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Kazuaki Tanaka

Publications and source records attributed to Kazuaki Tanaka.

16 recordsLinked to original sources
arXiv2026

A Green's Function-Based Enclosure Framework for Poisson's Equation and Generalized Sub- and Super-Solutions

This paper presents a novel framework for enclosing solutions of Poisson's equation based on generalized sub- and super-solutions constructed using fundamental solutions. The conventional definition of sub- and super-solutions based on variational inequalities often fails for natural function classes such as piecewise linear functions and encounters theoretical difficulties in non-convex polygonal domains, where H^2 regularity is lost because of corner singularities. To overcome these limitations, we introduce the concept of ``Green-representable solutions'' utilizing test functions constructed from fundamental solutions. This framework enables a new formulation of sub- and super-solutions that permits rigorous pointwise evaluation. For one-dimensional problems, we derive explicit constructions of the test functions. For two-dimensional polygonal domains, we employ the Method of Fundamental Solutions to generate test functions. The approach is validated through numerical experiments in both settings, including non-convex polygons. The results demonstrate that the proposed method yields strict and accurate pointwise enclosures of the true solution, even for problems with discontinuous source terms or geometric singularities.

math.NA↗
arXiv2026

Learn and Verify: A Framework for Rigorous Verification of Physics-Informed Neural Networks

The numerical solution of differential equations using neural networks has become a central topic in scientific computing, with Physics-Informed Neural Networks (PINNs) emerging as a powerful paradigm for both forward and inverse problems. However, unlike classical numerical methods that offer established convergence guarantees, neural network-based approximations typically lack rigorous error bounds. Furthermore, the non-deterministic nature of their optimization makes it difficult to mathematically certify their accuracy. To address these challenges, we propose a "Learn and Verify" framework that provides computable, mathematically rigorous error bounds for the solutions of differential equations. By combining a novel Doubly Smoothed Maximum (DSM) loss for training with interval arithmetic for verification, we compute rigorous a posteriori error bounds as machine-verifiable proofs. Numerical experiments on nonlinear Ordinary Differential Equations (ODEs), including problems with time-varying coefficients and finite-time blow-up, demonstrate that the proposed framework successfully constructs rigorous enclosures of the true solutions, establishing a foundation for trustworthy scientific machine learning.

cs.LG↗
arXiv2025

Error evaluation of partial scattering functions obtained from contrast variation small-angle neutron scattering

Contrast variation small-angle neutron scattering (CV-SANS) is a powerful tool to evaluate the structure of multi-component systems by decomposing scattering intensities $I$ measured with different scattering contrasts into partial scattering functions $S$ of self- and cross-correlations between components. The measured $I$ contains a measurement error, $ΔI$, and $ΔI$ results in an uncertainty of partial scattering functions, $ΔS$. However, the error propagation from $ΔI$ to $ΔS$ has not been quantitatively clarified. In this work, we have established deterministic and statistical approaches to determine $ΔS$ from $ΔI$. We have applied the two methods to (i) computational data of a core-shell sphere and experimental CV-SANS data of (ii) clay/polyethylene glycol (PEG) aqueous solutions and (iii) polyrotaxane solutions, and have successfully estimated the errors of \(S\). The quantitative error estimation of \(S\) offers us a strategy to optimize the combination of scattering contrasts to minimize error propagation.

cond-mat.soft↗
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↗
arXiv2022

Construction and Evaluation of a Self-Attention Model for Semantic Understanding of Sentence-Final Particles

Sentence-final particles serve an essential role in spoken Japanese because they express the speaker's mental attitudes toward a proposition and/or an interlocutor. They are acquired at early ages and occur very frequently in everyday conversation. However, there has been little proposal for a computational model of acquiring sentence-final particles. This paper proposes Subjective BERT, a self-attention model that takes various subjective senses in addition to language and images as input and learns the relationship between words and subjective senses. An evaluation experiment revealed that the model understands the usage of "yo", which expresses the speaker's intention to communicate new information, and that of "ne", which denotes the speaker's desire to confirm that some information is shared.

cs.CL↗
arXiv2021

A posteriori verification of the positivity of solutions to elliptic boundary value problems

The purpose of this paper is to develop a unified a posteriori method for verifying the positivity of solutions of elliptic boundary value problems by assuming neither $H^2$-regularity nor $ L^{\infty} $-error estimation, but only $ H^1_0 $-error estimation. In [J. Comput. Appl. Math, Vol. 370, (2020) 112647], we proposed two approaches to verify the positivity of solutions of several semilinear elliptic boundary value problems. However, some cases require $ L^{\infty} $-error estimation and, therefore, narrow applicability. In this paper, we extend one of the approaches and combine it with a priori error bounds for Laplacian eigenvalues to obtain a unified method that has wide application. We describe how to evaluate some constants required to verify the positivity of desired solutions. We apply our method to several problems, including those to which the previous method is not applicable.

math.NA↗
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

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↗
arXiv2021

A posteriori verification for the sign-change structure of solutions of elliptic partial differential equations

This paper proposes a method for rigorously analyzing the sign-change structure of solutions of elliptic partial differential equations subject to one of the three types of homogeneous boundary conditions: Dirichlet, Neumann, and mixed. Given explicitly estimated error bounds between an exact solution $ u $ and a numerically computed approximate solution $ \hat{u} $, we evaluate the number of sign-changes of $ u $ (the number of nodal domains) and determine the location of zero level-sets of $ u $ (the location of the nodal line). We apply this method to the Dirichlet problem of the Allen-Cahn equation. The nodal line of solutions of this equation represents the interface between two coexisting phases.

math.AP↗
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

Numerical verification method for positive solutions of elliptic problems

The purpose of this paper is to propose methods for verifying the positivity of a weak solution $ u $ of an elliptic problem assuming $ H^1_0 $-error estimation $ \left\|u-\hat{u}\right\|_{H_{0}^{1}} \leq ρ$ given some numerical approximation $ \hat{u} $ and an explicit error bound $ ρ$. We provide a sufficient condition for the solution to be positive and analyze the range of application of our method for elliptic problems with polynomial nonlinearities. We present numerical examples where our method is applied to some important problems.

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↗