A Recursive Algorithm for MultiFrequency Acoustic Inverse Source Problems

Abstract

An iterative/recursive algorithm is studied for recovering unknown sources of acoustic field with multifrequency measurement data. Under additional regularity assumptions on source functions, the first convergence result toward multifrequency inverse source problems is obtained by assuming the background medium is homogeneous and the measurement data is noise-free. Error estimates are also provided when the observation data is contaminated by noise. Numerical examples verify the reliability and efficiency of our proposed algorithm. Introduction.Identification and recovery of acoustic sources from measurable radiated waves have attracted considerable attention in both forward and inverse scattering theory. Such inverse source problems have wide applications in antenna synthesis [7], biomedical imaging [3], and, in particular, tomographic problems [6,34]. For inverse source and other scattering problems with stochastic phenomena we mention [8,9] and references therein.A theoretical understanding of the inverse source problems starts with the expansion of wave fields generated by a localized source which can be represented in the form of an integral equation, for instance, in [32] and [13] independently. A closer investigation on this integral equation yields the definition of radiating and nonradiating sources; cf. [20,31]. The latter paper collects an infinite number of sources with vanishing radiating fields outside of certain domains. Of course, such results are based on a well-known observation, as we show in subsection 2.2.Such observations further strengthen the theoretical analysis in inverse source problems where nonuniqueness within finite frequency observation is commonly accepted, i.e., [21]. Recent results in [22] prove that unique identification holds by measuring the acoustic field of an infinite and unbounded set of frequencies on a closed boundary. Further refinement is provided in [11], where the unknown source

Fast acoustic source imaging using multi-frequency sparse data

“…These methods can be classified into two categories: iterative methods and non-iterative methods. While very successful in many cases, iterative methods are usually computationally expensive since they require the solution of a direct problem in every step [2,19]. In contrast, the second group of reconstruction methods, i.e., non-iterative methods, avoids this problem, e.g., [10,11,13,14,17,18].…”

Linearized Inverse Schrödinger Potential Problem at a Large Wavenumber

“…The second one is the numerical error of approximated linearized DtN map Λ in (4.4) where the elliptic equation solvers of large wavenumbers and the numerical differentiation of the Neumann boundary data may induce some additional ill-posedness. The error estimate of Algorithm 1 may be carried out by choosing the length and angle sets following the analysis in [4,5]. We intend to report such results in a separate work.…”

Locating Multiple Multipolar Acoustic Sources Using the Direct Sampling Method

“…Over recent years, intensive attention [6,8,9,16,22,23,24,36,37,38,39] has been focused on the inverse source problem of determining a source F in the Helmholtz equation ∆u + k 2 u = F in Ω, (1.1) from boundary measurements u| Γ and ∂ ν u| Γ , where k > 0 is the wavenumber, Ω ⊂ R N (N = 2, 3) is a bounded Lipschitz domain with boundary Γ and ν denotes the outward unit normal to Γ. A main difficulty of the inverse source problem with a single wavenumber is the non-uniqueness of the source due to the existence of nonradiating sources [3,10,11,15,17], and several numerical methods with multi-frequency measurements [8,9,24,42] have been proposed to overcome it for the source with a arXiv:1801.05584v1 [math.AP] 17 Jan 2018 compact support in the L 2 sense. However, fortunately, with a single wavenumber, the uniqueness can be obtained if a priori information on the source is available [18,23].…”