On the Kohn–Vogelius formulation for solving an inverse source problem

Abstract

An inverse source problem related to the Poisson equation is the main concern of this work. Specifically, we deal with the reconstruction of a mass distribution in a geometrical domain from a partial boundary measurement of the associated potential. The considered problem is motivated by various applications such as the identification of geological anomalies underneath the Earth's surface. The proposed approach is based on the Kohn–Vogelius formulation and the topological derivative method. An explicit second-order sensitivity related to circular shaped anomalies is calculated for different examples of the Kohn–Vogelius type functional. Then, the optimal location and size of the unknown support of the mass distribution are characterized as the solution to a minimization problem. The resulting reconstruction procedure is non-iterative and robust with respect to noisy data. Finally, we produce numerical results from four different examples of the Kohn–Vogelius type functional. The results first demonstrate the method and then compare the robustness of each functional in solving the inverse source problem.

Keywords:

• Inverse source problem
• Kohn–Vogelius formulation
• topological derivative method

2010 Mathematics Subject Classifications:

• 49M15
• 31A25
• 35C20
• 35A15

Acknowledgments

This research was partly supported by CNPq (Brazilian Research Council), CAPES (Brazilian Higher Education Staff Training Agency) and FAPERJ (Research Foundation of the State of Rio de Janeiro). The support is gratefully acknowledged.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was partly supported by CNPq (Brazilian Research Council), CAPES (Brazilian Higher Education Staff Training Agency) and FAPERJ (Research Foundation of the State of Rio de Janeiro).