Uniqueness of an inverse source problem in experimental aeroacoustics

Abstract

This paper is concerned with the mathematical analysis of experimental methods for the estimation of the power of an uncorrelated, extended aeroacoustic source from measurements of correlations of pressure fluctuations. We formulate a continuous, infinite dimensional model describing these experimental techniques based on the convected Helmholtz equation in R 3 or R 2 . As a main result we prove that an unknown, compactly supported source power function is uniquely determined by idealized, noise-free correlation measurements. Our framework further allows for a precise characterization of state-of-the-art source reconstruction methods and their interrelations.

Theoretical analysis of the DAMAS algorithm and efficient implementation of the covariance matrix fitting method for large-scale problems

“…A similar equivalence was proven in [21], where the authors show that, in an ideal setting where the spatial covariance matrix is perfectly known in an open, continuous domain, the infinite dimensional equations to be solved to match the unknown source power distribution to the beamforming map, or to the spatial covariance matrix, are equivalent. This result does not apply in practice, where measurements are noisy and discrete, covariances are estimated, and the problems are frequently illconditioned.…”

A model reduction approach for inverse problems with operator valued data

“…Problems of this kind arise in a variety of applications, e.g. in fluorescence tomography [1,30], inverse scattering [6,13], or source identification [17], but also as linearizations of related nonlinear inverse problems, see e.g., [8,34] or [23] and the references given there. In such applications, U typically models the propagation of excitation fields generated by the sources, D describes the interaction with the medium to be probed, and V models the emitted fields which can be recorded by the detectors.…”

Analysis of an Inverse Source Problem with Correlation Data in Experimental Aeroacoustics

“…Proof. The proof boils down to an application of Proposition 2.1 and the asymptotic behavior of the no-flow Green's function, see [HRS20].…”

“…In particular this includes a formulation of the forward operator on infinite dimensional vector spaces. Essentially we treat the mathematical framework that was presented in [HRS20] in some more detail and generality. The main result consists of a uniqueness theorem which is valid for both free-field and waveguide geometries.…”

“…This section presents a derivation of the forward operator of the inverse source problem. Some parts are taken from [HRS20], where the forward operator was derived for a free field geometry. As in the last chapter we denote by D ⊂ R d with d ∈ {2, 3} the open domain of definition (or propagation domain) and we will restrict the analysis to the geometries discussed so far.…”