Kohzaburo Ohnaka

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.

Boundary element approach for identification of point forces of distributed parameter systems

Abstract Recently the identification of an external force applied to distributed parameter systems has become increasingly important in relation to the analysis and control of geophysical and environmental phenomena. A boundary element approach is developed for identifying the external force applied to a number of points on the domain of a distributed parameter system, which is described by a non- homogeneous partial differential equation of parabolic (diffusion) type. Firstly, an integral equation corresponding to the given partial differential equation is derived. The solution of the integral equation at several reference points is computed by using the boundary element method, and then the locations and magnitudes of the external force are identified with the number of application points by minimizing the sum of the squares of relative errors. Numerical examples are presented to illustrate the usefulness of the proposed method.

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.

A reliable estimation method for locations of point sources for an n-dimensional Poisson equation

An inverse source problem is discussed for an n-dimensional Poisson equation with several point sources. This paper presents a reliable method of estimating unknown locations of point sources under the condition that each location is bounded in a certain region. Our method is based on a weighted residual approach using harmonic functions for weighting functions and carried out by reducing the region containing each location. We also show that these regions are converged to true locations. The effectiveness of the method is illustrated by some numerical examples.

Boundary element approach for identification of boundary conditions of distributed-parameter systems

An identification problem is discussed for the boundary conditions of a deterministic distributed-parameter system governed by a partial differential equation of parabolic type. A method based on the fundamental ideas of the boundary element method is proposed for identification of unknown boundary conditions. The identification problem is formulated by using the ideas of boundary partition and weighted residual expression corresponding to the given partial differential equation. The boundary conditions are estimated by the method of least squares using the state observations taken at the interior points. A numerical example illustrates the applicability of the proposed method for boundary identification.

Identification of the external input of distributed-parameter systems by the boundary-element approach

An identification method is proposed for the unknown external input of distributed-parameter systems governed by a non-homogeneous partial differential equation of parabolic type. The mathematical formulation of the identification problem is based on a boundary partition and a weighted residual expression corresponding to the given partial differential equation, both of which are fundamental ideas of the boundary-element method. The unknown external input is estimated from state observations taken at points on the boundary by minimizing a suitable criterion function. The applicability of the proposed method is illustrated by a numerical example

Simultaneous identification of the external input and parameters of diffusion type distributed parameter systems

A simultaneous identification method is proposed for an unknown diffusion constant and an unknown external input of a distributed parameter system governed by a non-homogeneous partial differential equation of parabolic (diffusion) type. The mathematical formulation of the identification problem is presented using the weighted residual expression and the boundary element partition, both of which are fundamental to the boundary element method. The unknowns of the distributed parameter systems are identified from the noisy or noise-free state observations taken at the points on the boundary by minimizing a certain criterion function based on the above mathematical formulation. Numerical examples illustrate the usefulness of the proposed identification method.

A reliable estimation method of a dipole for three-dimensional Poisson equation

We discuss an inverse source problem for three-dimensional Poisson equation. The source term is expressed as a dipole that consists of two contiguous point sources. Our problem is estimating the dipole moment and its location under the condition that the potential and flux can be observed on the boundary. We propose a numerical method based on a weighted residual approach, which need not use direct analysis for the governing equation. The method should be reliable since the error caused by the numerical integration and observation noise can be evaluated theoretically. The effectiveness of the method is shown by numerical examples.