Homepage of Hehu Xie

Hehu Xie

Software Curriculum Vitae Research Topics Research Reports Some Publications

Institute of Computational Mathematics

Academy of Mathematics and Systems Sciences Chinese Academy of Sciences

Beijing 100190, P.R. China

Telephone: 0086-10-82541026

Fax: 0086-10-82541993

E-Mail: hhxie@lsec.cc.ac.cn

My Research Introduction (中文版)

Software

FEM_MATLAB: A Matlab software that implements the multilevel correction method.
Full_Eigen_Multigrid: A baltamatica software that implements the multilevel correction method for solving eigenvalue problems.
GCGE: a software for solving large scale eigenvalue problems.
北太天元版本的GCGE插件: 基于北太天元SDK建立的特征值解法器GCGE_bex, 计算效率请见GCGE_bex的测试报告
OpenPFEM: a parallel finite element analysis software(1D,2D,3D).
PASE: a parallel augmented subspace eigensolver.
特征值问题数值算法讲义: 课程特征值问题数值算法的讲义, 相应的视频如下: 特征值问题数值算法视频1 (Bilibili网站) 特征值问题数值算法视频2 (Bilibili网站)
TNN_MachineLearning: A Pytorch software for tensor neural network (TNN) and its corresponding machine learning methods for solving hihg dimensional parital differential equations. About TNN, please check its introudction: English Version (中文版)
all distributions
个人服务器计算环境搭建教程

Curriculum Vitae

Working experience

2016, March - Now, Professor,
Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences
2011, March - 2016, February, Associate Professor,
Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences
2008, July-2011, February, Assistant Professor,
Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences

Education

2003,Sep.-2008,June Ph.D candidate
Academy of Mathematics and Systems Science, Chinese Academy of Sciences
Supervisor: Professor Qun Lin
1999,Sep.-2003,July B.Sc.
Peking University, Beijing, China

Research Topics

Numerical Methods for Eigenvalue Problems
Neural Networks and Machine Learning Methods(TNN算法获得2023年"大湾区"粤港澳AI for Science科技竞赛特等奖(获得了最好的精度))
Numerical Methods for Partial Differential Equations
Finite Element Methods: Accuracy and Improvement
Computational Softwares and Theirs Applications
Numerical Method for Differential-Integral Equations and Ordinary Differential Equations

Research Introduction:

Research Reports

Yifan Wang and Hehu Xie, Computing multi-eigenpairs of high-dimensional eigenvalue problems using tensor neural networks, arXiv:2305.12656, 2023.
Yifan Wang and Hehu Xie, Solving high dimensional partial differential equations using tensor type discretization and optimization process, arXiv:2211.16548, 2022.
Yifan Wang, Zhongshuo Lin, Yangfei Liao, Haochen Liu and Hehu Xie, Solving high dimensional partial differential equations using tensor neural network and a posteriori error estimators, arXiv:2311.02732, 2023.
Yifan Wang, Yangfei Liao and Hehu Xie, Solving Schrödinger equation using tensor neural network. arXiv:2209.12572, 2022.
Yifan Wang, Pengzhan Jin and Hehu Xie, Tensor neural network and its numerical integration. arXiv:2207.02754, 2022 (accepted by Journal of Computational Mathematics).
Yunhui He, Qichen Hong, Hehu Xie, Meiling Yue and Chunguang You, Energy Error Estimates of Subspace Method and Multigrid Algorithm for Eigenvalue Problems. arXiv:1705.03038.
Xiaole Han, Yunhui He, Hehu Xie, Chunguang You, An Algebraic Multigrid Method for Eigenvalue Problems arXiv:1503.08462, 2015.
Hehu Xie and Chunguang You, A Small Note About Lower Bound of Eigenvalues, report, 2016.
Hehu Xie and Zhimin Zhang, A multilevel correction scheme for nonsymmetric eigenvalue problems by finite element methods, arXiv:1505.06288
Hehu Xie, A Multigrid Method for Nonlinear Eigenvalue Problems, arXiv:1405.7158， 2014.

Xiaole Han and Hehu Xie, Local a Priori Estimate on the General Scale Subdomains, arXiv:1401.5373, 2014.

Some Publications

87. Guanghui Hu, Hehu Xie and Fei Xu, On accelerating a multilevel correction adaptive finite element method for Kohn-Sham equation, Journal of Computational Physics, 472 (2023) 111674.

86. Qiang Du, Hehu Xie and Xiaobo Yin, On the convergence to local limit of nonlocal models with approximated interaction neighborhood, SIAM Journal On Numerical Analysis, 60(4) (2022), 2046-2068.

85. Ning Zhang, Xiaole Han, Yunhui He, Hehu Xie and Chunguang You, An algebraic Multigrid Method for Eigenvalue Problems and Its Numerical Tests, East Asian Journal on Applied Mathematics,11(1) (2021), 1-19

84. Fei Xu, Hehu Xie, Manting Xie and Meiling Yue, A Multigrid Method for the Ground State Solution of Bose–Einstein Condensates Based on Newton Iteration, BIT Numerical Mathematics,61 (2021), 645-663.

83. 张宁、李瑜、谢和虎、徐然、游春光, 一种求解特征值问题的广义共轭梯度算法,中国科学:数学, 51(8) (2021), 1297-1320.

82. Fei Xu, Hehu Xie and Ning Zhang, A parallel augmented subspace method for eigenvalue problems, SIAM Journal on Scientific Computing, 42(5) (2020), A2655-A2677.

81. 谢和虎, 子空间扩展算法及其应用,数值计算与计算机应用,41(3) (2020),169-191.

80. Yu Li, Hehu Xie, Ran Xu, Chunguang You and Ning Zhang, A parallel generalized conjugate gradient method for large scale eigenvalue problems, CCF Transactions on High Performance Computing, 2 (2020) , 111–122.

79. Jian Tian, Hehu Xie, Kai Yang and Ran Zhang, Analysis of continuous collocation solutions for nonlinear functional equations with vanishing delays, Computational and Applied Mathematics 39 (2020) 39:23.

78. Fusheng Luo, Hehu Xie, Manting Xie and Fei Xu, Adaptive time-stepping algorithms for molecular beam epitaxy: Based on energy or roughness, Applied Mathematics Letters, 99(2020), 105991.

77. 李瑜、谢和虎，基于特征线法的群体平衡系统的数值模拟, 数值计算与计算机应用, 40(4) (2019), 261-278.

76. Hehu Xie, Lei Zhang, Houman Owhadi, Fast eigenpairs computation with operator adapted wavelets and hierarchical subspace corrections, SIAM Journal on Numerical Analysis, 57(6) (2019), 2519-2550.

75. Qilong Zhai, Hehu Xie, Ran Zhang and Zhimin Zhang, Acceleration of weak Galerkin methods for the Laplacian eigenvalue problem, Journal of Scientific Computing, 79 (2019), 914-934.

74. Ning Zhang, Fei Xu and Hehu Xie, An efficient multigrid method for ground state solution of Bose-Einstein condensates, International Journal of Numerical Analysis and Modeling, 16(5) (2019), 789-803.

(Contribution: in this paper, we solve a nonlinear eigenvalue problem with the absolute asymptotically optimal complexity independent from the nonlinearity.)

73. Hehu Xie, Manting Xie and Ning Zhang, An efficient multigrid method for semilinear elliptic equation, Journal of Numerical Methods and Computer Applications, 40(2) (2019), 143-160.

谢和虎、谢满庭、张宁, 一种求解半线性椭圆问题的快速多重网格法，数值计算与计算机应用, 40(2) (2019), 143-160.

72. Qilong Zhai, Hehu Xie, Ran Zhang and Zhimin Zhang, The weak Galerkin method for elliptic eigenvalue problems, COmmun. Comput. Phys., 26(1) (2019), 160-191.

71. Chun'guang You, Hehu Xie and Xuefeng Liu, Guaranteed eigenvalue bounds for the Steklov eigenvalue problem, SIAM J. Numer. Anal., 57(3) (2019), 1395-1410.

70. Meiling Yue, HehuXie and Manting Xie, A cascadic multigrid method for nonsymmetric eigenvalue problem, Applied Numerical Mathematics, 146 (2019), 55-72.

69. Qichen Hong, Hehu Xie, Meiling Yue and Ning Zhang, Fully computable error bounds for eigenvalue problem, International Journal of Numerical Analysis and Modeling, 15 (2018), 2560-2576.

68. Guanghui Hu, Hehu Xie and Fei Xu, A multilevel correction adaptive finite element method for Kohn–Sham equation, Journal of Computational Physics, 355(15), (2018), 436-449.

67. Y. He, Y. Li, H. Xie, C. You and N. Zhang, A multilevel Newton's method for eigenvalue problems, Applications of Mathematics, 63(3) (2018), 281-303.

66. M. Xie, H. Xie and X. Liu, Explicit lower bounds for Stokes eigenvalue problems by using nonconforming finite elements, Japan Journal of Industrial and Applied Mathematics, 35(1) (2018), 335-354.

65. Y. Di, Hehu Xie and X. Yin, Anisotropic meshes and stabilization parameter design of linear SUPG method for 2D convection-dominated convection-diffusion equations, Journal of Scientific Computing, 76(1) (2018), 48-68.

64. Q. Hong, H. Xie, M. Yue and N. Zhang, Fully computable error bounds for eigenvalue problem, International Journal of Numerical Analysis and Modleing, 15(1-2) (2018), 260–276.

63. Fei Xu and Hehu Xie, A full multigrid method for semilinear elliptic equation, Applications of Mathematics, 62(3) (2017), 225-241.

62. Hehu Xie and Xinming Wu, A multilevel correction method for interior transmission eigenvalue problem, Journal of Scientific Computing, 72 (2017), 586-604.

61. Wei Gong, Hehu Xie and Ningning Yan, Adaptive multilevel correction method for finite element approximations of elliptic optimal control problems, Journal of Scientific Computing, 72 (2017), 820-841.

60. Xia Ji, Yingxia Xi and Hehu Xie, Nonconforming finite element method for the transmission eigenvalue problem, Advances in Applied Mathematics and Mechanics, 9(1) (2017), 92-103.

59. Xiaole Han, Hehu Xie and Fexi Xu, A cascadic multigrid method for eigenvalue problem, Journal of Computational Mathematics, 35(1) (2017), 75-91.

58. Hehu Xie and M. Xie, A multigrid method for ground state solution of Bose-Einetein condensates, Commun. Comput. Phys., 19(3) (2016),648-662.

57. S. Jia, Hehu Xie, M. Xie and F. Xu, A full multigrid method for nonlinear eigenvalue problems, Sci China Math, 59 (2016), 2037–2048.

56. F. Luo, T. Tang and Hehu Xie, Parameter-free time adaptivity based on energy evolution for the Cahn-Hilliard equation, Commun. Comput. Phys., 19(05) (2016), 1542-1563.

55. H. Chen, Hehu Xie and F. Xu, A full multigrid method for eigenvalue problems, Journal of Computational Physics,322 (2016),747-759.

54. Xiaole Han, Yu Li, Hehu Xie and Chunguang You, Local and parallel finite element algorithm based on multilevel discretization for eigenvalue problems, International Journal of Numerical Analysis & Modeling, 13(1) (2016),73-89.

53. Zhonghua Qiao, Tao Tang and Hehu Xie, Error analysis of a mixed finite element method for molecular beam epitaxy model,SIAM J. Numer. Anal., 53(1) (2015）, 184–205.

52. Hehu Xie and Tao Zhou, A multilevel finite element method for Fredholm integral eigenvalue problems,Journal of Computational Physics,303 (2015),173-184

51. Hehu Xie, A type of multi-level correction scheme for eigenvalue problems by nonconforming finite element methods, BIT Numerical Mathematics, 55(4) (2015), 1243-1266.

50. Hongtao Chen,Yunhui He, Yu Li and Hehu Xie, A multigrid method for eigenvalue problems based on shifted-inverse power technique. European Journal of Mathematics, 1(1) (2015), 207-228.

49. Xiaole Han, Yu Li and Hehu Xie, A multilevel correction method for Steklov eigenvalue problem by nonconforming finite element methods, Numerical Mathematics: Theory, Methods and Applications, 8(03) (2015), 383-405.

48. Hehu Xie and Xiaobo Yin, Acceleration of stabilized finite element discretizations for the Stokes eigenvalue problem, Adv. Comput. Math., 41(2015), 799–812

47. 谢和虎, 非线性特征值问题的多重网格算法,中国科学：数学，45（8）（2015），1193-1204.

Hehu Xie, A multigrid method for nonlinear eigenvalue problems (in Chinese), Sci Sin Math, 45 (2015), 1193-1204

46. Qun Lin, Hehu Xie and Fei Xu, Multilevel correction adaptive finite element method for semilinear elliptic equation, Application of Mathematics, 60(5) (2015), 527-550.

45. Wei Gong, Hehu Xie, Ningning Yan, A multilevel correction method for optimal controls of elliptic equation, SIAM J. Sci. Comput., 37(5) （2015), A2198-A2221, 2015.

44. Qun Lin, Fusheng Luo and Hehu Xie, A multilevel correction method for Stokes eigenvalue problems and its application, Mathematical Methods in the Applied Sciences, 38(18) (2015), 4540-4552.

43. Shanghui Jia, Fusheng Luo and Hehu Xie, A Posterior Error Analysis for the Nonconforming Discretization of Stokes Eigenvalue Problem, Acta Mathematica Sinica, English Series, 30(6) (2014), 949–967.

42. X. Feng, Z, Wen and Hehu Xie, Acceleration of two-grid stabilized mixed finite element method for the Stokes eigenvalue problem, Application of Mathematics, 59(6) (2014), 615-630.

41. Hehu Xie, A type of multilevel method for the Steklov eigenvalue problem, IMA J. Numer. Anal., 34(2) (2014), 592-608.

40. Hehu Xie, A multigrid method for eigenvalue problem, Journal of Computational Physics, 274(1), 2014, pages 550-561.

39. Xia Ji, Jiguang Sun and Hehu Xie, A multigrid method for Helmholtz transmission eigenvalue problems, Journal of Scientific Computing, 60(2), pages: 276-294, 2014.

38. Qun Lin, Fusheng Luo and Hehu Xie, A posteriori error estimator and lower bound of a nonconforming finite element method, Journal of Computational and Applied Mathematics, 256(1), pages 243-254, 2014.

37. Qun Lin and Hehu Xie, A multi-level correction scheme for eigenvalue problems, Mathematics of Computation, 84(291) (2015), 71-88.

36. Xiaobo Yin and Hehu Xie, Metric tensors for the interpolation error and its gradient in $L^p$ norm, Journal of Computational Physics, 256(1), Pages: 543-562, 2014.

35. Qun Lin, Hehu Xie and Jinchao Xu, Lower Bounds of the Discretization for Piecewise Polynomials, Math. Comp., 83 (2014), 1-13.

34. Yu Li, Qun Lin and Hehu Xie, A parallel method for population balance equations based on the method of characteristics, Proceedings of the International Conference Applications of Mathematics, 2013, pp. 140-149.

33. Ran Zhang, Benxi Zhu and Hehu Xie, Spectral methods for weakly singular Volterra integral equations with pantograph delays，Frontiers of Mathematics in China, Vol. 8, Issue 2, 2013, pp 281-299

32. Tao Tang, Hehu Xie and Xiaobo Yin, High-Order Convergence of Spectral Deferred Correction Methods on General Quadrature Nodes, Journal of Scientific Computing Volume 56, Issue 1, 2013, pp 1-13

31. Qiumei Huang, Hehu Xie and Hermann Brunner, The $hp$ Discontinuous Galerkin Method for Delay Differential Equations with Nonlinear Vanishing Delay, SIAM J. Sci. Comput., 35(3), 2013, A1604–A1620.

30. Qin Li, Qun Lin and Hehu Xie,Nonconforming finite element approximations of the Steklov eigenvalue problem and its lower bound approximations, Applications of Mathematics, Volume 58, Issue 2, 2013, pp 129-151.

29. Shanghui Jia， Hongtao Chen and Hehu Xie, A posteriori error estimator for eigenvalue problems by mixed finite element method, Science China Mathematics,Volume 56, Issue 5, 2013, pp 887-900

28. Qun Lin and Hehu Xie, Extrapolation of the finite element method on general meshes, International Journal of Numerical Analysis and Modeling Computing and Information, Vol. 10, Number 1, 2013, pp. 139-153.

27. Qun Lin and Hehu Xie, Recent results on Lower bounds of eigenvalue problems by nonconforming finite element methods, Inverse Problems and Imaging, Volume 7, No. 3, 2013, 795-811.

26. F.Luo, Qun Lin and Hehu Xie (谢和虎), Computing the lower and upper bounds of Laplace eigenvalue problem: by combining conforming and nonconforming finite element methods, Science China Mathematics, Volume 55, Issue 5, pp 1069-1082, 2012.

25. Qun Lin and Hehu Xie (谢和虎), A Superconvergence result for mixed finite element approximations of the eigenvalue problem^,ESAIM: Mathematical Modelling and Numerical Analysis, Volume 46/Issue 04, pp 797-812, 2012.

24、 Qun Lin and Hehu Xie (谢和虎), A multilevel correction type of adaptive finite element method for Steklov eigenvalue problems, Proceedings of the International Conference Applications of Mathematics 2012, pp. 134-143.

23. Q. Lin and H. Xie, The asymptotic lower bounds of eigenvalue problems by nonconforming finite element methods, Mathematics in Practice and Theory, 42 (11) (2012), 219–226.(In Chinese with English title and abstract)

22. H. Xie, R. Zhang and H. Brunner, Collocation Methods for General Volterra Functional Integral Equations with Vanishing Delays, SIAM Journal on Scientific Computing, 33(6) (2011), 3303-3332.

21. Qun Lin and Hehu Xie, An observation on Aubin-Nitsche Lemma and its applications, Mathematics in Practice and Theory, 41(17) (2011), 247-258.(In Chinese, English title and abstract)

20. Qun Lin and Hehu Xie, A type of finite element gradient recovery method based on vertex-edge-face interpolation: The recovery rechnique and superconvergence property, East Asian Journal on Applied Mathematics, 1(3) (2011), 248-263.

19. S. Jia, H. Xie and X. Yin, A type of finite element gradient recovery method based on vertex-edge interpolation, Mathematics in Practice and Theory, 41(6) (2011), 239-243. (In Chinese, English title and abstract)

18. Q. Huang, H. Xie and H. Brunner, Superconvergence of a discontinuous Galerkin solutions for delay differential equations of pantograph type, SIAM Journal on Scientific Computing， 33(5) (2011), 2664–2684.

17. Qun Lin and Hehu Xie, Superconvergence measurement for General meshes by linear finite element method, Mathematics in Practice and Theory, 41(1)(2011), 138-152. (In Chinese, English title and abstract)

16. N. Ahmed, G. Matthies, L. Tobiska and H. Xie, Discontinuous Galerkin time stepping with local projection stabilization for transient convection-diffusion-reactionproblems，Comput. Methods Appl. Mech. Engrg., 200 (2011), 1747-1756.

15. H. Eichel, L. Tobiska and H. Xie, Supercloseness and superconvergence of stabilized low order finite element discretizations of the Stokes problem, Mathematics of Computation, 80(274)(2011), 697–722.

14. H. Brunner, Q. Huang and H. Xie, Discontinuous Galerkin methods for delay differential equations of pantograph type, SIAM J. NUMER. ANAL., Vol. 48, No. 5, pp. 1944–1967, 2010.

13. Q. Lin, H. Xie, F. Luo, Y. Li and Y. Yang, Stokes eigenvalue approximations from below with nonconforming mixed finite element methods, Mathematics in Practice and Theory, 40(19)(2010), 157-168. (In Chinese, English title and abstract)

12. H. Brunner, Hehu Xie and R. Zhang, Analysis of collocation solutions for a class of functional equations with vanishing delays, IMA Journal of Numerical Analysis, 31(2011), 698-718.

11. Qun Lin and Hehu Xie, New expansions of numerical eigenvalue for $-\Delta u = \lambda\rho u$ by linear finite element on different triangular meshes, International Journal of Information and Systems Sciences, 6(1)(2010), 10-34

10. Qun Lin and Hehu Xie, Asymptotic error expansion and Richardson extrapolation of eigenvalue approximations for second order elliptic problems by the mixed finite element method, Applied Numerical Mathematics, 59 (8)(2009), 1884—1893.

9. Q. Huang and Hehu Xie, Superconvergence of Galerkin solutions for Hammerstein equations, International Journal of Numerical Analysis & Modeling, 2009， Volume 6, Number 4, Pages: 696-710.

8. H. Chen, S. Jia and Hehu Xie, Postprocessing and higher order convergence for the mixed finite element approximations of the Stokes eigenvalue problems, Application of Mathematics, 54(3)(2009), 237-250.

7. S. Jia, Hehu Xie, X. Yin and S. Gao, Approximation and eigenvalue extrapolation of Stokes eigenvalue problem by nonconforming finite element methods, Applications of Mathematics, 54(1)(2009), 1-15.

6. Hehu Xie and S. Jia, Extrapolation for the second order elliptic problems by mixed finite element methods in three dimensions, International Journal of Numerical Analysis and Modeling, 5(1)(2008), 112-131.

5. S. Jia, Hehu Xie, X. Yin and S. Gao, Approximation and eigenvalue extrapolation of biharmonic eigenvalue problem by nonconforming finite element methods, Numerical Methods for Partial Differential Equations, 24(2) ,435-448,2008.

4. X. Yin, Hehu Xie, S. Jia and S. Gao, Asymptotic expansions and extrapolations of eigenvalues for the stokes problem by mixed finite element methods, Journal of Computational and Applied Mathematics, 215(1)(2008), 127-141

3. Hehu Xie, Extrapolation of the Nédélec element for the Maxwell equations by the mixed finite element method, Advances in Computational Mathematics, 29(2) (2008), 135-145.

2. Hehu Xie and S. Gao, Superconvergence of the least-squares mixed finite element approximations for the second order elliptic problems, International Journal of Information and Systems Sceiences, Vol.3 (2) (2007), 277-282.

1. V. Shaidurov and Hehu Xie, Richardson extrapolation for eigenvalue of discrete spectral problem on general mesh, Computational Technologies, Vol. 12 No. 3(2007), 24-37.

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