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Shin'ichi Oishi

Publications and source records attributed to Shin'ichi Oishi.

17 recordsLinked to original sources
arXiv2024

Polygonal Sequence-driven Triangulation Validator: An Incremental Approach to 2D Triangulation Verification

Two-dimensional Delaunay triangulation is a fundamental aspect of computational geometry. This paper presents a novel algorithm that is specifically designed to ensure the correctness of 2D Delaunay triangulation, namely the Polygonal Sequence-driven Triangulation Validator (PSTV). Our research highlights the paramount importance of proper triangulation and the often overlooked, yet profound, impact of rounding errors in numerical computations on the precision of triangulation. The primary objective of the PSTV algorithm is to identify these computational errors and ensure the accuracy of the triangulation output. In addition to validating the correctness of triangulation, this study underscores the significance of the Delaunay property for the quality of finite element methods. Effective strategies are proposed to verify this property for a triangulation and correct it when necessary. While acknowledging the difficulty of rectifying complex triangulation errors such as overlapping triangles, these strategies provide valuable insights on identifying the locations of these errors and remedying them. The unique feature of the PSTV algorithm lies in its adoption of floating-point filters in place of interval arithmetic, striking an effective balance between computational efficiency and precision. This research sets a vital precedent for error reduction and precision enhancement in computational geometry.

cs.CG↗
arXiv2021

Rigorous numerical enclosures for positive solutions of Lane-Emden's equation with sub-square exponents

The purpose of this paper is to obtain rigorous numerical enclosures for solutions of Lane-Emden's equation $-Δu=|u|^{p-1} u$ with homogeneous Dirichlet boundary conditions. We prove the existence of a nondegenerate solution $u$ nearby a numerically computed approximation $\hat{u}$ together with an explicit error bound, i.e., a bound for the difference between $ u $ and $\hat{u}$. In particular, we focus on the sub-square case in which $1<p<2$ so that the derivative $p|u|^{p-1}$ of the nonlinearity $|u|^{p-1} u$ is not Lipschitz continuous. In this case, it is problematic to apply the classical Newton-Kantorovich theorem for obtaining the existence proof, and moreover several difficulties arise in the procedures to obtain numerical integrations rigorously. We design a method for enclosing the required integrations explicitly, proving the existence of a desired solution based on a generalized Newton-Kantorovich theorem. A numerical example is presented where an explicit solution-enclosure is obtained for $ p=3/2 $ on the unit square domain $Ω=(0,1)^2$.

math.NA↗
arXiv2021

Numerical verification for asymmetric solutions of the Hénon equation on bounded domains

The Hénon equation, a generalized form of the Emden equation, admits symmetry-breaking bifurcation for a certain ratio of the transverse velocity to the radial velocity. Therefore, it has asymmetric solutions on a symmetric domain even though the Emden equation has no asymmetric unidirectional solution on such a domain. We discuss a numerical verification method for proving the existence of solutions of the Hénon equation on a bounded domain. By applying the method to a line-segment domain and a square domain, we numerically prove the existence of solutions of the Hénon equation for several parameters representing the ratio of transverse to radial velocity. As a result, we find a set of undiscovered solutions with three peaks on the square domain.

math.NA↗
arXiv2021

Computer-assisted proof for the stationary solution existence of the Navier-Stokes equation over 3D domains

This paper proposes a computer-assisted solution existence verification method for the stationary Navier-Stokes equation over general 3D domains. The proposed method verifies that the exact solution as the fixed point of the Newton iteration exists around the approximate solution through rigorous computation and error estimation. The explicit values of quantities required by applying the fixed point theorem are obtained by utilizing newly developed quantitative error estimation for finite element solutions to boundary value problems and eigenvalue problems of the Stokes equation.

math.NA↗
arXiv2021

Inverse norm estimation of perturbed Laplace operators and corresponding eigenvalue problems

In numerical existence proofs for solutions of the semi-linear elliptic system, evaluating the norm of the inverse of a perturbed Laplace operator plays an important role. We reveal an eigenvalue problem to design a method for verifying the invertibility of the operator and evaluating the norm of its inverse based on Liu's method and the Temple-Lehman-Goerisch method. We apply the inverse-norm's estimation to the Dirichlet boundary value problem of the Lotka-Volterra system with diffusion terms and confirm the efficacy of our method.

math.NA↗
arXiv2020

Sharp numerical inclusion of the best constant for embedding $H_{0}^{1}(Ω) \hookrightarrow L^{p}(Ω)$ on bounded convex domain

In this paper, we propose a verified numerical method for obtaining a sharp inclusion of the best constant for the embedding $H_{0}^{1}(Ω) \hookrightarrow L^{p}(Ω)$ on bounded convex domain in $\mathbb{R}^{2}$. We estimate the best constant by computing the corresponding extremal function using a verified numerical computation. Verified numerical inclusions of the best constant on a square domain are presented.

math.NA↗
arXiv2020

Explicit a posteriori and a priori error estimation for the finite element solution of Stokes equations

For the Stokes equation over 2D and 3D domains, explicit a posteriori and a priori error estimation are novelly developed for the finite element solution. The difficulty in handling the divergence-free condition of the Stokes equation is solved by utilizing the extended hypercircle method along with the Scott-Vogelius finite element scheme. Since all terms in the error estimation have explicit values, by further applying the interval arithmetic and verified computing algorithms, the computed results provide rigorous estimation for the approximation error. As an application of the proposed error estimation, the eigenvalue problem of the Stokes operator is considered and rigorous bounds for the eigenvalues are obtained. The efficiency of proposed error estimation is demonstrated by solving the Stokes equation on both convex and non-convex 3D domains.

math.NA↗
arXiv2019

A new formulation for the numerical proof of the existence of solutions to elliptic problems

Infinite-dimensional Newton methods can be effectively used to derive numerical proofs of the existence of solutions to partial differential equations (PDEs). In computer-assisted proofs of PDEs, the original problem is transformed into the infinite Newton-type fixed point equation $w = - {\mathcal L}^{-1} {\mathcal F}(\hat{u}) + {\mathcal L}^{-1} {\mathcal G}(w)$, where ${\mathcal L}$ is a linearized operator, ${\mathcal F}(\hat{u})$ is a residual, and ${\mathcal G}(w)$ is a local Lipschitz term. Therefore, the estimations of $\| {\mathcal L}^{-1} {\mathcal F}(\hat{u}) \|$ and $\| {\mathcal L}^{-1}{\mathcal G}(w) \|$ play major roles in the verification procedures. In this paper, using a similar concept as the `Schur complement' for matrix problems, we represent the inverse operator ${\mathcal L}^{-1}$ as an infinite-dimensional operator matrix that can be decomposed into two parts, one finite dimensional and one infinite dimensional. This operator matrix yields a new effective realization of the infinite-dimensional Newton method, enabling a more efficient verification procedure compared with existing methods for the solution of elliptic PDEs. We present some numerical examples that confirm the usefulness of the proposed method. Related results obtained from the representation of the operator matrix as ${\mathcal L}^{-1}$ are presented in the appendix.

math.NA↗
arXiv2017

Accurate method of verified computing for solutions of semilinear heat equations

We provide an accurate verification method for solutions of heat equations with a superlinear nonlinearity. The verification method numerically proves the existence and local uniqueness of the exact solution in a neighborhood of a numerically computed approximate solution. Our method is based on a fixed-point formulation using the evolution operator, an iterative numerical verification scheme to extend a time interval in which the validity of the solution can be verified, and rearranged error estimates for avoiding the propagation of an overestimate. As a result, compared with the previous verification method using the analytic semigroup, our method can enclose the solution for a longer time. Some numerical examples are presented to illustrate the efficiency of our verification method.

math.NA↗
arXiv2016

Numerical validation of blow-up solutions of ordinary differential equations

This paper focuses on blow-up solutions of ordinary differential equations (ODEs). We present a method for validating blow-up solutions and their blow-up times, which is based on compactifications and the Lyapunov function validation method. The necessary criteria for this construction can be verified using interval arithmetic techniques. Some numerical examples are presented to demonstrate the applicability of our method.

math.NA↗
arXiv2016

Estimation of Sobolev embedding constant on a domain dividable into bounded convex domains

This paper is concerned with an explicit value of the embedding constant from $W^{1,q}(Ω)$ to $L^{p}(Ω)$ for a bounded domain $Ω\subset\mathbb{R}^N~(N\in\mathbb{N})$, where $1\leq q\leq p\leq \infty$. To obtain this value, we previously proposed a formula for estimating the embedding constant on bounded and unbounded Lipschitz domains by estimating the norm of Stein's extension operator, in the article (K. Tanaka, K. Sekine, M. Mizuguchi, and S. Oishi, Estimation of Sobolev-type embedding constant on domains with minimally smooth boundary using extension operator, Journal of Inequalities and Applications, Vol. 389, pp. 1-23, 2015). This formula is also applicable to a domain that can be divided into Lipschitz domains. However, the values computed by the previous formula are very large. In this paper, we propose several sharper estimations of the embedding constant on a bounded domain that can be divided into convex domains.

math.FA↗
arXiv2016

Numerical verification method for positiveness of solutions to elliptic equations

In this paper, we propose a numerical method for verifying the positiveness of solutions to semilinear elliptic equations. We provide a sufficient condition for a solution to an elliptic equation to be positive in the domain of the equation, which can be checked numerically without requiring a complicated computation. We present some numerical examples.

math.NA↗
arXiv2013

Verified computations for hyperbolic 3-manifolds

For a given cusped 3-manifold $M$ admitting an ideal triangulation, we describe a method to rigorously prove that either $M$ or a filling of $M$ admits a complete hyperbolic structure via verified computer calculations. Central to our method are an implementation of interval arithmetic and Krawczyk's Test. These techniques represent an improvement over existing algorithms as they are faster, while accounting for error accumulation in a more direct and user friendly way.

math.GT↗
arXiv2012

Verified eigenvalue evaluation for Laplacian over polygonal domain of arbitrary shape

The finite element method(FEM) is applied to bound leading eigenvalues of Laplace operator over polygonal domain. Compared with classical numerical methods, most of which can only give concrete eigenvalue bounds over special domain of symmetry, our proposed algorithm can provide concrete eigenvalue bounds for domain of arbitrary shape, even in the case that eigenfunction has singularity. The problem of eigenvalue estimation is solved in two steps. First, we construct computable a priori error estimation for FEM solution of Poisson's problem even for non-convex domain with re-entrant corner. Second, a new computable lower and upper bounds is developed for eigenvalues. As the interval arithmetic is implemented in the FEM computation, the desired eigenvalue bounds can be expected to be mathematically correct. The Lehmann's theorem is also adopted to sharpen the eigenvalue bounds with high precision. At the end of this paper, we illustrate several computation examples, such as the case of L-shaped domain and crack domain, to demonstrate the efficiency and flexibility of proposed method.

math.NA↗
arXiv2012

Space guaranteeing a primitive chaotic behavior

This study describes such a situation that a Cantor set emerges as a result of the exploration of sufficient conditions for the property which is generalized from fundamental chaotic maps, and the Cantor set even guarantees infinitely many varieties of the behavior with the property, as well as a typical continuum.

nlin.CD↗