Nonlinear Theory and Its Applications, IEICE
Online ISSN : 2185-4106
ISSN-L : 2185-4106
Volume 5 , Issue 1
Showing 1-9 articles out of 9 articles from the selected issue
Special section on Recent Progress in Verified Numerical Computations
• Nobito Yamamoto, Takeshi Ogita
Type: FOREWORD
Subject area: -
2014 Volume 5 Issue 1 Pages 1
Published: 2014
Released: January 01, 2014
JOURNALS FREE ACCESS
• Hidenori Ogata
Type: INVITED PAPER
Subject area: -
2014 Volume 5 Issue 1 Pages 2-14
Published: 2014
Released: January 01, 2014
JOURNALS FREE ACCESS
In this paper, we propose to apply the dipole simulation method proposed by Katsurada to two-dimensional potential problems in general regions. The dipole simulation method is an extension of the charge simulation method (the method of fundamental solutions) and gives an approximate solution by a linear combination of the dipole potentials. We also present an application of the proposed method to numerical conformal mapping. Some numerical examples show the effectiveness of the proposed method.
• Naoya Yamanaka, Shin'ichi Oishi
Type: PAPER
Subject area: -
2014 Volume 5 Issue 1 Pages 15-34
Published: 2014
Released: January 01, 2014
JOURNALS FREE ACCESS
An efficient format and fast algorithms of basic operations for 4-fold working precision are proposed. The proposed format is an unevaluated sum of four double precision numbers, capable of representing at least 203 bits of mantissa. Hence, it is slightly less accurate than quad-double format proposed by Hida et. al. [1], however presented algorithms based on the format are faster than those algorithms. By numerical experiments it is shown that the proposed algorithms are efficient.
• Yuka Yanagisawa, Takeshi Ogita, Shin'ichi Oishi
Type: PAPER
Subject area: -
2014 Volume 5 Issue 1 Pages 35-46
Published: 2014
Released: January 01, 2014
JOURNALS FREE ACCESS
This paper is concerned with an inverse matrix factorization based on Cholesky factorization for ill-conditioned matrices. Recently, Ogita and Oishi derived an iterative algorithm to calculate an accurate approximate inverse of the exact Cholesky factor for such matrices. In this paper, a modified version of the algorithm is proposed. It is explained that the proposed algorithm gives more accurate results than the original one by a numerical analysis. Numerical evidence is also shown.
• Takehiko Kinoshita, Yoshitaka Watanabe, Mitsuhiro T. Nakao
Type: PAPER
Subject area: -
2014 Volume 5 Issue 1 Pages 47-52
Published: 2014
Released: January 01, 2014
JOURNALS FREE ACCESS
This paper presents a numerical method to verify the invertibility of a linear elliptic operator. The invertibility of a linearized operator is useful information when verifying the existence of a solution for the corresponding nonlinear elliptic partial differential equations (PDEs). The proposed method is proved on the function spaces more suitable than previous methods. The a posteriori estimate, which is expected to converge to the exact operator norm of the inverse elliptic operators, is obtained by less computational cost than existing methods.
• Akitoshi Takayasu, Xuefeng Liu, Shin'ichi Oishi
Type: PAPER
Subject area: -
2014 Volume 5 Issue 1 Pages 53-63
Published: 2014
Released: January 01, 2014
JOURNALS FREE ACCESS
For Poisson's equation over a polygonal domain of general shape, the solution of which may have a singularity around re-entrant corners, we provide an explicit a priori error estimate for the approximate solution obtained by finite element methods of high degree. The method used herein is a direct extension of the one developed in preceding paper of the second and third listed authors, which provided a new approach to deal with the singularity by using linear finite elements. In the present paper, we also give a detailed discussion of the dependency of the convergence order on solution singularities, mesh sizes and degrees of the finite element method used.
• Kouta Sekine, Akitoshi Takayasu, Shin'ichi Oishi
Type: PAPER
Subject area: -
2014 Volume 5 Issue 1 Pages 64-79
Published: 2014
Released: January 01, 2014
JOURNALS FREE ACCESS
This paper presents an algorithm of identifying parameters satisfying a sufficient condition of Plum's Newton-Kantorovich like theorem. Plum's theorem yields a numerical existence test of solutions for nonlinear partial differential equations. The sufficient condition of Plum's theorem is given by the nonemptiness of a region defined by one dimensional nonlinear inequalities. The aim of this paper is to develop a systematic method of constructing an inner inclusion of this region. If $\underline\rho \in {\mathbb R}^+$ is the minimum included in this region, $\underline\rho$ gives the minimum of the error bounds. Moreover, if $\overline\rho \in {\mathbb R}^+$ is the maximum included in this region, then $\overline\rho$ gives the maximum radius of a ball in which the exact solution is unique. In this paper, an algorithm is developed for finding $\rho_e$ and $\rho_u$ such that they belong to this region and become close approximations of $\underline\rho$ and $\overline\rho$, respectively. Finally, to illustrate features of Plum's theorem with our proposed algorithm, some numerical results compared with results by Plum's theorem with the Newton method are presented. In addition to this, Plum's theorem with our algorithm is also compared with Newton-Kantorovich's theorem. One of the most important facts found in this paper is that, for some examples, $\rho_e$ become smaller than error bounds obtained by Newton-Kantorovich's theorem. Moreover, also for these examples, $\rho_u$ become greater than regions indicating uniqueness of the exact solution derived by Newton-Kantorovich's theorem. This implies that Plum's theorem can be seen as a modification of Newton-Kantorovich's theorem.
Regular section
• Yuya Shinde, Takahide Oya
Type: PAPER
2014 Volume 5 Issue 1 Pages 80-88
Published: 2014
Released: January 01, 2014
JOURNALS FREE ACCESS
We propose a single-electron “slime mold” circuit that can find the shortest path of a maze based on the behaviors of slime molds. The circuit consists of a single-electron reaction-diffusion (SE-RD) circuit and logic gate circuits. An SE-RD circuit can be used to represent the dilatation behaviors of slime molds. In this study, a slime mold model that was originally developed for cellular automata is used for the logic gate circuit. We used a Monte Carlo simulation to confirm that the proposed circuit can operate like slime molds. Moreover, it was able to find the optimal path of any maze.