Presents the latest groundbreaking theoretical foundation to shape optimization in a form accessible to mathematicians, scientists and engineers. Shapes and Geometries: Metrics, Analysis.-Google Books with shapes and geometries in problems where the geometry is. shape optimization and have been steeped in fields of nonlinear metric spaces of domains that are the images of. the differential calculus in vector spaces. Chapter 7 is a metrics,
The nonlinearized reconstruction of the cross-sectional contour
of a homogeneous, possibly multiply connected obstacle buried in
a half-space from time-harmonic wave field data collected above
this half-space in both transverse magnetic (TM) and transverse
electric (TE) polarization cases is investigated. The
reconstruction is performed via controlled evolution of a level
set that was pioneered by Litman et al (Litman A, Lesselier D
and Santosa F 1998 Inverse Problems 14 685-706) but at this time
restricted to free space for TM data collected all around the
sought obstacle. The main novelty of the investigation lies in
the following points: from the rigorous contrast-source domain
integral formulation (TM) and integral-differential formulation
(TE) of the direct scattering problems in the buried obstacle
configuration, and from appropriately cast adjoint scattering
problems, we demonstrate, by processing min-max formulations of
an objective functional J made of the data error to be
minimized, that its derivatives with respect to the evolution
time t are given in closed form. They are contour integrals
involving the normal component of the velocity field of
evolution times the product of direct and adjoint fields (TM),
or of the scalar product of gradients of such fields (TE) at
t. This approach only calls for the analysis of the well posed
direct and adjoint scattering problems formulated from the TM
and TE Green systems of the unperturbed layered environment and,
unusually, it avoids the differentiation of state fields.
Other contributions of the paper come from exhibiting and
analysing via comprehensive numerical experimentation how and
under which conditions evolutions of level sets involving
velocities opposite to shape gradients perform in demanding
configurations including two disjoint obstacles, constitutive
materials strongly less or more refractive than the embedding
material, aspect-limited and noisy monochromatic data, in the
severe TE case as well as in the more menial TM case. A
comparison with a binary-specialized modified-gradient solution
method is also led for several, more and more lossy embedding
half-spaces. Rules of thumb for effectiveness of the inversions
and pending theoretical and computational questions are outlined
in conclusion.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.