Numerical solution of the inverse source problem for the Helmholtz equation with multiple frequency data

Abstract

The inverse source problem of the Helmholtz Equation with multiple frequency data is investigated. Three cases are considered: (1) both the magnitude and phase of measurements on the whole boundary (full aperture data) are available; (2) only limited aperture measurements of the field are available; (3) only the magnitude information of the fields on the boundary is available. A continuation method is introduced which can successfully capture both the macro structures and the small scales of the source function. Numerical examples are presented to demonstrate the efficiency of the method.

A numerical study on the stability of a class of Helmholtz problems

“…Our numerical results also show that the stability constants are large only for countable values of k (depend on the geometry of the domain ), and behave like constants for most values of k. Hence, they can be considered independent of k in an average value sense. Therefore, it is not surprising that stable and accurate reconstruction methods can be obtained for the inverse problems with multiple frequency data [7][8][9][10].…”

“…A related topic for the Helmholtz equation is the inverse source problem [8][9][10]. There are two main difficulties associated with the inverse source problem at fixed frequency: non-uniqueness and the ill-posedness.…”

A Recursive Algorithm for MultiFrequency Acoustic Inverse Source Problems

“…Meanwhile, the nonuniqueness source identification allows minimum norm (energy) solutions [19,30,29] where physical constraints are naturally included. To uniquely identify the unknown sources, multifrequency measurements are employed in [1,2,12,22]. In all these works the unknown sources are represented by a linear combination of different basis functions, for instance, the standard Fourier basis functions in [2], finite element basis functions in [12], and eigenfunctions of Dirichlet eigenvalue problems for homogeneous and nonhomogeneous media in [1,22], respectively.…”

“…To uniquely identify the unknown sources, multifrequency measurements are employed in [1,2,12,22]. In all these works the unknown sources are represented by a linear combination of different basis functions, for instance, the standard Fourier basis functions in [2], finite element basis functions in [12], and eigenfunctions of Dirichlet eigenvalue problems for homogeneous and nonhomogeneous media in [1,22], respectively. To numerically reconstruct the unknown sources, one needs to recursively recover the coefficients by measurements taken for different frequencies.…”

“…Since the problem is inherently ill-conditioned some regularization step must be taken. For example, in [2,12] standard Tikhonov regularization and steepest descent methods are considered but without any convergence analysis.…”