Reconstruction Algorithms of an Inverse Coefficient Identification Problem for the Schrödinger Equation

Abstract

In this paper, we consider an inverse problem of coefficient identification for the Schr¨odinger equation from the observation data on the exterior boundary. Our aim is to detect the number, the location, the size and the shape of the coefficient with piecewise constant within a body. This problem is nonlinear and ill-posed, thus we should apply stable and elegant reconstruction algorithms in order to improve the corresponding approximation. We give several examples to show the viability of our proposed methods.

Keywords:

• Schrödinger equation
• coefficient identification
• Levenberg-Marquardt algorithm
• Trust-Region-Reflective optimization algorithm
• ill-posed problem

AMS Subject Classification:

• 65N20
• 65N21

Additional information

Funding

The research of Ji-Chuan Liu was supported by the Fundamental Research Funds for the Central Universities (2014QNA57) and the NSF of China (11601512).