Stability for inverse random source problems in electromagnetic and biharmonic waves

Abstract

This paper is concerned with inverse random source problems for electromagnetic and biharmonic wave equations. The driven sources are assumed to be generalized microlocally isotropic Gaussian random fields such that the covariances are classical pseudo-differential operators. Uniqueness and stability are established for both inverse random source problems. The stability estimates consist of a Lipschitz type data discrepancy and a logarithmic stability, which decreases as the upper bound of wavenumbers increases. These increasing stability results reveal that ill-posedness can be overcome by using multi-wavenumber data. The analysis is based on integral equations and analytical continuation, which only requires multi-frequency Dirichlet data on the boundary in a finite interval and removes the limitation of data to be collected at all high wavenumbers. For the first time, the stability is established on the inverse source problems for both the Maxwell and biharmonic equations by Dirichlet boundary measurements.