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Japan Journal of Industrial and Applied Mathematics
Japan Journal of Industrial and Applied Mathematics
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.
Nakao, Mitsuhiro T.; Oishi, Shin'ichi
Sugihara, Masaaki
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.
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.
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.
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.
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.
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.
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...
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.
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...
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.
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.
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.
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.
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.
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.
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.
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.