The author extends earlier results on the application of the linear sampling method to inverse Dirichlet problems for harmonic vector fields to the case of Neumann boundary conditions. This problem can be interpreted as determining the shape \(D\) of a superconductor from the total magnetostatic fields of dipoles located at a surface surrounding \(D\).
Linear sampling methods are a recent approach to overcome the nonlinearity of inverse problems: a parameter point \(z\) is introduced and a linear operator equation is solved to determine whether \(z\) belongs to \(D\). For the given problem the author analyses two integral operators over the smooth surface surrounding \(D\). These operators are compact, self-adjoint and positive semi-definite, which follows from classical results on boundary integral operators for the Laplace equation and new factorization results. Finally it is proved that operator equations with their square roots and right-hand sides depending on \(z\) are solvable in \(L_2\) if and only if \(z\) belongs to \(D\).