143 research outputs found
New Stability Estimates for the Inverse Medium Problem with Internal Data
A major problem in solving multi-waves inverse problems is the presence of
critical points where the collected data completely vanishes. The set of these
critical points depend on the choice of the boundary conditions, and can be
directly determined from the data itself. To our knowledge, in the most
existing stability results, the boundary conditions are assumed to be close to
a set of CGO solutions where the critical points can be avoided. We establish
in the present work new weighted stability estimates for an electro-acoustic
inverse problem without assumptions on the presence of critical points. These
results show that the Lipschitz stability far from the critical points
deteriorates near these points to a logarithmic stability
Stability estimates for the fault inverse problem
We study in this paper stability estimates for the fault inverse problem. In
this problem, faults are assumed to be planar open surfaces in a half space
elastic medium with known Lam\'e coefficients. A traction free condition is
imposed on the boundary of the half space. Displacement fields present jumps
across faults, called slips, while traction derivatives are continuous. It was
proved in \cite{volkov2017reconstruction} that if the displacement field is
known on an open set on the boundary of the half space, then the fault and the
slip are uniquely determined. In this present paper, we study the stability of
this uniqueness result with regard to the coefficients of the equation of the
plane containing the fault. If the slip field is known we state and prove a
Lipschitz stability result. In the more interesting case where the slip field
is unknown, we state and prove another Lipschitz stability result under the
additional assumption, which is still physically relevant, that the slip field
is one directional
H\"older Stability for an Inverse Medium Problem with Internal Data
We are interested in an inverse medium problem with internal data. This
problem is originated from multi-waves imaging. We aim in the present work to
study the well-posedness of the inversion in terms of the boundary conditions.
We precisely show that we have actually a stability estimate of H\"older type.
For sake of simplicity, we limited our study to the class of Helmholtz
equations +V with bounded potential V
Coefficient identification in parabolic equations with final data
In this work we determine the second-order coefficient in a parabolic
equation from the knowledge of a single final data. Under assumptions on the
concentration of eigenvalues of the associated elliptic operator, and the
initial state, we show the uniqueness of solution, and we derive a Lipschitz
stability estimate for the inversion when the final time is large enough. The
Lipschitz stability constant grows exponentially with respect to the final
time, which makes the inversion ill-posed. The proof of the stability estimate
is based on a spectral decomposition of the solution to the parabolic equation
in terms of the eigenfunctions of the associated elliptic operator, and an ad
hoc method to solve a nonlinear stationary transport equation that is itself of
interest
Small Perturbations of an Interface for Elastostatic Problems
We consider solutions to the Lam\'e system in two dimensions. By using
systematic way, based on layer potential techniques and the field expansion
(FE) method (formal derivation), we establish a rigorous asymptotic expansion
for the perturbations of the displacement field caused by small perturbations
of the shape of an elastic inclusion with C2-boundary. We extend these
techniques to determine a relationship between traction-displacement
measurements and the shape of the object and derive an asymptotic expansion for
the perturbation in the elastic moments tensors (EMTs) due to the presence of
small changes in the interface of the inclusion.Comment: 42pages,0 figures. arXiv admin note: text overlap with
arXiv:1601.0677
Stability for quantitative photoacoustic tomography revisited
This paper is concerned with the stability issue in determining absorption
and diffusion coefficients in quantitative photoacoustic imaging. Assuming that
the optical wave is generated by point sources in a region where the optical
coefficients are known, we derive pointwise H{\"o}lder stability estimate of
the inversion. This result shows that the reconstruction of the optical
coefficients is stable in the region close to the optical illumination sources
and deteriorate exponentially far away. Our stability estimate is therefore in
accordance with known experimental observations. Mathematics subject
classification : 35R30
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