Homotopy, polynomial equations, Gross boundary data, and small Helmholtz systems (Q1913770)

Inverse problems of the boundary measurement type appear in several geophysical contexts including DC resistivity, electromagnetic induction and ground water flow. This paper considers the inverse problem of recovering a spatially varying coefficient in Helmholtz and modified Helmholtz equations from possibly imprecise measurements made at the boundary. It is shown that a two-dimensional discrete Helmholtz inverse problem with enough boundary measurements reduces to a well-determined system of polynomial equations. A homotopy procedure decides whether real positive solutions exist and, if so, generates the entire solution list. The procedure applies to the solution of large scale problems but is tenable computationally at present only for small inverse problems. Numerical results are presented to confirm the theory for both perfect and noisy data.