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Research topics

• High-precision eigenvalue estimation for differential operators

  The finite element method is adopted to give concrete bounds for eigenvalue of differential operators. The upper bound for eigenvalues has been well done in the history. However, to give lower bound eigenvalue is quite a difficult task. Compared with the classical research on eigenvalue evaluation, which only focused on approximate value of eigenvalue, new theories and computation methods are developed to give exact and sharp lower and upper bounds for the eigenvalues.

• Quantitative error estimation for finite element method

  The classical error analysis theory for finite element method mainly focuses on the qualitative one, such as the convergence order, the stability. However, in many cases, for example the computer-assisted mathematical proof, we need the concrete values of approximation errors. For this purpose, we need to challenge the following problems, which have close relation with the eigenvalue problem of differential operators.

  • to give not only the boundedness but also the concrete values of many error constants in the error analysis.
  • to deal with the singularity of PDE solutions in case that the domain has an re-entrant corner.

  I have interest in studying the finite element methods including,

  • conforming FEM, non-conforming FEM, mixed FEM and hybrid FEM.

  
  

• Verified computation

  "Verified computation" means solving various problems to provide veirified solutions, i.e., mathematically correct results.

  In floating-point number based numerical computation, there comes rounding error, which makes the computation result to be an approximate solution. To give mathematically exact result, the interval airthmetic proves to be powerful tool to bound the rounding error.

  One application of verified computaton is to investigate the existence and uniqueness of solution to non-linear partial differential equations.

• Four-probe resistence measurement

  
  Supported by JKA RING!RING! Project.
  競輪・オートレース補助事業

  The Four-probe measurement of resistivity of semi-product material is related to the following Poisson's boundary value problem.
  

  \( -\Delta u = f \text{ in } \Omega, {\partial{u}}/{{\partial n}} =0 \text{ on } \partial \Omega \)

  By solving the above problem in 3D with proper processing of singularities, the resistivity can be calculated. See detail.

  • This is research is supported by JKA RING!RING! Project.

Papers

Journal papers

• Google Scholar Citations Report .
  

1. Qin LI, Xuefeng LIU, Explicit Finite Element Error Estimates for Nonhomogeneous Neumann problems, Applications of Mathematics, Vol. 63, Issue 3, pp. 367-379, 2018. Link at arxiv.org, 2018/06/05.
2. Xuefeng LIU, Fumio KIKUCHI, Explicit Estimation of Error Constants Appearing in Non-conforming Linear Triangular Finite Element, Applications of Mathematics, Vol. 63, Issue 4, pp. 381–39, 2018. Link at arxiv.org, 2018/05/27.
3. Manting Xie, Hehu Xie, Xuefeng Liu, Explicit lower bounds for Stokes eigenvalue problems by using nonconforming finite elements, Vol. 35, Issue 1, pp. 335–354, 2018, March (link pdf)
4. Xuefeng Liu, Chun'guang You, Explicit Bound for Quadratic Lagrange Interpolation Constant on Triangular Finite Elements, Applied Mathematics and Computation, Vol.319, pp. 693-701, 2018. (arXiv URL)
5. Xuefeng LIU, Yuan LIU, and Zhouwang Yang, Paper review assignment problem under the minimum cost flow network model (ネットワークフローのモデルで論文査読者割当の問題を考える), Proceedings of the Twenty-Eighth RAMP Symposium, Niigata University, Niigata, October 13–14, 2016. PDF
6. Xuefeng Liu, A framework of verified eigenvalue bounds for self-adjoint differential operators, Applied Mathematics and Computation, 267, pp.341–355, 2015 (PDF)
7. Xuefeng Liu, Tomoaki Okayama and Shin’ichi Oishi High-Precision Eigenvalue Bound for the Laplacian with Singularities, Editors: Ruyong Feng, Wen-shin Lee and Yosuke Sato, Computer Mathematics, 311-323, Spinger, 2014. (PDF)
8. Kazuaki Tanaka, Akitoshi Takayasu, Xuefeng Liu and Shin'ichi Oishi, Verified norm estimation for the inverse of linear elliptic operators using eigenvalue evaluation, Japan J. Indust. Appl. Math. 31, pp.665–679, 2014 (PDF)
9. Akitoshi Takayasu, Xuefeng Liu and Shin'ichi Oishi,Remark on computable a priori error estimation for higher degree finite element solution of elliptic problem, NOLTA, IEICE, Vol.5, No.1, pp.53-63, 2014. PDF
10. Xuefeng Liu and Shin'ichi Oishi, Guaranteed high-precision estimation for P0 interpolation constants on triangular finite elements, Japan Journal of Industrial and Applied Mathematics 30 (3), 635-652 PDF
11. Xuefeng Liu and Shin'ichi Oishi, Verified eigenvalue evaluation for Laplacian over polygonal domain of arbitrary shape, SIAM J. Numer. Anal., 51(3), 1634-1654. (link PDF)
12. Xuefeng Liu and Shin'ichi Oishi, Verified eigenvalue evaluation for Laplace operator on arbitrary polygonal domain, RIMS Kokyuroku, 2011. (link, PDF)
13. A. Takayasu, X. Liu and S. Oishi, Verified computations to semilinear elliptic boundary value problems on arbitrary polygonal domains, NOLTA, IEICE, Vol.E96-N, No.1, pp.34-61, Jan. 2013. (PDF)
14. Xuefeng Liu and Fumio Kikuchi, Analysis and estimation of error constants for P0 and P1 interpolations over triangular finite elements, J. Math. Sci. Univ. Tokyo, 17, p.27-78, 2010 (PDF)
15. Fumio Kikuchi and Xuefeng Liu, Estimation of interpolation error constants for the P0 and P1 triangular finite elements, Computer Methods in Applied Mechanics and Engineering, Vol.196, Issues 37-40, (2007), pp. 3750-3758. PDF
16. Xuefeng Liu and Fumio Kikuchi, Estimation of Error Constants Appearing in Non-conforming Linear Triangular Finite Element, Proceeding of APCOM’07 in conjunction with EPMESC XI (digital version only), December 3-6, 2007, Kyoto, JAPAN. PDF
17. Fumio Kikuchi and Xuefeng Liu, Determination of the Babuska-Aziz constant for the linear triangular finite element, Japan Journal of Industrial and Applied Mathematics (JJIAM), Vol.23, No.1, (2006), pp. 75-82. PDF

(Paper finished in USTC, China.)
18. Ghulam Mustafa and Xuefeng Liu, A subdivision scheme for volumetric models. Appl. Math.　J. Chinese Univ. Ser. B, Vol 20, no.2, p.213–224, 2005
19. Ghulam Mustafa and Xuefeng Liu, A New Solid Subdivision Scheme, Journal of University of Science and Technology of China, Vol.35(3), p.285-300, 2005
20. Xuefeng Liu, Blending Pipe Surface by Ringed Surface and Its Continuity, The Journal of　Uni. of Sci. & Tech. of China, Vol.34 No.1, P.20-P28, 2004
21. Xuefeng Liu, Jin Xu and Ronggang Bai, 3D Rebuilding of Vessel, Chinese Journal of Engineering Mathematics, Vol. 19 Supp., p.35-40, 2002.

学位論文

1. 博士論文： 適合および非適合1 次有限要素の誤差定数の解析(Analysis of error constants for linear　 conforming and nonconforming finite elements)　東京大学数理科学研究科2009年3月 （指導教官：菊地文雄） ( PDF download )
2. 修士論文：０次および１次三角形有限要素の補間誤差定数の評価 (Estimation of error constants for P0 and P1 interpolations over triangular finite elements)　東京大学数理科学研究科2006年3月（指導教官：菊地文雄）

Presentations:

招待講演

1. Xuefeng LIU, Computable error estimation for finite element method and computer-assisted proof, University of Science and Technology of China, Hefei, China, 2011/12/28
2. Xuefeng LIU, Mini-Course: Computer-assisted proof based on finite element method--the challenge from finite to infinite, University of Science and Technology of China, Hefei, China, 2011/12/23-27
3. Xuefeng LIU, On computable eigenvalue evaluation for elliptic eigen-problem with singularity, Japanese-German Workshop on Computer-Assisted Proofs and Verification Methods, Karlsruhe, Germany, 2011/09
4. 劉雪峰：任意多角形領域での有限要素法の事前誤差評価と楕円型微分作用素の固有値評価，線形計算研究会，東京大学(2010/12/20)．
5. 劉雪峰：任意多角形領域上での楕円型作用素の精度保証付き評価，日本応用数理学会 若手の会，国立情報学研究所(2010/11/25)．
6. 劉雪峰：任意多角形領域上での楕円型作用素に対する精度保証付き評価, 東京大学数値解析セミナー，東京大学大学院数理科学研究科(2010/11/16)．

学会発表

国際
1. Xuefeng Liu, Shin'ichi Oishi, On High Precision Eigenvalue Estimation for Self-adjoint Elliptic Differential Operator and Its Application, The Tenth Asian Symposium on Computer Mathematics (ASCM 2012), Beijing, China, 2012/10.
2. Xuefeng Liu, Shin'ichi Oishi, On guaranteed eigenvalue estimation of compact differential operator with singularity, 2012 International Symposium on Nonlinear Theory and its Applications(NOLTA 2012), Palma Majorca, Spain, 2012/10.
3. Xuefeng Liu, Shin'ichi Oishi, A framework of high precision eigenvalue estimation for selfadjoint elliptic differential operator, 15th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Verified Numerics (SCAN'2012), Novosibirsk,Russia, 2012/9.
4. Xuefeng Liu, A framework on high precision eigenvalue value evaluation for self-adjoint elliptic operator, 4th China-Japan-Korea Conference on Numerical Mathematics, 2012/08
5. Xuefeng Liu, Shin'ichi Oishi, On verified eigenvalue evaluation of self-ajoint elliptic differential operator, The 8th EASIAM, Taipei, Taiwan, 2012/06
6. Xuefeng Liu, Shin'ichi Oishi, On Verified Computation of laplacian Eigenvalues Over Polygonal Domain, 2011 International Symposium on Nonlinear Theory and its Applications (NOLTA 2011), Kobe, Hyogo,Japan, 2011/09
7. Xuefeng Liu, Shin'ichi Oishi, Numerical verification for solution existence of elliptic PDE on arbitrary polygonal domain, The 7th EASIAM, Kitakyushu Campus, Waseda, Japan, 2011/06
8. Xuefeng Liu, Shin'ichi Oishi, Computer Assisted Eigenvalue Bounds, ICIAM 2011, Vancouver, Canada, 2011/07/22
9. Xuefeng Liu, High precision eigenvalue bounds for laplacian over general polygonal domain, Workshop on Analytic and Computational Techniques in Spectral Theory and 2. Related Topics, Gregynog Hall, Cardiff University, UK. 2011/06/18-24
10. Verified Eigenvalue Evaluation for Elliptic Operator and Its Application, Applied Mathematics International Conference 2010 & The Sixth East Asia SIAM Conference, Kuala Kumpur, Malaysia (2010/06/23).
11. Guaranteed Eigenvalue Evaluation for Elliptic Problem over General Polygonal Domain, 14th GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic, and Validated Numerics (SCAN 2010), ENS de Lyon, Lyon, France 2010/9.
12. Estimation of Error Constants Appearing in Non-conforming Linear Triangular Finite Element, APCOM’07-EPMESC XI, Kyoto, Japan, 2007/12

国内
1. 南畑淳史,劉雪峰,大石進一, 非凸領域における特異関数とスプライン関数を用いたラプラス作用素の高精度な固有値評価, 日本応用数理学会2012年度年会, 北海道稚内市, 2012/8
2. 田中一成,高安亮紀, 劉雪峰, 大石進一, 線形楕円型作用素のNeumann条件下における精度保証付き逆作用素ノルム評価, 日本応用数理学会2012年度年会, 北海道稚内市, 2012/8
3. 菊井知美,劉雪峰,大石進一, Laplace作用素の固有値評価を用いた2D形状認識と応用, 日本応用数理学会2012年度年会, 北海道稚内市, 2012/8
4. 劉雪峰,大石進一, 自己共役楕円型微分作用素の高精度な固有値評価のについて, 日本応用数理学会2012年度年会, 北海道稚内市, 2012/8.
5. 劉雪峰,大石進一, On verified evaluation of several interpolation error constants, 日本応用数理学会2011年度年会, 京都, 2011年9月.
6. 劉雪峰,有限要素法の計算的な誤差評価とその応用,第33回発展方程式若手セミナー, 茨城県つくば, 2011年8月　
7. 劉雪峰,荻田武史,摂動理論を用いた一般化固有値問題に対する精度保証法の改善，応用数理学会，研究部会連合発表会，電気通信大学，2011年3月
8. 劉雪峰,大石進一,楕円型作用素の固有値の精度保証付き評価とその応用，RIMS研究集会「科学技術計算アルゴリズムの数理的基盤と展開」，京都大学数理解析研究所，2010年10月19日
9. 劉雪峰,大石進一,楕円型作用素の固有値の精度保証付き評価とその応用，日本応用数理学会2010 年度年会，明治大学，2010年９月８日
10. 劉雪峰,大石進一,楕円型作用素の固有値の精度保証付き評価とその応用，第29回 日本シミュレーション学会大会，山形大学工学部，2010年6月19日
11. 劉雪峰,大石進一,楕円型微分作用素の精度保証付き固有値評価と応用，第39回数値解析シンポジウム，鳥羽シーサイドホテル，2010年5月27日
12. 劉雪峰,菊地文雄,Estimation of error constants appearing in FEM, Seminar on "Accuracy and quality of numerical methods", 東京大学,2008年8月
13. 劉雪峰,菊地文雄,非適合三角形1 次有限要素法の解析と適用, 応用数学合同研究集会, 龍谷大学, 2007年12月
14. 劉雪峰,菊地文雄,非適合三角形１次有限要素に現れる誤差定数, 応用数学合同研究集会, 龍谷大学, 2006年12月
15. 劉雪峰,菊地文雄,０次および1 次三角形有限要素の補間誤差定数の評価, 応用数学合同研究集会, 龍谷大学, 2005年12月

Others

1. Scilab Toolbox Contest 最優秀賞［一般カテゴリ］、劉　雪峰、2012年10月29日
2. 若手研究（B）23740092　楕円型作用素の精度保証つき固有値評価と非線形問題への応用，2011－2013．
3. 独立行政法人情報処理推進機構（IPA）2008年度下期未踏本体,「A Cloud Computing-based Information Sharing System for Mobiles」研究開発のチーフクリエータ， 2008年12月～2009年8月, 700万円.