2-D inverse problems for the Laplacian: A meromorphic approximation approach

Abstract

We consider a classical inverse problem: detecting an insulating crack inside a homogeneous 2-D conductor, using overdetermined boundary data. Our method involves meromorphically approximating the complexified solution to the underlying Dirichlet–Neumann problem on the outer boundary of the conductor, and relating the singularities of the approximant (i.e., its poles) to the singular set of the approximated function (i.e., the crack). This approach was introduced in [L. Baratchart, J. Leblond, F. Mandréa, E.B. Saff, How can the meromorphic approximation help to solve some 2-D inverse problems for the Laplacian?, Inverse Problems 15 (1999) 79–90] when the crack is a real segment embedded in the unit disk. Here we show, more generally, that the best L 2 and L ∞ meromorphic approximants to the complexified solution on the outer boundary of the conductor have poles that accumulate on the hyperbolic geodesic arc linking the endpoints of the crack if the latter is analytic and “not too far” from a geodesic. The extension of the method to the case where the crack is piecewise analytic is briefly discussed. We provide numerical examples to illustrate the technique; as the computational cost is low, the results may be used to initialize a heavier local search. The bottom line of the approach is to regard the problem of “optimally” discretizing a potential using finitely many point masses as a regularization scheme for the underlying inverse potential problem. This point of view may be valuable in higher dimension as well.