This article describes a survey on numerical verification methods for second-order semilinear elliptic boundary value problems introduced by authors and their colleagues. Here “numerical verification” means a computer-assisted numerical method for proving the existence of a solution in a close and explicit neighborhood of an approximate solution. Three kinds of methods based on the infinite dimensional fixed-point theorems using Newton-like operator will be presented. In each verification method, a projection into a finite dimensional subspace and constructive error estimates of the projection play an important and essential role. It is shown that these methods are really useful for actual problems by illustrating numerical examples.

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We present an approach to robust geometric algorithms, which we call the principle of independence. In this approach, we distinguish between independent judgments and dependent judgments, and use numerical computation only for independent judgments. The result of judgments is always consistent and hence algorithms behave stably even in the presence of large numerical errors. The basic idea of this principle is described with three examples.

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Two random number generation methods based on a double-scroll chaotic system are introduced. Numerical models for the proposed designs have been developed where bootstrap method is utilized which allows us to estimate the statistical characteristics of underlying chaotic signals. Offset compensation loops have been developed in order to maximize the statistical quality of the output sequence and to be robust against external interference and parameter variations. Numerical results, verifying the feasibilities and correct operations of the random number generators are given such that numerically generated binary sequences fulfill FIPS-140-2 statistical test suites for randomness without post-processing.

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Multiple-precision arithmetic with interval variables has been developed for computation with guaranteed high accuracy. There are several computer program packages which deal with interval variables of the inf-sup form, e.g. MPFI, etc. On the other hand, it is impressed by INTLAB that interval multiple-precision arithmetic using the center-radius form has advantages on memory size and computing time. However, arithmetic of the center-radius form sometimes makes the radius of an interval larger than the inf-sup form does, which would be one of the reasons why there is no practical program package for interval multiple-precision arithmetic with the center-radius form.
The authors intend to establish a computer program package for multiple-precision arithmetic using intervals of the center-radius form which is still under construction. The present paper treats the problem of the center-radius form about the expansion of radii caused by the fundamental rules and the operation of square root. We propose several methods for calculation of multiplication, division, and square root, among which one can choose an appropriate method according to one's situation. Theoretical consideration and numerical examples are given for these methods.

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We present a model problem for global optimization in a specified number of unknowns. We give constraint and unconstraint formulations. The problem arose from structured condition numbers for linear systems of equations with Toeplitz matrix. We present a simple algorithm using additional information on the problem to find local minimizers which presumably are global. Without this it seems quite hard to find the global minimum numerically. For dimensions up to n=18 rigorous lower bounds for the problem are presented.

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Present authors have presented with Takayuki Kubo at University of Tsukuba a method of a computer assisted proof for the existence and uniqueness of solutions to two-point boundary value problems of nonlinear ordinary differential equations in the paper submitted for NOLTA, IEICE. This method uses piecewise linear finite element base functions and sometimes requires fine mesh. To overcome this difficulty, in this paper, an improved method is presented for the norm estimation of the residual to the operator equation. In this refined formulation, piecewise quadratic finite element base functions are used. A kind of the residual technique works sophisticatedly well. It is stated that the estimation of the residual can be expected smaller than that of the previous method. Finally, four examples are presented. Each result demonstrates that a remarkable improvement is achieved in accuracy of the guaranteed error estimation.

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Using techniques introduced by Nakao [4], Oishi [5, 6] and applied by Takayasu, Oishi, Kubo [11, 12] to certain nonlinear two-point boundary value problems (see also Rump [7], Chapter 15), we provide a numerical method for verifying the existence of weak solutions of two-point boundary value problems of the form

-u″ = a(x, u) + b(x, u)u′, 0 < x < 1,
u(0) = 0 = u(1),

in the vicinity of a given approximate numerical solution, where a and b are functions that fulfill some regularity properties. The numerical approximation is done by cubic spline interpolation. Finally, the method is applied to the Duffing, the van der Pol and the Toda oscillator. The rigorous numerical computations were done with INTLAB [8].

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An enclosure method for complex eigenvalues of ordinary differential operatos is presented. We formulate the eigenvalue problem as a nonlinear system and apply a fixed point theorem to enclose eigenvalues and eigenfunctions or a basis of the corresponding invariant subspaces in case of multiple eigenvalues. Some enclosure examples are given.

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This short note describes a computer-assisted stability proof for the Orr-Sommerfeld problem with Poiseuille flow. It is an application of a numerical verification technique for second-order elliptic boundary value problems introduced by a part of the authors.

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We have devised a method for hypothesis testing for the major features of multivariate data on the basis of collective synchronization in a network of nonsymmetrically coupled phase oscillators subject to a variant of Kuramoto's dynamics. We show through numerical experiments that nonsymmetric coupling allows testing whether the given test vectors match the major features.

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One of the onset mechanisms of the propagating pulse wave observed in a ring of coupled bistable oscillators is investigated. In particular, a formation mechanism of the propagating pulse wave solution and its related dynamics are discussed. For the 6-coupled oscillator case, it is verified that a global bifurcation of maps based on the heteroclinic tangle converts the fixed point corresponding to the standing pulse wave into the invariant circle corresponding to the propagating pulse wave.

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