Elliptical support with minimal boundary data
Agah D. Garnadi

Ronin Mathematician

Abstract

We consider the problem of determining the interface separating regions of constant density within a body, given only boundary measurements of the corresponding potential equation [1]. This inverse problem, which arises in gravimetry, aims to find an internal domain, D, within a reference domain,-\Omega-, based on external boundary measurements of the gravitational force. Isakov & Titi [2] showed that, in practical situations with noisy data, five parameters of the unknown domain D can be stably determined. An ellipse, for example, can be uniquely identified using five parameters. They proved the uniqueness and stability of recovering an ellipse in this inverse problem using minimal data at just three boundary points of potential measurement at the boundary.

For simplicity, we will addresses the problem in two dimensions (the plane) as a model [3]. We will present numerical examples with point measurements taken on the boundary of -\Omega-- (-\partial \Omega-).
References

[1] Ring, W., 1995. Identification of a core from boundary data. SIAM Journal on Applied Mathematics, 55(3), pp.677-706.
[2] Isakov, V. and Titi, A., 2022. On the inverse gravimetry problem with minimal data. Journal of Inverse and Ill-posed Problems, 30(6), pp.807-822.
[3] Agah D. Garnadi, A Lepskij-type stopping rule for simplified iteratively regularized Gauss-Newton method, The 6th SEAMS-UGM Conference 2011, p.317--322.

Keywords: Inverse Problem, Support Identification

Topic: Functional Analysis