Calderón's inverse problem with a finite number of measurements II: independent data

ABSTRACT

We prove a local Lipschitz stability estimate for Gel'fand-Calderón's inverse problem for the Schrödinger equation. The main novelty is that only a finite number of boundary input data is available, and those are independent of the unknown potential, provided it belongs to a known finite-dimensional subspace of $L^{\mathrm{\infty }}$. A similar result for Calderón's problem is obtained as a corollary. This improves upon two previous results of the authors on several aspects, namely the number of measurements and the stability with respect to mismodeling errors. A new iterative reconstruction scheme based on the stability result is also presented, for which we prove exponential convergence in the number of iterations and stability with respect to noise in the data and to mismodeling errors.

Keywords:

• Gel'fand-Calderón problem
• Calderón problem
• inverse conductivity problem
• electrical impedance tomography
• local uniqueness
• complex geometrical optics solutions
• Lipschitz stability
• reconstruction algorithm

2010 Mathematics Subject Classification:

• 35R30

Acknowledgments

This work has been carried out at the Machine Learning Genoa (MaLGa) center, Università di Genova (IT). The authors are members of the ‘Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni’ (GNAMPA), of the ‘Istituto Nazionale per l'Alta Matematica’ (INdAM). GSA is supported by a UniGe starting grant ‘curiosity driven’.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

GSA is supported by a UniGe starting grant “curiosity driven”.