Monotonicity of the Laplace Transform for dissipative systems: Magnetic Induction Tomography

Magnetic Induction Tomography (MIT) is a proven technique for looking inside conductive materials, using low-frequency electromagnetic fields. However, analyzing MIT data is mathematically complex because of the non-linear and ill-posed nature of the inverse problem. In this framework, the Monotonicity Principle is recognized as an effective approach to solve the inverse obstacle problem in realistic situations and real-time operations. To date, the Monotonicity Principle has been found for elliptic and hyperbolic PDEs, even in the presence of nonlinearities. Magnetic Induction Tomography is modeled by a parabolic PDE. The physics of this dissipative system pose severe challenges from the theoretical and applied perspectives. To date, the Monotonicity Principles for MIT have been found only for proper asymptotic approximations or source-free problems. This article recognizes that the operator mapping the Laplace transform of the applied source onto the measured quantity (Transfer Operator) satisfies a Monotonicity Principle on a real semi-axis of the complex plane. Therefore, this article introduces the concept of transfer operator in Magnetic Induction Tomography, proves a Monotonicity Principle for the transfer operator evaluated on a proper real semi-axis of the complex plane, and provides a description of the related (real-time) imaging method.