Masaru Ikehata

This paper is concerned with the reconstruction issue of an inverse crack problem in a two-dimensional bounded domain, which may have a possible application to the nondestructive evaluation of materials. It is assumed that the domain consists of two elastic plates welded together and contains unknown cracks on the junction line. The governing equation for the in-plane displacement is the Navier equation. The problem is to extract information about the location of cracks from observation data consisting of a single set of applied surface traction and the resulting in-plane displacement field measured on the boundary of the domain. It is shown that the enclosure method combined with the Kelvin transform yields explicit formulae for extracting such information from the observation data.

The integrated theory of the probe and singular sources methods (IPS) is developed for an inverse obstacle problem governed by the stationary Schrödinger equation in a bounded domain. The unknown obstacles are penetrable, and their surface is modeled by a part of the support of the potential in the governing equation. The main results concern an analytical detection method for these obstacles from the Dirichlet-to-Neumann map. They consist of three parts: a singular sources method via the probe method using a solution with higher-order singularity for the governing equation of the background medium; the discovery of an IPS function whose two ways of decomposition give us the indicator functions for both the probe and singular sources methods; a completely integrated version of both methods, which means their indicator functions coincide. Furthermore, a result on Side B of IPS is also given, concerning the blowing-up property of a sequence calculated from the Dirichlet-to-Neumann map.

The main purpose of this paper is to develop further the integrated theory of the probe and singular sources methods (IPS) which may work for a group of inverse obstacle problems. Here as a representative and typical member of the group, an inverse obstacle problem governed by the Helmholtz equation with a fixed wave number in a bounded domain is considered. It is assumed that the solutions of the Helmholtz equation outside the set of unknown obstacles satisfy the homogeneous Dirichlet or Neumann boundary conditions on each surface of obstacles. This is the case when two extreme types of obstacles are embedded in a medium. By considering this case, not only a concise technique for IPS is introduced but also a general correspondence principle from IPS to the probe method is suggested. Besides, as a corollary it is shown that the probe method together with the singular sources method reformulated in terms of the probe method has the Side B under a smallness conditions on the wave number k, which is the blowing up property of a sequence computed from the associated Dirichlet-to-Neumann map.

The probe and singular sources methods are two well-known classical direct reconstruction methods in inverse obstacle problems governed by partial differential equations. In this paper, by considering an inverse obstacle problem governed by the Laplace equation in a bounded domain as a prototype case, an integrated theory of the probe and singular sources methods is proposed. The theory consists of three parts: (i) introducing the singular sources method combined with the notion of the probe method; (ii) finding a third indicator function whose two ways decomposition yields the indicator functions in the probe and singular sources methods; (iii) finding the completely integrated version of the probe and singular sources methods.

In this paper, an integrated theory of the probe and singular sources methods for an inverse obstacle problem governed by the Stokes system in a bounded domain is developed. The main results consist of: the probe method for the Stokes system; the singular sources method by using the notion of the probe method; the completely integrated version of the probe and singular sources methods. In establishing the singular sources method, a third indicator function which is called the IPS function plays the central role.

What should you do when you re-read the paper you intended to cite and found an error in the proof? In section 4 of Ikehata (2022 Inverse Problems 38 075009) the author pointed out that the proof of lemma 3.5 in Sini and Yoshida (2012 Inverse Problems 28 055013; 2013 Inverse Problems 29 039501) does not work for a Lipschitz obstacle case, by applying their approach to an essentially same lemma (lemma 3.2 in Ikehata (2022 Inverse Problems 38 075009)). For this, Sini (2023 Inverse Problems accepted) gave a comment. This is a reply to their comment.

An inverse obstacle problem governed by the Stokes system in the time domain is considered. Two types of extraction formulae about the geometry of an unknown obstacle are given by using the most recent version of the time domain enclosure method in a unified style. Each of the formulae employs only a single set of velocity and the Cauchy stress fields over a finite time interval on the surface of the region where a viscous incompressible fluid occupies.

The time domain enclosure method is one of the analytical methods for inverse obstacle problems governed by partial differential equations in the time domain. This paper considers the case when the governing equation is given by the Maxwell system and consists of two parts. The first part establishes the base of the time domain enclosure method for the Maxwell system using a single set of the solutions over a finite time interval for a general (isotropic) inhomogeneous medium in the whole space. It is a system of asymptotic inequalities for the indicator function which may enable us to apply the time domain enclosure method to the problem of finding unknown penetrable obstacles embedded in various background media. As a first step of its expected applications, the case when the background medium is homogeneous and isotropic is considered and the time domain enclosure method is realized. This is the second part.

This is a review article on the development of the probe and enclosure methods from past to present, focused on their central ideas together with various applications.

This paper is concerned with a new type of inverse obstacle problem governed by a variable-order time-fraction diffusion equation in a bounded domain. The unknown obstacle is a region where the space dependent variable-order of fractional time derivative of the governing equation deviates from a known homogeneous background one. The observation data is given by the Neumann data of the solution of the governing equation for a specially designed Dirichlet data. Under a suitable jump condition on the deviation, it is shown that the most recent version of the time domain enclosure method enables one to extract information about the geometry of the obstacle and a qualitative nature of the jump, from the observation data.