4,010 research outputs found
Numerical studies of the Lagrangian approach for reconstruction of the conductivity in a waveguide
We consider an inverse problem of reconstructing the conductivity function in
a hyperbolic equation using single space-time domain noisy observations of the
solution on the backscattering boundary of the computational domain. We
formulate our inverse problem as an optimization problem and use Lagrangian
approach to minimize the corresponding Tikhonov functional. We present a
theorem of a local strong convexity of our functional and derive error
estimates between computed and regularized as well as exact solutions of this
functional, correspondingly. In numerical simulations we apply domain
decomposition finite element-finite difference method for minimization of the
Lagrangian. Our computational study shows efficiency of the proposed method in
the reconstruction of the conductivity function in three dimensions
Kirchhoff equations from quasi-analytic to spectral-gap data
In a celebrated paper (Tokyo J. Math. 1984) K. Nishihara proved global
existence for Kirchhoff equations in a special class of initial data which lies
in between analytic functions and Gevrey spaces. This class was defined in
terms of Fourier components with weights satisfying suitable convexity and
integrability conditions.
In this paper we extend this result by removing the convexity constraint, and
by replacing Nishihara's integrability condition with the simpler integrability
condition which appears in the usual characterization of quasi-analytic
functions.
After the convexity assumptions have been removed, the resulting theory
reveals unexpected connections with some recent global existence results for
spectral-gap data.Comment: 15 page
Globally strongly convex cost functional for a coefficient inverse problem
A Carleman Weight Function (CWF) is used to construct a new cost functional
for a Coefficient Inverse Problems for a hyperbolic PDE. Given a bounded set of
an arbitrary size in a certain Sobolev space, one can choose the parameter of
the CWF in such a way that the constructed cost functional will be strongly
convex on that set. Next, convergence of the gradient method, which starts from
an arbitrary point of that set, is established. Since restrictions on the size
of that set are not imposed, then this is the global convergence
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