O método das soluções fundamentais aplicado à reconstrução de fontes concentradas para problemas elípticos

Abstract

The inverse problem studied in this work is to reconstruct a concentrated source written by a finite linear combination of pointwise Dirac sources, based on information observed at the boundary of the domain. As an example of applications, we can point out: identi cation of hypocenters and epicenters of earthquakes, knowing a priori their effects on the Earth's surface; detection of monopoles and dipoles in magnetoencephalography and electroencephalography, aiding in the diagnosis of brain disorders such as tumors or stroke, for example. In this work, the inverse source problem associated with elliptical operators, such as the Laplace or Helmholtz operator, is solved through an optimization problem. In particular, the inverse source problem is reformulated as a minimization problem of a functional with respect to a set of admissible sources. The Method of Fundamental Solutions (MFS) is used to solve the direct auxiliary problems arising from the reformulation of the inverse problem, in view of all the advantages of this meshfree numerical method, as compared to domain discretization techniques, such as the Finite Di erence Method (MDF) and the Finite Element Method (MEF), for example. In addition, the MFS is used to represent the pointwise Dirac sources that make up the concentrated source by a single point, eliminating the noise that is characteristic when using discretization of the domain in the reconstruction algorithm for the representation of the source. With the numerical results obtained, it is possible to prove the e ciency, effectiveness and robustness of the proposed reconstruction algorithm, even considering noisy data.