On the Inverse Source Problem with Boundary Data at Many Wave Numbers

Abstract

We review recent results on inverse source problems for the Helmholtz type equations from boundary measurements at multiple wave numbers combined with new results including uniqueness of obstacles. We consider general elliptic differential equations of the second order and arbitrary observation sites. We present some new results and outline basic ideas of their proofs. To show the uniqueness we use the analytic continuation, the Fourier transform with respect to the wave numbers and uniqueness in the lateral Cauchy problem for hyperbolic equations. To derive the increasing stability we utilize sharp bounds of the analytic continuation for higher wave numbers, the Huygens’ principle, and boundary energy estimates in the initial boundary value problems for hyperbolic equations. Some numerical examples, based on a recursive Kaczmarz-Landweber iterative algorithm, shed light on theoretical results.