Monotonicity Principle in tomography of nonlinear conducting materials ^*

Abstract

We treat an inverse electrical conductivity problem which deals with the reconstruction of nonlinear electrical conductivity starting from boundary measurements in steady currents operations. In this framework, a key role is played by the Monotonicity Principle, which establishes a monotonic relation connecting the unknown material property to the (measured) Dirichlet-to-Neumann operator (DtN). Monotonicity Principles are the foundation for a class of non-iterative and real-time imaging methods and algorithms. In this article, we prove that the monotonicity principle for the Dirichlet Energy in nonlinear problems holds under mild assumptions. Then, we show that apart from linear and p-Laplacian cases, it is impossible to transfer this monotonicity result from the Dirichlet Energy to the DtN operator. To overcome this issue, we introduce a new boundary operator, identified as an average DtN operator.

Reconstruction of singular and degenerate inclusions in Calderón's problem

“…The main theoretical developments and related properties of monotonicity-based reconstruction in connection to Calderón's problem and EIT can be found in [4,10,12,15,16,17,20,21,23] for the continuum model and in [13,14,15,18] for related practical electrode models. For an extensive list of references to papers employing similar monotonicity-based arguments in other inverse coefficient problems, we refer to [15].…”

Simplified reconstruction of layered materials in EIT

“…See [1,3] and the references therein for more in-depth information on the methods and on reconstruction in electrical impedance tomography (EIT) in general. The main theoretical developments of monotonicity-based reconstruction in EIT can be found in [1,2,3,4,7,8,9,11,12,13] for the continuum model and in [5,6,7,10] for electrode models.…”

Monotonicity in inverse scattering for Maxwell's equations

“…Inverse obstacle scattering problems have been considered in [1,14], and an inverse crack detection problem has been studied in [13]. Monotonicity based reconstruction techniques for fractional order Schrödinger equations have been considered in [27,28], nonlinear materials have been studied in [7,12], and applications to eddy current problems as well as magnetic induction tomography have been discussed in [50,51]. In this work we generalize the concepts and the rigorous analysis from [20] to an inverse medium scattering problem governed by time-harmonic Maxwell's equations.…”

“…p(θ)e ikθ•x ds(θ) (12) for x ∈ R 3 . The latter is called a Herglotz wave pair with density p. We denote by (E q,p , H q,p ), (E s q,p , H s q,p ), and (E ∞ q,p , H ∞ q,p ) the corresponding total field, scattered field, and far field pattern, respectively.…”

“…We discuss the two integrals on the right hand side of (32) separately. Using (12) we find for the first integral that…”