Abstract
We derive some bounds which can be viewed as an evidence of increasing stability in the problem of recovery of the potential coefficient in the Schrodinger equation from the Dirichlet-to-Neumann map, when frequency (energy level) is growing. These bounds hold under certain a-priori bounds on the unknown coefficient. Proofs use complex- and real-valued geometrical optics solutions. We outline open problems and possible future developments.
Highlights
We consider the problem of recovery of the potential in the Schrodinger equation from many boundary measurements, i.e. from the Neumann data given for all Dirichlet data
It is clear that difficulties in theory and applications of many important inverse problems are due to their notorious instability with respect to data changes
Logarithmic stability permits as a rule to find only 10-20 Fourier coefficients of unknown coefficients or source terms
Summary
We consider the problem of recovery of the potential in the Schrodinger equation from many boundary measurements, i.e. from the Neumann data given for all Dirichlet data (the Dirichlet-to-Neumann map). In the two-dimensional case this problem is less overdetermined, and the methods of [23] are not applicable, but one enjoys advantages of the methods of inverse scattering and of theory of complex variables Using these methods Nachman [19] and Astala and Paivarinta [4] showed uniqueness of the conductivity coefficient, and using the results of [19], Isakov and Nachman [16] proved uniqueness of c resulting from a transformation of the conductitivy equation with a real-valued coefficient. The logarithmic stability is quite discouraging for applications, since small errors in the data of the inverse problem result in large errors in numerical reconstruction of physical properties of the medium. In the proofs we use complex and real valued geometrical optics solutions and explicit sharp bounds on fundamental solutions of elliptic operators with parameter k
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
