Stability of the inverse source problem for the Helmholtz equation in R³

We consider the reconstruction of a compactly supported source term in the constant coefficient Helmholtz equation in R³, from the measurement of the outgoing solution at a source-enclosing sphere. The measurement is taken at a finite number of frequencies. We explicitly characterize certain finite-dimensional spaces of sources that can be stably reconstructed from such measurements. The characterization involves only the measurement frequencies and the problem geometry parameters. We derive a singular value decomposition of the measurement operator, and prove a lower bound for the spectral bandwidth of this operator. By relating the singular value decomposition and the eigenvalue problem for the Dirichlet-Laplacian on the source support, we devise a fast and stable numerical method for the source reconstruction. We do numerical experiments to validate the stability and efficiency of the numerical method.