Computational Physics 12 (2012): pp. 14341460.  G. Hu.
Inverse wave scattering by unbounded obstacles: Uniqueness for the twodimensional Helmholtz equation,
WIAS Preprint No. 1592, 2011. Appl. Anal. 91 (2012): pp. 703717.  J. Elschner, G. Hu.
Uniqueness in inverse scattering of elastic waves by threedimensional polyhedral diffraction gratings,
WIAS Preprint No. 1591, 2011. Inverse Problems and Illposed Problems 19 (2011): pp. 717768.  J. Elschner, G. Hu.
Scattering of plane elastic waves by threedimensional diffraction gratings,
WIAS Preprint No. 1565, 2010. Mathematical Models and Methods in Applied Sciences 22 (2012): pp. 1150019/11150019/34.  J. Elschner, G. Hu.
Inverse scattering of elastic waves by periodic structures: Uniqueness under the third or fourth kind boundary conditions,
WIAS Preprint No. 1528, 2010. Appeared in: Methods and Applications of Analysis 18 pp.215244.  J. Elschner, G. Hu.
Inverse scattering of electromagnetic waves by multilayered structures: Uniqueness in TM mode,
WIAS Preprint No. 1549, 2010. To appear in: Inverse Problems and Imaging 5 (2011) 793813, with the title "Uniqueness in inverse transmission scattering problems for multilayered obstacles".  G. Hu, B. Zhang.
The linear sampling method for the inverse electromagnetic scattering by a partially coated biperiodic structure,
Mathematical Methods in the Applied Sciences, 34 (2011) pp. 509519.  G. Hu, J. Yang, B. Zhang.
An inverse electromagnetic scattering problem for a biperiodic inhomogeneous layer on a perfectly conducting plate,
WIAS Preprint No. 1499, 2010. Appeared in: Appl. Anal., 90 (2011) pp. 317333.  J. Elschner, G. Hu.
Global uniqueness in determining polygonal periodic structures with a minimal number of incident plane waves,
WIAS Preprint No. 1498, 2010. Appeared in: Inverse Problems, 26 (2010) pp. 115002/1115002/23.  J. Elschner, G. Hu.
Variational approach to scattering of plane elastic waves by diffraction gratings,
WIAS Preprint No. 1466, 2009. Appeared in: Math. Meth. Appl. Sci. 33 (2010) pp. 19241941  S. N. ChandlerWilde, J. Elschner.
Variational approach in weighted Sobolev spaces to scattering by unbounded rough surface,
WIAS Preprint No. 1455, 2009. Appeared in: SIAM J. Math. Anal., 42 (2010) pp. 25542580.  J.Elschner, M.Yamamoto.
Uniqueness in inverse elastic scattering with finitely many incident waves. WIAS Preprint No. 1449, 2009. Appeared in:
Inverse Problems, 26 (2010) pp. 045005/1045005/8