209 research outputs found
Generalized Impedance Boundary Conditions for Strongly Absorbing Obstacles: the full Wave Equations
This paper is devoted to the study of the generalized impedance boundary
conditions (GIBCs) for a strongly absorbing obstacle in the {\bf time} regime
in two and three dimensions. The GIBCs in the time domain are heuristically
derived from the corresponding conditions in the time harmonic regime. The
latters are frequency dependent except the one of order 0; hence the formers
are non-local in time in general. The error estimates in the time regime can be
derived from the ones in the time harmonic regime when the frequency dependence
is well-controlled. This idea is originally due to Nguyen and Vogelius in
\cite{NguyenVogelius2} for the cloaking context. In this paper, we present the
analysis to the GIBCs of orders 0 and 1. To implement the ideas in
\cite{NguyenVogelius2}, we revise and extend the work of Haddar, Joly, and
Nguyen in \cite{HJNg1}, where the GIBCs were investigated for a fixed frequency
in three dimensions. Even though we heavily follow the strategy in
\cite{NguyenVogelius2}, our analysis on the stability contains new ingredients
and ideas. First, instead of considering the difference between solutions of
the exact model and the approximate model, we consider the difference between
their derivatives in time. This simple idea helps us to avoid the machinery
used in \cite{NguyenVogelius2} concerning the integrability with respect to
frequency in the low frequency regime. Second, in the high frequency regime,
the Morawetz multiplier technique used in \cite{NguyenVogelius2} does not fit
directly in our setting. Our proof makes use of a result by H\"ormander in
\cite{Hor}. Another important part of the analysis in this paper is the
well-posedness in the time domain for the approximate problems imposed with
GIBCs on the boundary of the obstacle, which are non-local in time
A dissipative time reversal technique for photo-acoustic tomography in a cavity
We consider the inverse source problem arising in thermo- and photo-acoustic
tomography. It consists in reconstructing the initial pressure from the
boundary measurements of the acoustic wave. Our goal is to extend versatile
time reversal techniques to the case of perfectly reflecting boundary of the
domain. Standard time reversal works only if the solution of the direct problem
decays in time, which does not happen in the setup we consider. We thus propose
a novel time reversal technique with a non-standard boundary condition. The
error induced by this time reversal technique satisfies the wave equation with
a dissipative boundary condition and, therefore, decays in time. For larger
measurement times, this method yields a close approximation; for smaller times,
the first approximation can be iteratively refined, resulting in a convergent
Neumann series for the approximation
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