Takashi Ohe

A numerical method for finding the convex hull of polygonal cavities using the enclosure method

A numerical method for finding cavities that is based on a formula in the enclosure method is discussed. It is a combination of: a method of estimating the value of the support function of cavities at a given direction by the slope of the logarithm of the indicator function; a rule to avoid poor estimation of the value of the support function at directions near a non-regular direction. We present the numerical testing of the method and show its effectiveness.

A precise estimation method for locations in an inverse logarithmic potential problem for point mass models

Abstract A numerical method is discussed for an inverse logarithmic potential problem for a point mass model. We first show the relation between locations of point masses and Fourier expansion of the logarithmic potential, and show a method using discrete Fourier coefficients. Next a modified method is proposed based on a property of the discrete Fourier transform of the logarithmic potential. We give error estimates for our modified method and compare it with the method using only discrete Fourier coefficients. Numerical examples illustrate the applicability and advantages of our modified method.

Uniqueness and convergence of numerical solution of the cauchy problem for the laplace equation by a charge simulation method

We consider the Cauchy problem for the Laplace equation in the neighborhood of the circle. The charge simulation method is applied to the problem, and a theoretical analysis for the numerical solution is given. The analysis for the numerical solution of the charge simulation method can be found in some papers, but the approach in these papers is only for well-posed problems such as the Dirichlet problem. Since our problem is ill-posed, a different approach is required to analyze the convergence of the numerical solution. In this paper, we prove the unique existence of the numerical solution and its exponential convergence to the exact solution. Our result agrees well with numerical experiments.

An estimation method for the number of point masses in an inverse logarithmic potential problem using discrete Fourier transform

Abstract A numerical method is discussed for the estimation of the number of point masses in an inverse logarithmic potential problem. We consider the case of some pieces of point masses being located in an “indeterminacy domain”, and consider a problem to estimate the number of point masses using observation data of the logarithmic potential on the boundary. A uniqueness theorem is derived for this problem from an algebraic relation between parameters of a point mass model and Fourier coefficients of the logarithmic potential. Applying this theorem, we propose a numerical method for our problem using a criterion function computed from the discrete Fourier transform. The applicability of our method is illustrated by numerical examples.

An identification method of electric current dipoles in spherically symmetric conductor

A numerical method is proposed for the identification of electric current dipoles in a spherically symmetric conductor. We use observations of the magnetic induction outside of the conductor. Our idea is to probe electric current dipoles using an indicator function that satisfies the Laplace equation. In our method, any a priori information for the number of dipoles and initial estimates are not required. The effectiveness of the method is shown by numerical examples.

Real-time reconstruction of time-varying point sources in a three-dimensional scalar wave equation

This paper discusses a reconstruction of point sources in a three-dimensional scalar wave equation from boundary measurements. We assume that the number, locations and magnitudes of point sources are unknown. Under these assumptions, we propose a real-time reconstruction method of these point sources based on the concept of the reciprocity gap functional. In our method, the number, locations and magnitudes of point sources can be identified within small delay. The effectiveness of the proposed method is shown by numerical examples.

The enclosure method for an inverse crack problem and the Mittag-Leffler function

An inverse boundary value problem for the two-dimensional Laplace equation is considered. The problem is to extract information about the location and shape of unknown curves which are a mathematical model of perfectly insulated cracks embedded in a known bounded domain, from the associated Dirichlet-to-Neumann map or its partial knowledge on the boundary of the domain. For the purpose, a new indicator function which can be calculated from the Dirichlet-to-Neumann map acting on the trace of special harmonic function with a large parameter is introduced. The harmonic function comes from the Mittag?Leffler function which is a generalization of the exponential function. A theorem concerned with the asymptotic behaviour of the indicator function with respect to such a large parameter is obtained. As a corollary, a uniqueness theorem for some partial information about unknown cracks together with a characterization of the information is established. An algorithm based on those theoretical investigations is introduced and the numerical implementation is shown. Our method is examined by some numerical experiments.

Boundary element approach for an inverse source problem of the Poisson equation with a one-point-mass-like source

Abstract An inverse source problem of the Poisson equation is discussed with a boundary-element-like method. This method is based on a boundary integral expression of the Poisson equation. With this method, we can eliminate the harmonic term in the solution of the Poisson equation using the solution and flux on the boundary and consider only an inverse problem of the logarithmic potential. A simple model is used for the source term, and an effective numerical algorithm is presented for the determination of the source in this model. A precise error bound for the result is also given in the case of constant elements on the boundary. The applicability of our method and error estimates is illustrated by a numerical example.

Orientational phase transition of solid ortho-hydrogen

From the orientational dependence of the quadrupole energies, it is assumed that hydrogen molecules can take only four orientations characterized by the symmetry axis pointing to the [111] direction in the face-centered cubic lattice. It is shown that solid ortho-hydrogen undergoes the first-order phase transition for the molecular orientations. The transition temperature is derived in the mean-field approximation, and reexamined by means of the computer simulation. The intensity of x-ray diffuse scattering caused by orientational disorder shows an anomalous behavior at the transition temperature.