A Hölder-logarithmic stability estimate for an inverse problem in two dimensions
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Matteo Santacesaria
Abstract
The problem of the recovery of a real-valued potential in the two-dimensional Schrödinger equation at positive energy from the Dirichlet-to-Neumann map is considered. It is know that this problem is severely ill-posed and the reconstruction of the potential is only logarithmic stable in general. In this paper a new stability estimate is proved, which is explicitly dependent on the regularity of the potentials and on the energy. Its main feature is an efficient increasing stability phenomenon at sufficiently high energies: in some sense, the stability rapidly changes from logarithmic type to Hölder type. The paper develops also several estimates for a non-local Riemann–Hilbert problem which could be of independent interest.
© 2015 by De Gruyter
Articles in the same Issue
- Frontmatter
- Inverse problems for linear degenerate parabolic equations by “time-like” Carleman estimate
- Spectral problems and scattering on noncompact star-shaped graphs containing finite rays
- Stability estimates for Burgers-type equations backward in time
- A Hölder-logarithmic stability estimate for an inverse problem in two dimensions
- On convergence rates for iteratively regularized Newton-type methods under a Lipschitz-type nonlinearity condition
- Identification of an unknown spatial load distribution in a vibrating cantilevered beam from final overdetermination
Articles in the same Issue
- Frontmatter
- Inverse problems for linear degenerate parabolic equations by “time-like” Carleman estimate
- Spectral problems and scattering on noncompact star-shaped graphs containing finite rays
- Stability estimates for Burgers-type equations backward in time
- A Hölder-logarithmic stability estimate for an inverse problem in two dimensions
- On convergence rates for iteratively regularized Newton-type methods under a Lipschitz-type nonlinearity condition
- Identification of an unknown spatial load distribution in a vibrating cantilevered beam from final overdetermination
