Mostrando recursos 1 - 20 de 81

  1. Theory of an interval algebra and its application to numerical analysis [Reprint of Res. Assoc. Appl. Geom. Mem. 2 (1958), 29–46]

    Sunaga, Teruo
    This is reprinted, with permission from the author and the publisher, from the original paper published in Research Association of Applied Geometry (RAAG) Memoirs, Vol. 2 (1958) pp. 29-46, published by Gakujutsu Bunken Fukyu-kai, Tokyo, Japan.

  2. Guest editors' preface

    Nakao, Mitsuhiro T.; Oishi, Shin'ichi

  3. Preface

    Sugihara, Masaaki

  4. Optimal Control Problems for the Two Dimensional Rayleigh--Bénard Type Convection by a Gradient Method

    Lee, Hyung-Chun
    In this aricle, the author considers mathematical formulation and numerical solutions of distributed and Neumann boundary optimal control problems associated with the stationary Bénard problem. The solution of the optimal control problem is obtained by controlling of the source term of the equations and/or Neumann boundary conditions. Then the author considers the approximation, by finite element methods, of the optimality system and derive optimal error estimates. The convergence of a simple gradient method is proved and some numerical results are given.

  5. An Explicit Method for Convection-Diffusion Equations

    Ruas, Vitoriano; Brasil Jr., Antonio; Trales, Paulo
    An explicit scheme for time-dependent convection-diffusion problems is presented. It is shown that convenient bounds for the time step value ensure $L^{\infty}$ stability, in both space and time, for piecewise linear finite element discretizations in any space dimension. Convergence results in the same sense are also demonstrated under certain conditions. Numerical results certify the good performance of the scheme.

  6. DE-Sinc Method for Second Order Singularly Perturbed Boundary Value Problems

    Mori, Masatake; Nurmuhammad, Ahniyaz; Muhammad, Mayinur
    In this paper the sinc-Galerkin method, as well as the sinc-collocation method, based on the double exponential transformation (DE transformation) for singularly perturbed boundary value problems of second order ordinary differential equation is considered. A large merit of the present method exists in that we can apply the standard sinc method with only a small care for perturbation parameter. Through several numerical experiments we confirmed higher efficiency of the present method than that of other methods, e.g., sinc method based on the single exponential (SE) transformation, as the number of sampling points increases.

  7. Nonlinear and Linear Conservative Finite Difference Schemes for Regularized Long Wave Equation

    Koide, Satoshi; Furihata, Daisuke
    We propose four conservative schemes for the regularized long-wave (RLW) equation. The RLW equation has three invariants: mass, momentum, and energy. Our schemes are designed by using the discrete variational derivative method to inherit appropriate conservation properties from the equation. Two of our schemes conserve mass and momentum, while the other two schemes conserve mass and energy. With one of our schemes, we prove the numerical solution stability, the existence of the solutions, and the convergence of the solutions. Through some numerical computation examples, we demonstrate the efficiency and robustness of our schemes.

  8. Generalized Approximate Inverse Preconditioners for Least Squares Problems

    Cui, Xiaoke; Hayami, Ken
    This paper is concerned with a new approach for preconditioning large sparse least squares problems. Based on the idea of the approximate inverse preconditioner, which was originally developed for square matrices, we construct a generalized approximate inverse (GAINV) $M$ which approximately minimizes $\|I-MA\|_{\mathrm{F}}$ or $\|I-AM\|_{\mathrm{F}}$. Then, we also discuss the theoretical issues such as the equivalence between the original least squares problem and the preconditioned problem. Finally, numerical experiments on problems from Matrix Market collection and random matrices show that although the preconditioning is expensive, it pays off in certain cases.

  9. Representation Formula for the Critical Points of the Tadjbakhsh--Odeh Functional and its Application

    Watanabe, Kohtaro; Takagi, Izumi
    In order to study the buckled states of an elastic ring under uniform pressure, Tadjbakhsh and Odeh [14] introduced an energy functional which is a linear combination of the total squared curvature (elastic energy) and the area enclosed by the ring. We prove that the minimizer of the functional is not a disk when the pressure is large, and its curvature can be expressed by Jacobian elliptic $\cn({}\cdot{})$ function. Moreover, the uniqueness of the minimizer is proven for certain range of the pressure.

  10. Simulation Algorithm of Typical Modulated Poisson--Voronoi Cells and Application to Telecommunication Network Modelling

    Fleischer, F.; Gloaguen, C.; Schmidt, H.; Schmidt, V.; Schweiggert, F.
    We consider modulated Poisson--Voronoi tessellations, intended as models for telecommunication networks on a nationwide scale. By introducing an algorithm for the simulation of the typical cell of the latter tessellation, we lay the mathematical foundation for such a global analysis. A modulated Poisson--Voronoi tessellation has an intensity which is spatially variable and, hence, is able to provide a broad spectrum of model scenarios. Nevertheless, the considered tessellation model is stationary and we consider the case where the modulation is generated by a Boolean germ-grain model with circular grains. These circular grains may either have a deterministic or random but bounded radius. Furthermore, based on the introduced simulation algorithm for the typical cell...

  11. Deviation from the Predicted Wavenumber in a Mode Selection Problem for the Turing Patterns

    Kuwamura, Masataka
    In this paper, we investigate a mode selection problem for the Turing patterns generated from small random initial disturbances in one-dimensional reaction-diffusion systems on a sufficiently large domain. For this problem, it is widely accepted that the maximizer of the dispersion relation give rise to the wavenumber to be selected. Even in a small neighborhood of the bifurcation point, our numerical experiments show that this is not always true.

  12. On Computation of a Power Series Root with Arbitrary Degree of Convergence

    Kitamoto, Takuya
    Given a bivariate polynomial $f(x,y)$, let $\phi(y)$ be a power series root of $f(x,y)=0$ with respect to $x$, i.e., $\phi(y)$ is a function of $y$ such that $f(\phi(y),y)=0$. If $\phi(y)$ is analytic at $y=0$, then we have its power series expansion \begin{equation} \phi(y)=\alpha_{0}+\alpha_{1}y+\alpha_{2}y^{2}+\cdots+\alpha_{r}y^{r}+\cdots. \end{equation} Let $\phi^{(k)}(y)$ denote $\phi(y)$ truncated at $y^{k}$, i.e., \begin{equation} \phi^{(k)}(y)=\alpha_{0}+\alpha_{1}y+\alpha_{2}y^{2}+\cdots+\alpha_{k}y^{k}. \end{equation} Then, it is well known that, given initial value $\phi^{(0)}(y)=\alpha_{0}\in\mathbf{C}$, the symbolic Newton's method with the formula \begin{equation} \phi^{(2^{m}-1)}(y)\gets\phi^{(2^{m-1}-1)}(y) -\frac{f(\phi^{(2^{m-1}-1)}(y),y)}{\frac{\partial f}{\partial x}(\phi^{(2^{m-1}-1)}(y),y)} \quad (\mod y^{2^{m}}) \end{equation} computes $\phi^{(2^{m}-1)}(y)$ ($1\le m$) in (2) with quadratic convergence (the roots are computed in the order $\phi^{(0)}(y) \to \phi^{(2^{1}-1)}(y) \to \phi^{(2^{2}-1)}(y) \to \cdots \to \phi^{(2^{m}-1)}(y)$). References [1] and [3] indicate that the symbolic Newton's method can be generalized so that its convergence degree is an...

  13. Computer Algebra for Guaranteed Accuracy. How Does It Help?

    Kanno, Masaaki; Anai, Hirokazu
    Today simulation technologies (based on numerical computation) are definitely vital in many fields of science and engineering. The accuracy of numerical simulations grows in importance as simulation technologies develop and prevail, and many researches have been carried out for establishing numerically stable algorithms. In recent years, combining computer algebra and other guaranteed accuracy approaches draws much attention as one of promising directions for developing guaranteed accuracy algorithms for a wider class of problems. This paper illustrates several typical usages of symbolic and algebraic methods for guaranteed accuracy computation, highlighting some of the recent applications in control problems.

  14. A Constructive A Priori Error Estimation for Finite Element Discretizations in a Non-Convex Domain Using Singular Functions

    Kobayashi, Kenta
    In solving elliptic problems by the finite element method in a bounded domain which has a re-entrant corner, the rate of convergence can be improved by adding a singular function to the usual interpolating basis. When the domain is enclosed by line segments which form a corner of $\pi/2$ or $3\pi/2$, we have obtained an explicit a priori $H^{1}_{0}$ error estimation of $O(h)$ and an $L^{2}$ error estimation of $O(h^{2})$ for such a finite element solution of the Poisson equation. Particularly, we emphasize that all constants in our error estimates are numerically determined, which plays an essential role in the numerical verification of solutions to non-linear elliptic problems.

  15. Numerical Verification Method for Infinite Dimensional Eigenvalue Problems

    Nagatou, Kaori
    We consider an eigenvalue problem for differential operators, and show how guaranteed bounds for eigenvalues (together with eigenvectors) are obtained and how non-existence of eigenvalues in a concrete region can be assured. Some examples for several types of operators will be presented.

  16. Iterative Refinement for Ill-Conditioned Linear Systems

    Oishi, Shin'ichi; Ogita, Takeshi; Rump, Siegfried M.
    This paper treats a linear equation \begin{equation*} Av=b, \end{equation*} where $A \in \mathbb{F}^{n\times n}$ and $b \in \mathbb{F}^n$. Here, $\mathbb{F}$ is a set of floating point numbers. Let $\mathbf{u}$ be the unit round-off of the working precision and $\kappa(A)=\|A\|_{\infty}\|A^{-1}\|_{\infty}$ be the condition number of the problem. In this paper, ill-conditioned problems with \begin{equation*} 1 < \mathbf{u}\kappa(A) < \infty \end{equation*} are considered and an iterative refinement algorithm for the problems is proposed. In this paper, the forward and backward stability will be shown for this iterative refinement algorithm.

  17. Numerical Verification Method of Solutions for Elliptic Equations and Its Application to the Rayleigh--Bénard Problem

    Watanabe, Yoshitaka; Nakao, Mitsuhiro T.
    We first summarize the general concept of our verification method of solutions for elliptic equations. Next, as an application of our method, a survey and future works on the numerical verification method of solutions for heat convection problems known as Rayleigh--Bénard problem are described. We will give a method to verify the existence of bifurcating solutions of the two-dimensional problem and the bifurcation point itself. Finally, an extension to the three-dimensional case and future works will be described.

  18. Computer-Assisted Proofs for Semilinear Elliptic Boundary Value Problems

    Plum, Michael
    For second-order semilinear elliptic boundary value problems on bounded or unbounded domains, a general computer-assisted method for proving the existence of a solution in a ``close'' and explicit neighborhood of an approximate solution, computed by numerical means, is proposed. To achieve such an existence and enclosure result, we apply Banach's fixed-point theorem to an equivalent problem for the error, i.e., the difference between exact and approximate solution. The verification of the conditions posed for the fixed-point argument requires various analytical and numerical techniques, for example the computation of eigenvalue bounds for the linearization at the approximate solution. The method is used to prove existence and multiplicity results for some specific examples.

  19. Recent Development in Rigorous Computational Methods in Dynamical Systems

    Arai, Zin; Kokubu, Hiroshi; Pilarczyk, Paweł
    We highlight selected results of recent development in the area of rigorous computations which use interval arithmetic to analyse dynamical systems. We describe general ideas and selected details of different ways of approach and we provide specific sample applications to illustrate the effectiveness of these methods. The emphasis is put on a topological approach, which combined with rigorous calculations provides a broad range of new methods that yield mathematically reliable results.

  20. An Application of Taylor Models to the Nakao Method on ODEs

    Yamamoto, Nobito; Komori, Takashi
    The authors give short survey on validated computaion of initial value problems for ODEs especially Taylor model methods. Then they propose an application of Taylor models to the Nakao method which has been developed for numerical verification methods on PDEs and apply it to initial value problems for ODEs with some numerical experiments.

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