A method combining Legendre basis reduction in time with a Carleman-estimate-based contraction mapping reconstructs initial data for nonlinear Schrödinger equations from boundary observations, with proven stability under noise.
Carleman estimate for the Schr¨ odinger equation and application to magnetic inverse problems.Journal of Mathematical Analysis and Applications, 474(1):116–142
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Inverse initial data for nonlinear Schr\"odinger equation via Carleman estimates and the contraction principle
A method combining Legendre basis reduction in time with a Carleman-estimate-based contraction mapping reconstructs initial data for nonlinear Schrödinger equations from boundary observations, with proven stability under noise.