We consider the reconstruction of a compactly supported source term in the constant-coefficient Helmholtz equation in R 3 , 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.

中文翻译：

---

R 3中Helmholtz方程的反源问题的稳定性

我们考虑通过在源封闭球体处的传出解的测量，在R 3中的常数Helmholtz方程中重建紧支撑源项。以有限数量的频率进行测量。我们明确地描述了可以从这些测量值稳定重建的源的某些有限维空间。表征仅涉及测量频率和问题几何参数。我们推导了测量算子的奇异值分解，并证明了该算子的频谱带宽的下限。通过在源支持上将奇异值分解和Dirichlet–Laplacian的特征值问题联系起来，我们为源重构设计了一种快速稳定的数值方法。