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Masaru Ikehata

Publications and source records attributed to Masaru Ikehata.

At least 19 recordsLinked to original sources
arXiv2026

Removing small wavenumber constraints in Side B of the Probe Method

The Probe Method is an analytical reconstruction scheme for inverse obstacle problems utilizing the Dirichlet-to-Neumann map associated with the governing partial differential equation. It consists of two distinct parts: Side A and Side B. Both are based on the indicator sequence which is calculated from the Dirichlet-to-Neumann map acting on "needle-like" specialized solution of the governing equation for the background medium, whose energy is concentrated on an arbitrary given needle inside. In Side A, the limit of the indicator sequence-referred to as the indicator function-is computed before the needles touch the obstacle, and the boundary is identified as the point where this function first blows up. In contrast, Side B states the blow-up of the indicator sequence after the needles have come into contact with the obstacle. For the Helmholtz equation, the validity of Side B has long required a small wavenumber constraint. This paper finally removes this long-standing restriction, establishing the method's applicability for broader cases.

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arXiv2026

Integrating the probe and singular sources methods: IV. IPS function for the Schrödinger equation

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.

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arXiv2025

The enclosure method using a single point on the graph of the response operator for the Stokes system

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.

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arXiv2024

Integrating the probe and singular sources methods: III. Mixed obstacle case

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.

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arXiv2024

Integrating the probe and singular sources methods

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.

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arXiv2024

Integrating the probe and singular sources methods :II. The Stokes system

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.

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arXiv2022

Revisiting the probe and enclosure methods

This paper is concerned with reconstruction issue of inverse obstacle problems governed by partial differential equations and consists of two parts. (i) The first part considers the foundation of the probe and enclosure methods for an impenetrable obstacle embedded in a medium governed by the stationary Schrödinger equation. Under a general framework, some natural estimates for a quantity computed from a pair of the Dirichlet and Neumann data on the outer surface of the body occupied by the medium are given. The estimates enables us to derive almost immediately the necessary asymptotic behaviour of indicator functions for both methods. (ii) The second one considers the realization of the enclosure method for a penetrable obstacle embedded in an absorbing medium governed by the stationary Schrödinger equation. The unknown obstacle considered here is modeled by a perturbation term added to the background complex-valued $L^{\infty}$-potential. Under a jump condition on the term across the boundary of the obstacle and some kind of regularity for the obstacle surface including Lipschitz one as a special case, the enclosure method using infinitely many pairs of the Dirichlet and Neumann data is established.

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arXiv2022

On extracting the positions of multiple unknown cracks that occur on the junction line of two elastic plates

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 has some unknown cracks on the junction line and the governing equation of in-plane displacement is Navier's equation. The problem is to extract information about the location of cracks from the observation data which is a single set of a loading surface traction and the resulted in-plane displacement field on the boundary of the domain. It is shown that the enclosure method combined with the Kelvin transform yields explicit extraction formulae of such information from the observation data.

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arXiv2022

The enclosure method for the detection of variable order in fractional diffusion equations

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.

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arXiv2021

Extraction formulae for an inverse boundary value problem for the equation $\nabla\cdot(σ-iωε)\nabla u=0$

We consider an inverse boundary value problem for the equation $\nabla\cdot(σ-iωε)\nabla u=0$ in a given bounded domain $Ω$ at a fixed $ω>0$. $σ$ and $ε$ denote the conductivity and permittivity of the material forming $Ω$, respectively. We give some formulae for extracting information about the location of the discontinuity surface of $(σ,ε)$ from the Dirichlet-to-Neumann map. In order to obtain results we make use of two methods. The first is the enclosure method which is based on a new role of the exponentially growing solutions of the equation for the background material. The second is a generalization of the enclosure method based on a new role of Mittag-Leffler's function.

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arXiv2021

Reconstruction of a source domain from the Cauchy data: II. Three dimensional case

This paper is concerned with reconstruction issue of some typical inverse problems and consists of three parts. First a framework of the enclosure method for an inverse source problem governed by the Helmholtz equation at a fixed wave number in three dimensions is introduced. It is based on the nonvanishing of the coefficient of the leading profile of an oscillatory integral over a domain having a conical singularity. Second an explicit formula of the coefficient for a domain having a circular cone singularity and its implication under the framework are given. Third, an application under the framework to an inverse obstacle problem governed by an inhomogeneous Helmholtz equation at a fixed wave number in three dimensions is given.

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arXiv2021

The enclosure method for the heat equation

This paper shows how the enclosure method which was originally introduced for elliptic equations can be applied to inverse initial boundary value problems for parabolic equations. For the purpose a prototype of inverse initial boundary value problems whose governing equation is the heat equation is considered. An explicit method to extract an approximation of the value of the support function at a given direction of unknown discontinuity embedded in a heat conductive body from the temperature for a suitable heat flux on the lateral boundary for a fixed observation time is given.

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arXiv2021

On reconstruction of inclusions in a heat conductive body from dynamical boundary data over a finite time interval

The enclosure method was originally introduced for inverse problems of concerning non-destructive evaluation governed by elliptic equations. It was developed as one of useful approaches in inverse problems and applied for various equations. In this paper, an application of the enclosure method to an inverse initial boundary value problem for a parabolic equation with a discontinuous coefficient is given. A simple method to extract the depth of unknown inclusions in a heat conductive body from a single set of the temperature and heat flux on the boundary observed over a finite time interval is introduced. Other related results with infinitely many data are also reported. One of them gives the minimum radius of the open ball centered at a given point that contains the inclusions. The formula for the minimum radius is newly discovered.

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arXiv2021

The enclosure method for inverse obstacle scattering over a finite time interval: V. Using time-reversal invariance

The wave equation is time-reversal invariant. The enclosure method using a Neumann data generated by this invariance is introduced. The method yields the minimum ball that is centered at a given arbitrary point and encloses an unknown obstacle embedded in a known bounded domain from a single point on the graph of the so-called response operator on the boundary of the domain over a finite time interval. The occurrence of the lacuna in the solution of the free space wave equation is positively used.

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arXiv2021

Prescribing a heat flux coming from a wave equation

What happens when one prescribes a heat flux which is proportional to the Neumann data of a solution of the wave equation in the whole space on the surface of a heat conductive body? It is shown that there is a difference in the asymptotic behaviour of the indicator function in the most recent version of the time domain enclosure method, which aims at extracting information about an unknown cavity embedded in the body.

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