159 research outputs found
Inverse obstacle problem for the non-stationary wave equation with an unknown background
We consider boundary measurements for the wave equation on a bounded domain
or on a compact Riemannian surface, and introduce a method to
locate a discontinuity in the wave speed. Assuming that the wave speed consist
of an inclusion in a known smooth background, the method can determine the
distance from any boundary point to the inclusion. In the case of a known
constant background wave speed, the method reconstructs a set contained in the
convex hull of the inclusion and containing the inclusion. Even if the
background wave speed is unknown, the method can reconstruct the distance from
each boundary point to the inclusion assuming that the Riemannian metric tensor
determined by the wave speed gives simple geometry in . The method is based
on reconstruction of volumes of domains of influence by solving a sequence of
linear equations. For \tau \in C(\p M) the domain of influence is
the set of those points on the manifold from which the distance to some
boundary point is less than .Comment: 4 figure
Recovery of zeroth order coefficients in non-linear wave equations
This paper is concerned with the resolution of an inverse problem related to
the recovery of a scalar (potential) function from the source to solution
map, of the semi-linear equation on a globally hyperbolic
Lorentzian manifold . We first study the simpler model problem where the
geometry is the Minkowski space and prove the uniqueness of through the use
of geometric optics and a three-fold wave interaction arising from the cubic
non-linearity. Subsequently, the result is generalized to globally hyperbolic
Lorentzian manifolds by using Gaussian beams
Unique continuation for the Helmholtz equation using stabilized finite element methods
In this work we consider the computational approximation of a unique
continuation problem for the Helmholtz equation using a stabilized finite
element method. First conditional stability estimates are derived for which,
under a convexity assumption on the geometry, the constants grow at most
linearly in the wave number. Then these estimates are used to obtain error
bounds for the finite element method that are explicit with respect to the wave
number. Some numerical illustrations are given.Comment: corrected typos; included suggestions from reviewer
A stabilized finite element method for inverse problems subject to the convection-diffusion equation. I: diffusion-dominated regime
The numerical approximation of an inverse problem subject to the
convection--diffusion equation when diffusion dominates is studied. We derive
Carleman estimates that are on a form suitable for use in numerical analysis
and with explicit dependence on the P\'eclet number. A stabilized finite
element method is then proposed and analysed. An upper bound on the condition
number is first derived. Combining the stability estimates on the continuous
problem with the numerical stability of the method, we then obtain error
estimates in local - or -norms that are optimal with respect to the
approximation order, the problem's stability and perturbations in data. The
convergence order is the same for both norms, but the -estimate requires
an additional divergence assumption for the convective field. The theory is
illustrated in some computational examples.Comment: 21 pages, 6 figures; in v2 we added two remarks and an appendix on
psiDOs, and made some minor correction
A finite element data assimilation method for the wave equation
We design a primal-dual stabilized finite element method for the numerical
approximation of a data assimilation problem subject to the acoustic wave
equation. For the forward problem, piecewise affine, continuous, finite element
functions are used for the approximation in space and backward differentiation
is used in time. Stabilizing terms are added on the discrete level. The design
of these terms is driven by numerical stability and the stability of the
continuous problem, with the objective of minimizing the computational error.
Error estimates are then derived that are optimal with respect to the
approximation properties of the numerical scheme and the stability properties
of the continuous problem. The effects of discretizing the (smooth) domain
boundary and other perturbations in data are included in the analysis.Comment: 23 page
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