The factorization method for recovering cavities in a heat conductor

Abstract

In this paper, we develop a factorization method to reconstruct cavities in a heat conductor by knowing the Neumann-to-Dirichlet map at the boundary of this conductor. The factorization method is a very well known reconstruction method for inverse problems described by selfadjoint elliptic equations. It enables us to reconstruct the boundaries of unknown targets such as cavities and inclusions. This method was successfully applied for examples to the inverse scattering problem by obstacles for the acoustic wave and the electrical impedance tomography problem. The forward model equations for these problems are the Helmholtz equation and the conductivity equation which are both self-adjoint equations. On the other hand, the heat equation is a typical non-selfadjoint equation. This paper is giving the first attempt to study the factorization method for inverse problems whose forward model equations are non-selfadjoint. We emphasize that an auxiliary operator introduced in this paper is the key which makes this attempt successful. This operator connects the back projection operators for the forward and backward heat equations.

Regularization of the Factorization Method with Applications to Inverse Scattering

“…The factorization method (see for e.g. [6,11,12,18,19,21,23,25]) is a method used to solve inverse shape problems and fall under the category of qualitative methods. Qualitative methods are otherwise refereed to as non-iterative or direct methods.…”

Regularization of the factorization method applied to diffuse optical tomography

“…Non-destructive testing plays an important role in many medical and engineering applications. In general, the factorization method and similar qualitative methods such as the direct sampling method [9,18,27] can be used to derive analytically rigorous and computational simple methods for solving inverse shape problems coming from elliptic [13], parabolic [14] and hyperbolic [7] partial differential equations. One of the main advantages of using qualitative methods over a non-linear optimization techniques is the fact that in general qualitative methods require little a priori information about the region of interest.…”

Direct sampling methods for isotropic and anisotropic scatterers with point source measurements

“…The factorization method was initially introduced in [15] for far-field data but has been extended to many other models, see for e.g. [2,6,17]. Also we derive explicit decay rates for our imaging functionals by two ways.…”