Researcher Database
Last Updated :2024/10/17
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Profile and Settings
Affiliation
Degree
Profile and Settings
Name (Japanese)
KurodaName (Kana)
HirotoshiName
201001097732038232
Alternate Names
Affiliation
Achievement
Research Interests
- Homogenization method Nonlinear partial differential equations Variational problem singular diffusion equations nonlinear evolution equations Mosco convergence spectrum
Research Areas
Research Experience
- 2017/12 - Today Hokkaido University Department of Mathematics, Faculty of Science Associate Professor
- 2014/04 - 2017/11 Hokkaido University Faculty of science Specially Appointed Associate Professor
- 2012/04 - 2014/03 Osaka Prefecture University Faculty of Liberal Arts and Sciences Lecturer for Higher Education
- 2010/04 - 2012/03 Tohoku University Graduate School of Science Assistant Professor
- 2009/10 - 2010/03 Tohoku University Graduate School of Science
- 2009/04 - 2009/09 Hokkaido University Department of Mathematics, Faculty of Science Researcher
Education
- 2004/04 - 2009/03 Hokkaido University
- 2002/04 - 2004/03 Hokkaido University
- 1999/04 - 2002/03 Hokkaido University School of Science Mathematics
- 1998/04 - 1999/03 Hokkaido University School of Science
Awards
- 2024/09 北海道大学 高等教育推進機構 R5年度全学教育科目エクセレント・ティーチャー
- 2023/07 日本応用数理学会 2023年度日本応用数理学会論文賞（ノート部門）
受賞者: 山田 崇恭;正宗 淳;寺本 央;長谷部 高広;黒田 紘敏 - 2021/08 北海道大学 高等教育推進機構 R2年度全学教育科目エクセレント・ティーチャーズ
受賞者: 黒田 紘敏 - 2021/04 一般社団法人 日本機械学会 2020年度 日本機械学会賞（論文）
幾何学的特徴量に対する偏微分方程式系に基づく幾何学的特徴制約付きトポロジー 最適化（積層造形における幾何学的特異点を考慮したオーバーハング制約法）受賞者: 山田 崇恭;正宗 淳;寺本 央;長谷部 高広;黒田 紘敏 - 2014/04 大阪府立大学 高等教育推進機構 機構長教育奨励賞
受賞者: 黒田 紘敏 - 2002/03 北海道大学 クラーク記念財団クラーク賞
受賞者: 黒田 紘敏
Published Papers
- Yoshikazu Giga, Hirotoshi Kuroda, Michał ŁasicaMathematics in Engineering 5 (6) 1 - 45 2640-3501 2023 [Refereed]
<abstract><p>We define rigorously a solution to the fourth-order total variation flow equation in $ \mathbb{R}^n $. If $ n\geq3 $, it can be understood as a gradient flow of the total variation energy in $ D^{-1} $, the dual space of $ D^1_0 $, which is the completion of the space of compactly supported smooth functions in the Dirichlet norm. However, in the low dimensional case $ n\leq2 $, the space $ D^{-1} $ does not contain characteristic functions of sets of positive measure, so we extend the notion of solution to a larger space. We characterize the solution in terms of what is called the Cahn-Hoffman vector field, based on a duality argument. This argument relies on an approximation lemma which itself is interesting. We introduce a notion of calibrability of a set in our fourth-order setting. This notion is related to whether a characteristic function preserves its form throughout the evolution. It turns out that all balls are calibrable. However, unlike in the second-order total variation flow, the outside of a ball is calibrable if and only if $ n\neq2 $. If $ n\neq2 $, all annuli are calibrable, while in the case $ n = 2 $, if an annulus is too thick, it is not calibrable. We compute explicitly the solution emanating from the characteristic function of a ball. We also provide a description of the solution emanating from any piecewise constant, radially symmetric datum in terms of a system of ODEs.</p></abstract>
- Hasebe Takahiro, Kuroda Hirotoshi, Teramoto Hiroshi, Masamune Jun, Yamada TakayukiTransactions of the Japan Society for Industrial and Applied Mathematics 一般社団法人 日本応用数理学会 30 (3) 249 - 258 2020 [Refereed][Not invited]
Abstract. This article proves that each solution of Yamada’s partial differential equation and thermal equation gives normal vector filed of shapes. First, each problem setting is defined and numerical solutions of the equations are provided using the Finite Element Method. Next, each theorem is given and proved. - YAMADA Takayuki, MASAMUNE Jun, TERAMOTO Hiroshi, HASEBE Takahiro, KURODA HirotoshiTransactions of the JSME (in Japanese) 一般社団法人 日本機械学会 85 (877) 19 - 00129-19-00129 2019 [Refereed][Not invited]
This paper aims to develop a scheme for geometrical feature constraints in topology optimization for Additive Manufacturing (AM) without support structures based on the Partial Differential Equation (PDE) of geometrical shape features. To begin with, the basic concept of topology optimization and a level set-based topology optimization method are briefly described. Second, the PDE system for geometrical shape features is formulated. Here, aspects of the distribution of state variables are discussed using an analytical solution of the PDE. Based on the discussion, a function indicating the extended normal vector including geometrical singularity points is formulated. Third, geometrical requirements of product shape in AM without support structures – the so-called overhang constraint – are clarified in two-dimensions. A way of extending of the proposed concept to three-dimensional problems is also clarified. Additionally, geometrical singularities in the overhang constraint are discussed. Based on the PDE system and the clarified geometrical requirements, the overhang constraint including geometrical singularities is formulated. A topology optimization problem of the linear elastic problem is formulated considering the overhang constraint. A level set-based topology optimization algorithm is constructed where the Finite Element Method (FEM) is used to solve the governing equation of the linear elastic problem and the PDE, and to update the level set function. Finally, two-dimensional numerical examples are provided to confirm the validity and utility of the proposed method.
- Riku Takahashi, Yumihiko Ikura, Daniel R. King, Takayuki Nonoyama, Tasuku Nakajima, Takayuki Kurokawa, Hirotoshi Kuroda, Yoshihiro Tonegawa, Jian Ping GongSOFT MATTER 12 (23) 5081 - 5088 1744-683X 2016 [Refereed][Not invited]
Most studies on hydrogel swelling instability have been focused on a constrained boundary condition. In this paper, we studied the mechanical instability of a piece of disc-shaped hydrogel during free swelling. The fast swelling of the gel induces two swelling mismatches; a surface-inner layer mismatch and an annulus-disc mismatch, which lead to the formation of a surface crease pattern and a saddle-like bulk bending, respectively. For the first time, a stripe-like surface crease that is at a right angle on the two surfaces of the gel was observed. This stripe pattern is related to the mechanical coupling of surface instability and bulk bending, which is justified by investigating the swelling-induced surface pattern on thin hydrogel sheets fixed onto a saddle-shaped substrate prior to swelling. A theoretical mechanism based on an energy model was developed to show an anisotropic stripe-like surface crease pattern on a saddle-shaped surface. These results might be helpful to develop novel strategies for controlling crease patterns on soft and wet materials by changing their three-dimensional shape. - Yoshikazu Giga, Hirotoshi KurodaCOMMUNICATIONS ON PURE AND APPLIED ANALYSIS 14 (1) 121 - 125 1534-0392 2015/01 [Refereed][Not invited]
For a very strong diffusion equation like total variation flow it is often observed that the solution stops at a steady state in a finite time. This phenomenon is called a finite time stopping or a finite time extinction if the steady state is zero. Such a phenomenon is also observed in one-harmonic map flow from an interval to a unit circle when initial data is piecewise constant. However, if the target manifold is a unit two-dimensional sphere, the finite time stopping may not occur. An explicit example is given in this paper. - Fourth-order total variation flow with Dirichlet condition: Characterization of evolution and extinction time estimatesY. Giga, H. Kuroda, H. MatsuokaAdvances in Mathematical Sciences and Applications 24 (2) 499 - 534 2014 [Refereed][Not invited]
- Approximating problems of vectorial singular diffusion equations with inhomogeneous terms and numerical simulationsH. Kuroda, N. YamazakiDiscrete and Continuous Dynamical Systems 2009, Dynamical Systems, Differential Equations and Applications. 7th AIMS Conference, Supplement 486 - 495 2009 [Refereed][Not invited]
- The Dirichlet problems with singular diffusivity and inhomogeneous termsH. KurodaAdvances in Mathematical Sciences and Applications 19 (1) 269 - 284 2009 [Refereed][Not invited]
- Yoshikazu Giga, Hirotoshi Kuroda, Noriaki YamazakiFREE BOUNDARY PROBLEMS: THEORY AND APPLICATIONS 154 209 - + 2007 [Refereed][Not invited]
We consider a gradient flow system of total variation with constraint. Our system is often used in the color image processing to remove a noise from picture. In particular, we want to preserve the sharp edges of picture and color chromaticity. Therefore, the values of solutions to our model is constrained in some fixed compact Riemannian manifold. By this reason, it is very difficult to analyze such a problem, mathematically. The main object of this paper is to show the global solvability of a constrained singular diffusion equation associated with total variation. In fact, by using abstract convergence theory of convex functions, we shall prove the existence of solutions to our models with piecewise constant initial and boundary data. - Y. Giga, H. Kuroda, N. YamazakiInterdisciplinary Information Sciences 東北大学 11 (2) 199 - 204 1340-9050 2005 [Refereed][Not invited]
We consider a constrained gradient system of total variation flow. Our system is often used in color image processing to remove a noise from picture. In this paper, using abstract convergence theory of convex functions, we show the global existence of solutions to our problem with piecewise constant initial data. - Yoshikazu Giga, Hirotoshi KurodaBoletim da Sociedade Paranaense de Matematica 22 (1) 9 - 20 0037-8712 2004 [Refereed][Not invited]
A gradient system of total variation is considered for a mapping from the unit disk to the unit sphere in R3. For a class of initial data it is shown that a solution of its Dirichlet problem loses its smoothness in finite time. © SPM.
MISC
- グラフに退化する領域上での放物型方程式の解の特異極限について黒田紘敏 第7回数学総合若手研究集会, Hokkaido university technical report series in mathematics 148- 77 -84 2011 [Not refereed][Not invited]
- 特異拡散方程式の近似問題について黒田紘敏 第31回発展方程式若手セミナー報告集 161 -172 2009 [Not refereed][Not invited]
Presentations
- ４階全変動流方程式の解の挙動について [Invited]黒田紘敏日本応用数理学会 2024年度年会 2024/09
- The behavior of solutions of the fourth order total variation flow equations [Not invited]黒田紘敏応用解析研究会, 早稲田大学 2024/04
- 4階の全変動流方程式の解の挙動について [Invited]日本数学会 2024年度年会 実函数論分科会 特別講演 2024/03
- Fractional time differential equations as a singular limit of the Kobayashi-Warren-Carter system [Not invited]非線形現象の数値シミュレーションと解析2024 2024/03
- The fourth-order total variation flow in $\mathbb{R}^n$ [Not invited]The 25th Northeastern Symposium on Mathematical Analysis 2024/02
- The fourth-order total variation flow in $\mathbb{R}^n$ [Not invited]Yoshikazu Giga, Hirotoshi Kuroda, Michał Łasica10th International Congress on Industrial and Applied Mathematics 2023/08
- 全空間における4階の全変動流方程式の球対称解 [Not invited]黒田紘敏日本数学会2022年度秋季総合分科会 2022/09
- 特異拡散方程式の解の生存時間の評価について [Invited]第8回室蘭連続講演会 2022/03
- Characterization of Structure by Persistent Homology [Invited]北海道大学 理学部 × WPI-ICReDD合同シンポジウム 2020/12
- The behavior of the Laplacian with Neumann boundary condition on thin domains [Not invited]The 14th SNU-HU Symposium of Mathematics 2020 2020/11
- さまざまな特異拡散方程式の extinction-time とその評価 [Not invited]非線形現象の数値シミュレーションと解析2019 2019/03
- 確率のはなし～直感はどこまで信じられる？ [Not invited]北海道大学オープンキャンパス高校生限定プログラム 2018/08
- 均質化法の紹介 ―ミクロとマクロをつなぐ数学理論― [Not invited]学部1年生向けサイエンスグローブ「リガクの世界をのぞいてみない？」 2018/06
- 学位取得後に職を得るまでの流れについて [Not invited]博士院生・数学キャリアパスセミナー 2018/04
- 均質化法における有効拡散係数の決定法 [Not invited]均質化理論と局所体積平均理論の融合及びその新展開～構造体の迷路性と機械的分散効果に迫る～ 2017/12
- 北大物質科学リーディングプログラムにおける数理連携の取り組みの紹介 [Not invited]白田記念会 2016/08 北海道大学
- 数理連携を推進するアクティブラーニング－数学と化学の融合を目指して－ [Not invited]自然科学のためのアクティブラーニング 2016/03
- The behavior of the Laplace operator on a thin domain which degenerates into a graph [Not invited]第2回リーディングプログラム国際シンポジウム“Ambition Across the Disciplines” 2014/12
- An example of Non-finite time stopping solution for 1-harmonic map flow equation [Not invited]北海道大学偏微分方程式セミナー 2014/04
- The behavior of the Laplacian with Neumann boundary condition on thin domains [Not invited]The 37th Sapporo Symposium on Partial Differential Equations 2012/08
- グラフに退化する領域上のラプラシアンの特異極限 [Not invited]なかもず解析セミナー 2012/06
- グラフに退化する領域上のラプラシアンの挙動について [Not invited]作用素論セミナ－ 2012/05
- グラフに退化する領域上での放物型方程式の解の特異極限について [Not invited]第7回数学総合若手研究集会 2011
- グラフへ退化する細い領域上の Neumann Laplacian を定める quadratic form の Mosco 収束 [Not invited]日本数学会2011年度秋季総合分科会 2011
- ごく細い領域上にあるラプラシアンの特異極限に関する考察 [Not invited]JST「数学」第2回領域シンポジウム越境する数学～CREST 研究報告会～ 2011
- グラフに退化する領域上での放物型方程式の解の特異極限について [Not invited]第7回数学総合若手研究集会 2011
- グラフへ退化する細い領域上の Neumann Laplacian を定める quadratic form の Mosco 収束 [Not invited]日本数学会2011年度秋季総合分科会 2011
- ごく細い領域上にあるラプラシアンの特異極限に関する考察 [Not invited]JST「数学」第2回領域シンポジウム越境する数学～CREST 研究報告会～ 2011
- ごく細い領域における演算子の振る舞いについて [Not invited]異分野融合シンポジウム--新エネルギー・材料創生に向けて-- 2010
- 細い領域におけるラプラシアンの振る舞いについて [Not invited]日本応用数理学会2010年度年会 2010
- グラフに退化する領域上のラプラシアンの特異極限 [Not invited]信州数理物理セミナー 2010
- On the limit behavior of solutions to parabolic equations on thin domains [Not invited]CREST小谷チーム中間報告会 2010
- ごく細い領域における演算子の振る舞いについて [Not invited]異分野融合シンポジウム--新エネルギー・材料創生に向けて-- 2010
- 細い領域におけるラプラシアンの振る舞いについて [Not invited]日本応用数理学会2010年度年会 2010
- グラフに退化する領域上のラプラシアンの特異極限 [Not invited]信州数理物理セミナー 2010
- On the limit behavior of solutions to parabolic equations on thin domains [Not invited]CREST小谷チーム中間報告会 2010
- The stationary solutions of the singular diffusion equation with inhomogeneous terms [Not invited]The 10th Northeastern Symposium on Mathematical Analysis 2009
- 非斉次項をもつ特異拡散方程式の近似問題 [Not invited]第31回発展方程式若手セミナー 2009
- The stationary solutions of the singular diffusion equation with inhomogeneous terms [Not invited]The 10th Northeastern Symposium on Mathematical Analysis 2009
- 非斉次項をもつ特異拡散方程式の近似問題 [Not invited]第31回発展方程式若手セミナー 2009
- Evolution problems for singular diffusion equations with inhomogeneous terms [Not invited]The 9th Northeastern Symposium on Mathematical Analysis 2008
- Nonlinear problems with singular diffusivity and inhomogeneous terms [Not invited]パターンダイナミクスの数理とその周辺 2008
- 非斉次項をもつ特異拡散方程式の定常問題について [Not invited]日本数学会 2008年度 秋季総合分科会 実関数論分科会 2008
- Evolution problems for singular diffusion equations with inhomogeneous terms [Not invited]The 9th Northeastern Symposium on Mathematical Analysis 2008
- Nonlinear problems with singular diffusivity and inhomogeneous terms [Not invited]パターンダイナミクスの数理とその周辺 2008
- 非斉次項をもつ特異拡散方程式の定常問題について [Not invited]日本数学会 2008年度 秋季総合分科会 実関数論分科会 2008
- ここ数年の答案の変化から見た新教育課程 [Not invited]北海道算数数学教育会高等学校部会 石狩支部研究会 2007
- ここ数年の答案の変化から見た新教育課程 [Not invited]北海道算数数学教育会高等学校部会 石狩支部研究会 2007
- Global solvability of constrained singular diffusion equation associated with essential variation [Not invited]International School on Partial Differential Equations 2006
- Global solvability of constrained singular diffusion equation associated with essential variation [Not invited]International School on Partial Differential Equations 2006
- An existence result for a discretized 1-harmonic map flow [Not invited]The 6th Northeastern Symposium on Mathematical Analysis 2005
- 離散化された束縛条件付全変動流の解の存在と大域可解性 [Not invited]日本数学会 2005年度 秋季総合分科会 実関数論分科会 2005
- An existence result for a discretized 1-harmonic map flow [Not invited]The 6th Northeastern Symposium on Mathematical Analysis 2005
- 離散化された束縛条件付全変動流の解の存在と大域可解性 [Not invited]日本数学会 2005年度 秋季総合分科会 実関数論分科会 2005
- On breakdown of solutions of constrained gradient system of total variation [Not invited]The 5th Northeastern Symposium on Mathematical Analysis 2004
- On breakdown of solutions of constrained gradient system of total variation [Not invited]Variational Problems and Related Topics, Reserch Institute for Mathematical Sciences 2004
- 束縛条件付全変動流の解の爆発 [Not invited]日本数学会 2004年度 秋季総合分科会 函数方程式論分科会 2004
- A behavior of solutions of 1-harmonic map flow equation [Not invited]Mathematical Aspects of Image Processing and Computer Vision 2004 2004
- An existence result for a discretized 1-harmonic map flow [Not invited]第30回発展方程式研究会 2004
- On breakdown of solutions of constrained gradient system of total variation [Not invited]The 5th Northeastern Symposium on Mathematical Analysis 2004
- On breakdown of solutions of constrained gradient system of total variation [Not invited]Variational Problems and Related Topics, Reserch Institute for Mathematical Sciences 2004
- 束縛条件付全変動流の解の爆発 [Not invited]日本数学会 2004年度 秋季総合分科会 函数方程式論分科会 2004
- A behavior of solutions of 1-harmonic map flow equation [Not invited]Mathematical Aspects of Image Processing and Computer Vision 2004 2004
- An existence result for a discretized 1-harmonic map flow [Not invited]第30回発展方程式研究会 2004
Association Memberships
Research Projects
- Development of the theory of diffusion equations for analysis on data separationJapan Society for the Promotion of Science：Grants-in-Aid for Scientific Research Challenging Research (Pioneering)Date (from‐to) : 2018/06 -2024/03Author : 儀我 美一, 石毛 和弘, 行木 孝夫, 黒田 紘敏データ分離問題は機械学習の分野では基本的な問題であり、さまざまな解析手法が提案されている。この種の問題に対して、離散数学的手法を用いて何らかの評価関数を最小にするものを求める方法が主であった。しかし、データ数が増えると、離散的手法は計算量が増えて困難になる。流体力学の研究に見られるように、各分子の動きをすべて追跡するよりも、いわゆる連続体近似を行って、平均量を解析したほうが、少ない計算量で知りたい結果が多い。そこで、データサイエンスにかかわる基本的な問題である2値分離、クラスタリング、時系列分離問題に絞り、現在の研究状況を分析した。そのために2018年12月18日に「データ分離問題の基礎と新展開」というセミナーを開催した。その結果、偏微分方程式的な研究手法が、今後需要が伸びそうであることを確認し、具体的な問題に取り組んでいる。 例えばクラスタリングに用いようとしている小林-Warren-Caterモデルは、全変動エネルギーに対するAmbrosio-Tortorelli近似とみなせる。ディリクレエネルギーに対するAmbrosio-Tortorelli近似の極限は、Munford-Sheh汎関数になることが知られている。全変動エネルギーの場合どうなるかを現在研究しているところである。一方、全変動流方程式や、伝播モデルについての数学解析は着実に進行している。 例えば空間離散型の全変動写像流方程式について、それを近似する凸型変分的時間離散スキームを作り、その収束を証明することに成功した。これは例えば、回転群に値を取る全変動写像流方程式の解の計算を容易にするものとして注目されている。
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2019/04 -2022/03Author : Yamada TakayukiIn this study, the geometric constraints required from the manufacturing process are formulated by using partial differential equations. These partial differential equations are called fictitious physical model because it is a fictitious field introduced to represent manufacturability. We also integrated it with the topology optimization method to create a new design method that integrates design and production. Furthermore, by considering assemblability, we proposed a method that also allows topology optimization of mechanical structures composed of multiple parts.
- Japan Society for the Promotion of Science：Grants-in-Aid for Scientific ResearchDate (from‐to) : 2013/04 -2016/03Author : GIGA Yoshikazu, ASAI Tomoro, OHTSUKA Takeshi, GIGA Mi-Ho, KURODA Hirotoshi, NAKAYASU Atsushi, HAMAMUKI NaoWe consider the Eikonal equation in a space such as network or fractal, where the gradient of function is not well-defined in canonical way. We establish the theory of viscosity solutions in a general metric space. We also establish the theory of viscosity solutions for a curvature flow equation describing motion of a surface of a crystal or a grain boundary, especially a crystalline curvature flow, which has a strong anisotropy, when the surface is regarded as a curve. A curvature flow with strong anisotropy is regarded at least formally as a gradient flow of area measured by non-Euclidean metric in a suitable metric space. However, a general theory is not yet established so we study the problem individually.