The monotonicity method for the inverse crack scattering problem

Abstract

The monotonicity method for the inverse acoustic scattering problem is to understand the inclusion relation between an unknown object and artificial one by comparing the far field operator with artificial operator. This paper introduces the development of this method to the inverse crack scattering problem. Our aim is to give the following two indicators: One (Theorem 1.1) is to determine whether an artificial small arc is contained in the unknown arc. The other one (Theorem 1.2) is whether an artificial large domain contain the unknown one. Finally, numerical examples based on Theorem 1.1 are given.

Monotonicity in inverse obstacle scattering on unbounded domains

“…The analysis in [26] has been extended to inverse coefficient problems for the Helmholtz equation on bounded domains in [24,25], and in [16] the approach has been generalized to the inverse medium scattering problem on unbounded domains with plane wave incident fields and far field observations of scattered waves. An application of the monotonicity method to an inverse crack detection problem for the Helmholtz equation has recently been considered in [10]. For further recent contributions on monotonicity based reconstruction methods for various inverse problems for partial differential equations we refer to [1,2,20,21,32,36,38].…”

Monotonicity Principle for Tomography in Nonlinear Conducting Materials

“…For instance, such attempt was made by authors in [11,12,13] for the first time in history to study Electrical Resistance Tomography and Magnetic Induction Tomography. Interested readers can see [11,13,14,15,16,12,13,17,18,19,20,21,22,23,24,25,26,27,28,29] to be familiarize with the use of Monotonicity Principle for much more general settings and physical systems governed by different PDEs. The vitality of Monotonicity Principle Methods (MPM) is evident as, even in the presence of noise, it has magnificent property of providing rigorous upper and lower bounds of the unknown quantities with a suitable choice of hypothesises (See [30]).…”

Magnetic induction tomography via the monotonicity principle

“…This test is highly suitable for three main features: (i) the negligible computational cost for processing a given test inclusion, (ii) the processing on different test inclusions can be carried out in parallel, and (iii) the MP provides rigorous upper and lower bounds to the unknown defect, even in the presence of noise (see [10] and references therein), under proper assumptions. Subsequently, the MP was extended to many different settings [2,1,11,12,13,4,14,15,16,17,18,19,20,21]) and even to nonlinear materials [22].…”