The framework of the enclosure method with dynamical data and its applications

TL;DR

This paper develops a framework for the enclosure method to solve inverse problems involving parabolic equations with discontinuous coefficients, enabling the extraction of inclusion shapes within heat conductive bodies using boundary measurements.

Contribution

It establishes a new framework for applying the enclosure method to inverse parabolic problems with discontinuous coefficients, including explicit solution constructions and application to inhomogeneous bodies.

Findings

Framework reduces inverse problems to those with large parameters.

Explicit solutions constructed using complex exponentials for homogeneous bodies.

Application to inhomogeneous bodies with a new parameter called virtual slowness.

Abstract

The aim of this paper is to establish the framework of the enclosure method for some class of inverse problems whose governing equations are given by parabolic equations with discontinuous coefficients. The framework is given by considering a concrete inverse initial boundary value problem for a parabolic equation with discontinuous coefficients. The problem is to extract information about the location and shape of unknown inclusions embedded in a known isotropic heat conductive body from a set of the input heat flux across the boundary of the body and output temperature on the same boundary. In the framework the original inverse problem is reduced to an inverse problem whose governing equation has a large parameter. A list of requirements which enables one to apply the enclosure method to the reduced inverse problem is given. Two new results which can be considered as the application of the framework are given. In the first result the background conductive body is assumed to be homogeneous and a family of explicit complex exponential solutions are employed. Second an application of the framework to inclusions in an isotropic inhomogeneous heat conductive body is given. The main problem is the construction of the special solution of the governing equation with a large parameter for the background inhomogeneous body required by the framework. It is shown that, introducing another parameter which is called the virtual slowness and making it sufficiently large, one can construct the required solution which yields an extraction formula of the convex hull of unknown inclusions in a known isotropic inhomogeneous conductive body.