Lihu Xu (徐禮虎)

Associate Professor
3075, E11, Department of Mathematics,  Faculty of Science and Technology,  University of Macau
lihuxu'AT'um.edu.mo,  xulihu2007'AT'gmail.com

I received my PhD from Imperial College London in 01/2008, and did postdoctoral research at Bonn University (09/2007-08/2008), Technology University of Eindhoven (09/2008-08/2010), and Technology University of Berlin (10/2010-08/2011). I was a lecturer (permanent position, equal to assistant professor) at Brunel University London from 09/2011 to 11/2013, and a visiting scholar at Boston College from 10/2013 to 12/2013. I joined UM as an assistant professor in 01/2014 and was promoted to associate professor in 08/2019. I am also an associate member in Center for Applied Mathematics and  Center for Artificial Intelligence and Robotics. More information about me can be found at my google scholar page and my home page of UM.  

Teaching at UM  

  • 2023/2024

            MATH3025 Topics in Statistics and Data Science–Bayesian Inferences and MCMC

            MATH2010 Computational Statistics

            Sabbatical Leave (spring term) : I will not supervise final year projects.

  • 2022/2023

              MATH2010 Computational Statistics

              MATH3008 Introduction to Stochastic Calculus

              MATH3017  Data Driven Sampling Methods     

  • 2020/2021

              MATH2010 Computational Statistics

              MATH3008 Introduction to Stochastic Calculus

              MATH7004 Advanced Probability and Statistics–High Dimensional Probability with Applications to Data Science

  • 2019/2020

              MATH2005 Probability

              MATH3008 Introduction to Stochastic Calculus

              MATH7013 Stochastic Differential Equations 

  • 2018/2019

              MATH2006 Applied Statistics

              MATH3008 Introduction to Stochastic Calculus

              MATH7013 Stochastic Differential Equations

  • 2017/2018

              MATH2005 Probability

              MATH3008 Introduction to Stochastic Calculus

              MATH7013 Stochastic Differential Equations

  • 2016/2017

              MATH2005 Probability

              MATH3008 Introduction to Stochastic Calculus

              MATB325 Introduction to Real Analysis and Hilbert Space

  • 2015/2016

              MATH2005 Probability

              MATH3008 Introduction to Stochastic Calculus

              MATH7004 Topics in Probability and Statistics–Advanced Probability

  • 2014/2015

              MATH2005 Probability

              MATH3008 Introduction to Stochastic Calculus

              MATH111 Probability and Statistics for Engineering

  • 2013/2014

              MATH111 Probability and Statistics for Engineering

Teaching at Brunel University London

  • Probability (UG)
  • Analysis II (UG) 
  • Analysis I (UG) 
  • Financial Mathematics (UG, Final Year Project Course)

Teaching at Technology University of Berlin

  • Introduction to Stochastic Partial Differential Equations (PhD)

Final Year Projects at UM

  • Generative AI: diffusion models (2023)
  • Tensor data analysis and its applications (2022)
  • A test of machine learning method by MNIST dataset (2021)
  • Prediction of occurrence of influenza A and B in Macao using methodological factors (Joint with Professor Xiaohua Zhang, 2020)
  • Central limit theorem for high dimensional regressions (2019)
  • Fused Kolmogorov filter: a nonparametric model-free screening method (2018)
  •  Dictionary identification-sparse matrix-factorization via ℓ1 minimization (2017)
  • False discovery rate in statistics (2016)
  • Least angle regression and squre-root lasso (2015)

If you are interested in my research or being a PhD student under my supervision, please feel free to contact me by email:  lihuxu@um.edu.mo, xulihu2007@gmail.com. You may find the information of PhD program and financial support from here. 

Post-doc

 Dr. Huiming Zhang (2020–2022)
          First job: Tenure track Associate Professor at Beihang University (北京航空航天大學)
 

PhD Students

Dr. Fang Xie (2015–2018)

         Thesis: L1 penalized methods in high-dimensional regressions and its theoretical properties 

         First Job: Assistant Professor at UIC of BNU-HKBU

Dr. Peng Chen (2017–2021)

         Thesis: Probability approximations with their applications to limit theorems and stochastic algorithms

         First Job: Assistant Professor at Nanhang University (南京航空航天大學)

Dr. Xinghu Jin (2017–2021)

         Thesis: Approximation to steady states of M/Ph/M+n queques,   

         First Job: Assistant Professor at Technology University of Hefei

Dr. Xin Xu (2017–2021)

         Thesis: Aggregation in several chemotaxis models (Co-supervisor: Xuefeng Wang, Linlin Su)

         First Job: Postdoc at Nankai University

Dr. Jianya Lu (2018–2022)

         Thesis: Limit theorems of time series and stochastic algorithms

         First Job:  Lecturer  (Assistant Professor)  in Statistics at University of Essex, see the news of UM. 

Dr. Qiuran Yao (2019–2023)

        Thesis: Robust estimations in statistical learning and distribution estimations via DNN-based GAN (Co-supervisor: Lianjie Shu) 

Mr. Xiang Li (2020–):

Mr. Yu Wang (2021–):

Mr. Yingjun Mo (2022–):

Mr. Qiyang Pei (2023–): 

Master Students

Ms. Fang Xie (2013—2015)

        Thesis: High dimensional lasso regressions

        First Job: PhD at the University of Macau

Mr. Libang Deng (2013—2015):

       Thesis: Asset pricing models and an SPDE

       First Job: Huashi Finance Co. Ltd 

Mr. Zhilei Fan (2015—2017)

       Thesis: False discovery rate and self-normalized Cramer type moderate deviations

       First Job: Beijing Tao Capital Company Limited 

Ms. Qiuran Yao (2016–2019)

      Thesis: Lasso and estimation consistency of Tweedie’s compound Poisson model with elastic net

      First Job: PhD at the University of Macau

Mr. Xiang Li (2018–2020)

      Thesis: L1-regression of a truncated estimator of the mean and a stable approximation

      First Job: PhD at the University of Macau

Ms. Jinjin Yu (2018–2021)

      Thesis: Variable selection with square root loss function based on kernel methods

      First Job: PhD at the University of Macau

Mr.  Man-Nok Lou (2019-2021)

      Thesis: Applications of linear regressions

      First Job: Peidao High School

Mr. Qiyang Pei (2021–2023) 

      Thesis: Malliavin calculus and its applications to stochastic numerical analysis

      First Job: PhD at the University of Macau

Mr. Sibo Zhao (2023–)  

Ms. Yueren Zhao (2023–)

Ms. Chenyue Li (2023–)    

Research Interests

My research interests include several topics in the field of probability, statistics and data science: applied stochastic processes, large deviations (rare events), normal and non-normal approximations, Stein’s method, with their applications to high dimensional statistics and stochastic algorithms in machine learning. I am also interested in applying probability to machine learning and mathematical modellings. 

Research Grants

I have received 8 external grants from FDCT of S.A.R. Macau and NSFC of China. Among them,  the recent grant FDCT/0074/2023/R1A2 from FDCT and  two general project grants No. 12071499 and No. 11571390 from NSFC were very prestigious and competitive.   

Publications 

 XXX: current or former PhD students or postdoc

  1. Xinghu Jin, Guodong Pang, Lihu Xu, Xin Xu: An approximation to the invariant measure of the limiting diffusion of G/Ph/n+GI queues in the Halfin-Whitt regime and related asymptotics,  Mathematics of Operations Research (accepted). 
  2. Xiequan Fan, Haijuan Hu,  Lihu Xu: Normalized and self-normalized Cramer-type moderate deviations for Euler–Maruyama scheme for SDE, Science China Mathematics (to appear).
  3. Peng Chen, Ivan Nourdin, Lihu Xu, Xiaochuan Yang: Multivariate stable approximation in Wasserstein distance by Stein’s method, Journal of Theoretical Probability, 37 (1), 446-488, 2024.
  4. Peng Chen, Changsong Deng, Rene Schilling, Lihu Xu:  Approximation of the invariant measure of stable SDEs by an Euler–Maruyama scheme,  Stochastic Processes and Their Applications, 163,136-167, 2023. 
  5. Lihu Xu, Fang Yao, Qiuran Yao, Huiming Zhang: Non-asymptotic guarantees for robust statistical learning under infinite second moment assumption,  Journal of Machine Learning Research, 24, 1-46, 2023. 
  6. Vahagn Nersesyan, Xuhui Peng, Lihu Xu: Large deviations principle via Malliavin calculus for the Navier-Stokes system driven by a degenerate white-in-time noise, Journal of Differential Equations, 362, 230-249,2023. 
  7. Peng Chen, Qi-Man Shao and Lihu Xu: A probability approximation framework: Markov process approach,  Annals of Applied Probability, 33(2), 1619-1659, 2023. 
  8. Peng Chen, Jianya Lu and Lihu Xu: Approximation to stochastic variance reduced gradient Langevin dynamics by stochastic delay differential equations,  Applied Mathematics & Optimizations, 85, no.2, Paper No.15, 40pp, 2022.
  9. Changsong Deng, Rene Schilling and Lihu Xu: Singular integrals of subordinators with applications to structure properties of SPDEs,  Transactions of the American Mathematical Society,  375, no. 9, 6043-6073, 2022.
  10. Peng Chen, Ivan Nourdin, Lihu Xu, Xiaochuan Yang and Rui Zhang: Non-integrable stable approximation by Stein’s method, Journal of Theoretical Probability, 2, 1137-1186, 2022.
  11. Jianya Lu, Yuzhen Tan and Lihu Xu: Central limit theorem and self-normalized Cramer type moderate deviation for Euler-Maruyama scheme,  Bernoulli, 28 (2), 937-964, 2022. 
  12. Peng Chen, Xinghu Jin, Xiang Li and Lihu Xu: A generalized Catoni’s M-estimator under finite \alpha-th moment assumption with \alpha \in (1,2),  Electronic Journal of Statistics, 15 (2), 5523-5544, 2021.
  13. Amarjit Budhiraja, Yong Chen and Lihu Xu: Large deviations of the entropy production rate for a class of Gaussian processes,  
    Journal of Mathematical Physics, 62 (5), 052702, 2021. 
  14. Hao Shen, Jian Song, Rongfeng Sun and Lihu Xu: Scaling limit of a directed polymer among a Poisson field of independent walks, Journal of Functional Analysis, 281 (5), 109066, 2021.
  15. Peng Chen; Ivan Nourdin and Lihu Xu: Stein’s method for asymmetric alpha-stable distribution, with applications to stable CLT,  Journal of Theoretical Probability, 34 (3), 1382-1407, 2021. 
  16. Xiaobin Sun, Ran Wang, Lihu Xu and Xue Yang: Large deviation for two-time-scale stochastic Burgers equation, Stochastic and Dynamics, no. 5, Paper No. 2150023, 37 pp, 2021.
  17. Ran Wang, Jie Xiong and Lihu Xu: Large deviation principle of occupation measures for non-linear monotone SPDEs, Science China Mathematics, 64 (4), 799-822, 2021.
  18. Zhao Dong; Feng-Yu Wang and Lihu Xu: Irreducibility and asymptotics of stochastic Burgers equation driven by alpha-stable processes, Potential Analysis, 52, 371-392, 2020.
  19. Jinzhu Jia, Fang Xie and Lihu Xu: Sparse Poisson regression with penalized weighted score function. Electronic Journal of Statistics, 13(2), 2898–2920, 2019.
  20. Peng Chen and Lihu Xu: Approximation to stable law by the Lindeberg principle. Journal of Mathematical Analysis and Applications, 480(2), 123338, 28 pp, 2019.
  21. Xiaobin Sun, Yingchao Xie and Lihu Xu: Exponential mixing for SPDEs driven by highly degenerate Lévy noises. Illinois Journal of Mathematics, 64(1), 75-102, 2019.
  22. Lihu Xu: Approximation of stable law in Wasserstein-1 distance by Stein’s method, Annals of Applied Probability, 29(1), 458-504, 2019.

    This paper ‘breaks new grounds on the Stein’s famous method and also opens up many new interesting research problems’ (from a 5 pages’ referee’s report).

  23. Xiao Fang; Qi-Man Shao and Lihu Xu: Multivariate approximations in Wasserstein distance by Stein’s method and Bismut’s formula, Probability Theory and Related Fields, 174(3,4), 945–979, 2019.
  24. Lihu Xu: Singular integrals of stable subordinator, Statistics & Probability Letters, Vol. 139, 115-118, 2018.
  25. Xiaobin Sun, Yimin Xiao, Lihu Xu and Jianliang Zhai: Uniform dimension results for a family of Markov processes, Bernoulli, Vol. 24 4B, 3924-3951, 2018.
  26. Ran Wang and Lihu Xu: Asymptotics for stochastic reaction-diffusion equation driven by subordinate Brownian motions, Stochastic Processes and Their Applications, Vol. 128, 5, 1772-1796, 2018.
  27. Fang Xie, Lihu Xu and Youcai Yang: Lasso for sparse linear regression with exponentially β-mixing errors. Statistics and Probability Letters, 125 (2017), 64–70.
  28. Ran Wang, Jie Xiong and Lihu Xu: Irreducibility of stochastic real Ginzburg-Landau equation driven by alpha-stable noises and applications, Bernoulli, 23 (2017), no. 2, 1179–1201.
  29. Lihu Xu, Wen Yue and Tusheng Zhang: Smooth densities of the laws of perturbed diffusion processes, Statistics and Probability Letters, 119 (2016), 55–62.
  30. Yong Chen, Hao Ge, Jie Xiong and Lihu Xu: The large deviation principle and steady-state fluctuation theorem for the entropy production rate of a stochastic process in magnetic fields, Journal of Mathematical Physics, 57 (2016), no. 7, 073302, 16 pp.
  31. Xiao-Hong Chen, Qi-Man Shao, Wei Biao Wu and Lihu Xu: Self-normalized Cramer Type Moderate Deviations under Dependence, Annals of Statistics, 44 (2016), no. 4, 1593–1617.
  32. Feng-Yu Wang, Jie Xiong and Lihu Xu: Asymptotics of sample entropy production rate for stochastic differential equations, Journal of Statistical Physics, 163 (2016), no. 5, 1211–1234.
  33. Zhao Dong, Symon Peszat and Lihu Xu: On some  smoothening effects of the transition semigroup  of  a  Levy process, Journal of Mathematical Analysis and Applications, 434 (2016), no. 2, 1566–1580.
  34. Matthias Winter, Lihu Xu, Jianliang Zhai and Tusheng Zhang: The dynamics of the stochastic shadow Gierer-Meinhardt system, Journal of Differential Equations 260 (2016), no. 1, 84–114.
  35. Fang Li and Lihu Xu: Finite time blowup of the stochastic shadow Gierer-Meinhardt System, Electronic Communication on Probability, 20 (2015).
  36. Feng-Yu Wang, Lihu Xu and Xicheng Zhang: Gradient estimates for SDEs driven by multiplicative Levy noise, Journal of Functional Analysis, 269 (2015), 3195–3219.
  37. Ran Wang and Lihu Xu: Asymptotics of the Entropy Production Rate for d-Dimensional Ornstein–Uhlenbeck Processes. Journal of Statistical Physics, 160 (2015), no. 5, 1336–1353.
  38. Feng-Yu Wang and Lihu Xu: Log-Harnack inequality for Grushin type semigroup, Revista Mathematica Iberoamericna, 30 (2014), no. 2, 405-418.
  39. Lihu Xu: Exponential mixing for 2D SDEs forced by degenerate Levy noises, Journal of Evolution Equations14 (2014), no. 2, 249-272.
  40. Zhao Dong, Lihu Xu and Xicheng Zhang: Exponential ergodicity of stochastic Burgers equations forced by alpha-stable processes, Journal of Statistical Physics, (2014), no. 4, 929-949.
  41. Lihu Xu: Ergodicity of stochastic real Ginzburg-Landau equations forced by alpha-stable noises, Stochastic Processes and their Applications, 123, no 10, (2013), 3710-3736.
  42. Feng-Yu Wang and Lihu Xu: Derivative Formula and Applications for Hyperdissipative Stochastic Navier-Stokes/Burgers Equations, Infinite Dimensional Analysis Quantum Probability and Related Topics, Vol. 15, No. 3 (2012), pp 19.
  43. Sergio Albeverio, Arnaud Debussche and Lihu Xu: Exponential mixing of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noises, Applied Mathematics & Optimizations, 66, 2 (2012), 273-308.
  44. Enrico Priola, Armen Shirikyan, Lihu Xu and Jerzy Zabczyk: Exponential ergodicity and regularity for equations with Levy noise, Stochastic Processes and their Applications, 122, 1 (2012), 106-133.
  45. Zhao Dong, Lihu Xu and Xicheng Zhang: Invariance measures of stochastic 2D Navier-Stokes equations driven by alpha-stable processes, Electronic Communication on Probability, Vol. 16 (2011), 678-688.
  46. Lihu Xu: A modified log-Harnack inequality and asymptotically strong Feller property, Journal of Evolution Equations, 11 (2011), 925-942.
  47. Feng-Yu Wang, Jiang-Lun Wu and Lihu Xu: Log-Harnack inequality of stochastic Burgers equations, Journal of Mathematical Analysis and Applications, 384 (2011), 151–159.
  48. Enrico Priola, Lihu Xu and J. Zabczyk: Exponential mixing for SPDEs with Levy noises, Stochastic and Dynamics, 11 (2011), 521-534.
  49. Marco Romito and Lihu Xu: Ergodicity of 3D stochastic Navier-Stokes equations driven by mildly degenerate noises, Stochastic Processes and their Applications, 121, (2011), no. 4, 673-700.
  50. Lihu Xu and Boguslaw Zegarlinski: Existence and exponential mixing of the alpha-stable systems with unbounded interactions, Electronic Journal of Probability, Vol. 15 (2010), 1994-2018.
  51. Lihu Xu and Boguslaw Zegarlinski: Ergodicity of finite and infinite dimensional alpha-stable systems, Stochastic Analysis and its Applications, 27 (2009), no. 4, 797–824.
  52. Robert Olkiewicz, Lihu Xu and Boguslaw Zegarlinski: Nonlinear problems in infinite interacting particle systems, Infinite Dimensional Analysis Quantum Probability and Related Topics, 11 (2008), no. 2, 179–211.
  • FST Teaching Excellence Award, UM, 2023/2024.
  • FST Teaching Excellence Award, UM, 2018/2019.
  • Junior Scientific Leader, Banach International Mathematical Center, Poland, 2018.
  • Invited Mini Course Lecturer, Simon Semester, Banach International Mathematical Center, Poland, 2018.
  • FST Research Excellence Award, 2016/2017. 
  • DAAD Fellowship, Weierstrass Institute,  Berlin, Germany, 2006. 
  • Oversea Research Student (ORS) Award, Imperial College London, UK, 2004-2007.
  • International Student Scholarship, Imperial College London, UK, 2004-2007.