The factorization method (FM) is an attractive qualitative inverse scattering technique for the detection of geometrical features of unknown objects. This method depends on the selection of regularization parameters slightingly and has low calculation necessities. The aim of this work is to present a near-field FM for inverse source problems that have many applications. A modified test equation is obtained by converting the far-field term to Hankel's function. A different method has been proposed by manipulating the asymptotic approximation of Hankel's function in order to obtain near-field equations with incident angle and distance parameters. The novelty of this study is an integral equation based on the FM, which consists of multifrequency sparse near-field electric field measurements. We proved that the solution of the proposed integral equation gives information about the location of scatterers. The proposed algorithm is validated with simulation results and the capabilities of the presented method are assessed with several frequency regions and sources. Additionally, the presented method is compared with the direct sampling method in order to understand the performance of the proposed approach over a given scenario. The developed FM provides accurate results for electromagnetic source problems.

In this paper, an inverse source problem for the time-fractional diffusion equation is investigated. The observational data is on the final time and the source term is assumed to be temporally independent and with a sparse structure. Here the sparsity is understood with respect to the pixel basis, i.e., the source has a small support. By an elastic-net regularization method, this inverse source problem is formulated into an optimization problem and a semismooth Newton (SSN) algorithm is developed to solve it. A discretization strategy is applied in the numerical realization. Several one and two dimensional numerical examples illustrate the efficiency of the proposed method.

Knowledge of helium diffusion kinetics is critical for materials in which helium measurements are made, particulary for thermochronology. In most cases the helium ages were younger than expected, an observation attributes to diffusive loss of helium and the ejection of high energy alpha particles. Therefore it is important to accurately calculate the distribution of the source term within a sample. In this paper, the prediction of the helium concentrations as function of a spatially variable source term are considered. Both the forward and inverse solutions are presented. Under the assumption of radially symmetric geometry, an analytical solution is deduced based on the eigenfunction expansion. Two regularization methods, the Tikhonov regularization and the spectral cutoff regularization, are considered to obtain the regularized solution. Error estimates with optimal convergence order are shown between the exact solution and the regularized solution. Numerical examples are presented to illustrate the validity and effectiveness of the proposed methods