Research

My research focuses on numerical methods for partial differential equations (PDEs) and scientific computing. I have been working on the following several areas

new Eulerian techniques to approximate high frequency asymptotic solutions to wave phenomena and their related inverse problems,
numerical methods for interfacial motions, and
variational and statistical methods for image processing.

Publications

High Frequency Wave Propagation

  1. Wingfai Kwan, Shingyu Leung, Xiao-Ping Wang and Jianliang Qian. A Fast Huygens Sweeping Method for Capturing Paraxial Multi-color Optical Self-focusing in Nematic Liquid Crystals. Journal of Computational Physics, v348, 2017, Pages 108-138.
  2. Roland Glowinski, Shingyu Leung and Jianliang Qian. Operator-Splitting Based Fast Sweeping Methods for Isotropic Wave Propagation in a Moving Fluid. SIAM J. Sci. Comput., 38(2), A1195-A1223, 2016.
  3. Shingyu Leung, Jianliang Qian and Susana Serna, Fast Huygen Sweeping Methods for Schrodinger Equations in the Semi-Classical Regime. Methods and Applications of Analysis, 21(1), 2014, Pages 31-66. (UCLA CAM 13-71)
  4. Shingyu Leung and Jianliang Qian, The Backward Phase Flow and FBI-Transform-Based Eulerian Gaussian Beams for the Schrodinger Equation. UCLA-CAM 10-85. Journal of Computational Physics, Volume 229, Issue 23, November 20 2010, Pages 8888-8917. [pdf]
  5. Shingyu Leung and Jianliang Qian, Eulerian Gaussian Beams for Semi-Classical Solutions of Schrodinger Equations. Journal of Computational Physics, Volume 228, Issue 8, May 1 2009, Pages 2951-2977. [pdf]
  6. Shingyu Leung, Jianliang Qian and Robert Burridge, Eulerian Gaussian Beams for High Frequency Wave Propagation. Geophysics, Volume 72, Issue 5, September-October 2007, Pages SM61-SM76. [pdf]
  7. Shingyu Leung, Jianliang Qian and Stanley Osher, A Level Set Method for Three-dimensional Paraxial Geometrical Optics with Multiple Sources. Commun. Math. Sci., Volume 2, Issue 4, December 2004, Pages 643-672. [pdf]
  8. Jianliang Qian and Shingyu Leung, A Local Level Set Method for Paraxial Geometrical Optics. SIAM J. Sci. Comp., Volume 28, Issue 1, Pages 206-223. [pdf]
  9. Jianliang Qian and Shingyu Leung, A Level Set Based Eulerian Method for Paraxial Multivalued Traveltimes. Journal of Computational Physics, Volume 197, Issue 2, July 1 2004, Pages 711-736. [pdf]

Inverse Problems

  1. Roland Glowinski, Shingyu Leung, Jianliang Qian. A Penalization-Regularization-Operator Splitting Method for Eikonal-based Traveltime Tomography. SIAM J. on Imaging Sciences, Vol. 8, No. 2, 2015, pp. 1263-1292. (UCLA CAM 15-02)
  2. Wangtao Lu, Shingyu Leung and Jianliang Qian. An Improved Fast Local Level Set Method for Three-Dimensional Inverse Gravimetry. Inverse Problems and Imaging., Volume 9, No. 2, 2015, Pages 479-509. [pdf]
  3. Wenbin Li, Shingyu Leung and Jianliang Qian. A Level-Set Adjoint-State Method for Crosswell Transmission-Reflection Traveltime Tomography. Geophysical J. Int., 199(1), 2014, Pages 348-367. (UCLA CAM 13-76)
  4. Wenbin Li and Shingyu Leung, A fast local level set adjoint state method for first arrival transmission traveltime tomography with discontinuous slowness. Geophysical J. Int., 195(1), 2013, Pages 582-596. (CAM 13-75) [pdf]
  5. Victor Isakov, Shingyu Leung and Jianliang Qian, A Three-Dimensional Inverse Gravimetry Problem for Ice With Snow Caps. Inverse Problems and Imaging, Volume 7, No. 2, 2013. [pdf]
  6. Songting Luo, Shingyu Leung and Jianliang Qian, An Adjoint State Method for Numerical Approximation of Continuous Traffic Congestion Equilibria. Commun. Comput. Phys., Volume 10, Number 5, November 2011, Pages 1113-1131. [pdf]
  7. Victor Isakov, Shingyu Leung and Jianliang Qian, A Fast Local Level Set Method for Inverse Gravimetry. Commmun. Comput. Phys., Volume 10, Number 4, October 2011, Pages 1044-1070. [pdf]
  8. Shingyu Leung and Jianliang Qian, An Adjoint State Method for 3D Transmission Traveltime Tomography Using First Arrival. Commun. Math. Sci., Volume 4, Number 1, March 2006, Pages 249-266. UCLA CAM 06-06. [pdf]
  9. Shingyu Leung and Jianliang Qian, Transmission Traveltime Tomography Based on Paraxial Liouville Equations and Level Set Formulations. Inverse Problems 23 (2007) 799-821. [pdf]
  10. Shingyu Leung and Jianliang Qian, A Transmission Tomography Problem Based on Multiple Arrivals from Paraxial Liouville Equations, In Expanded Abstract for the SEG 75th Annual Meeting, Houston, USA, 2005. [pdf]

Interfaces Problems

  1. Meng Wang, Shingyu Leung and Hongkai Zhao, Modified Virtual Grid Difference for Discretizing the Laplace-Beltrami Operator on Point Clouds. Accepted by SIAM J. Sci. Comput. [arXiv]
  2. Tony Wong and Shingyu Leung, A Fast Sweeping Method for Eikonal Equations on Implicit Surfaces. J Sci Comput (2016) 67: 837. [pdf]
  3. Sean Y. Hon, Shingyu Leung and Hongkai Zhao, A Cell Based Particle Method for Modeling Dynamic Interfaces. Journal of Computational Physics, Volume 272, 2014, Pages 279-306. (UCLA CAM 13-74)
  4. Shingyu Leung, John Lowengrub and Hongkai Zhao, A Grid Based Particle Method for Solving Partial Differential Equations on Evolving Surfaces and Modeling High Order Geometrical Motion. UCLA-CAM 11-54. Journal of Computational Physics, Volume 230, Issue 7, April 1 2011, Pages 2540-2561. [pdf]
  5. Shingyu Leung and Hongkai Zhao, Gaussian Beam Summation for Diffraction in Inhomogeneous Media Based on the Grid Based Particle Method. UCLA-CAM 09-71. Commmun. Comput. Phys., Volume 8, Number 4, May 17 2010, Pages 758-796. [pdf]
  6. Shingyu Leung and Hongkai Zhao, A Grid Based Particle Method for Evolution of Open Curves and Surfaces. Journal of Computational Physics, Volume 228, Issue 20, November 1 2009, Pages 7706-7728. [pdf]
  7. Shingyu Leung and Hongkai Zhao, A Grid Based Particle Method for Moving Interface Problems. UCLA-CAM 08-08. Journal of Computational Physics, Volume 228, Issue 8, May 1 2009, Pages 2993-3024. [pdf]

Image Processing and Data Sciences

  1. Ke Wei, Jian-Feng Cai, T.F. Chan and Shingyu Leung. Guarantees of Riemannian Optimization for Low Rank Matrix Completion. Submitted. arXiv:1603.06610
  2. Hao Liu, Zhigang Yao, Shingyu Leung and Tony F. Chan. A Level Set Based Variational Principal Flow Method for Nonparametric Dimension Reduction on Riemannian Manifolds. SIAM J. Sci. Comput., 39(4), A1616-A1646, 2017. [pdf]
  3. Ke Wei, Jian-Feng Cai, T.F. Chan and Shingyu Leung. Guarantees of Riemannian Optimization for Low Rank Matrix Recovery. SIAM J. on Matrix Anal. & Appl., 37(3), 2016, Pages 1198-1222. arXiv:1511.01562
  4. Ke Wei, X.C. Tai, T.F. Chan and Shingyu Leung, Primal-Dual Method for Continuous Max-Flow Approaches . UCLA-CAM 15-67. Proceedings of the 5th Eccomas Thematic Conference on Computational Vision and Medical Image Processing (VipIMAGE 2015), 2015.
  5. Jun Liu, Xue-cheng Tai, Shingyu Leung and Haiyang Huang, A New Continuous Max-flow Algorithm for Multiphase Image Segmentation using Super-level Set Functions. UCLA-CAM 12-81. Journal of Visual Communication and Image Representation, Volume 25, Issue 6, 2014, Pages 1472-1488. [pdf]
  6. Jun Liu, Xue-Cheng Tai, and Shingyu Leung, A Generic Convexification and Graph Cut Method for Multiphase Image Segmentation. EMMCVPR Lecture Notes in Computer Science Volume 8081, 2013, pp 251-265. [pdf]
  7. Jun Liu and Shingyu Leung, A Splitting Algorithm for Image Segmentation on Manifolds Represented by the Grid based Particle Method. UCLA-CAM 12-80. Journal of Scientific Computing, Volume 56, 2013, Pages 243-266. [pdf].
  8. Jun Liu, Yin Bon Ku and Shingyu Leung, Expectation-Maximization Algorithm with Total Variation Regularization for Vector-Valued Image Segmentation. UCLA-CAM 12-79. Journal of Visual Communication and Image Representation, Volume 23, No. 8, 2012, Pages 1234-1244. [pdf]
  9. J.H. Ha, A. Hokugo, C.M. Mengatto, S. Leung, S. Osher, I. Nishimura, Mathematical Restoration Techniques to Improve Micro-CT Images of Implant Osseointegration. In: IADR/AADR/CADR 89th General Session, 2011, San Diego, Calif. Journal of Dental Research - Spec Issue A. US: JDR, 2011. v.90.
  10. Shingyu Leung, Gang Liang, Knut Solna and Hongkai Zhao, Expectation-Maximization Algorithm with Local Adaptivity. SIAM J. Imaging Sci., Volume 2, Issue 3, Pages 834-857, 2009. [pdf]
  11. Shingyu Leung and Stanley Osher, Fast Global Minimization of the Active Contour Model with TV-Inpainting and Two-phase Denoising. Proceeding of the 3rd IEEE Workshop on Variational, Geometric and Level Set Methods in Computer Vision, 2005, Pages 149-160. [pdf]

Computational Dynamical Systems

  1. Shingyu Leung, Guoqiao You, Tony Wong and Yu Keung Ng, Recent Developments in Eulerian Approaches for Visualizing Continuous Dynamical Systems. Submitted. [pdf]
  2. Guoqiao You and Shingyu Leung. Eulerian Based Interpolation Schemes for Flow Map Construction and Line Integral Computation with Applications to Coherent Structures Extraction. J. of Sci. Comput. [pdf]
  3. Guoqiao You, Tony Wong and Shingyu Leung, Eulerian Methods for Visualizing Continuous Dynamical Systems using Lyapunov Exponents. SIAM J. Sci. Comput., 39(2), A415-A437, 2017. [arXiv:1603.06446]
  4. Guoqiao You and Shingyu Leung, A Fast Semi-Implicit Level Set Method for Curvature Dependent Flows with an Application to Limit Cycles Extraction in Dynamical Systems. Commu. Comp. Phys., 18(1), 2015, pp. 203-229. [pdf]
  5. Guoqiao You and Shingyu Leung, VIALS: An Eulerian Tool Based on Total Variation and the Level Set Method for Studying Dynamical Systems. Journal of Computational Physics, Volume 266, 2014, Pages 139-160. [pdf]
  6. Guoqiao You and Shingyu Leung, An Eulerian Method for Computing the Coherent Ergodic Partition of Continuous Dynamical Systems. Journal of Computational Physics, Volume 264, 2014, Pages 112-132. (UCLA CAM 13-73) [pdf]
  7. Shingyu Leung, The Backward Phase Flow Method for the Eulerian Finite Time Lyapunov Exponent Computations. Chaos 23, 043132, 2013. [pdf]
  8. Shingyu Leung, An Eulerian Approach for Computing the Finite Time Lyapunov Exponent. UCLA-CAM 11-53. Journal of Computational Physics, Volume 230, Issue 29, May 1 2011, Pages 3500-3524. [pdf]

Postdoc

  1. Boxi Xu (2014-).
  2. Jun Liu (2011-2012). Beijing Normal University.

Graduate Students

  1. Yu-Keung NG. MPhil Program (2016-), Department of Mathematics, HKUST.
  2. Wing-Fai Andy Kwan. MPhil Program (2015-), Department of Mathematics, HKUST.
  3. Hao Liu. PhD Program (2014-), Department of Mathematics, HKUST.
  4. Ningchen Ying. PhD Program (2013-), Department of Mathematics, HKUST.
  5. Ka-Wah Tony Wong. MPhil Program (2014-2016), Department of Mathematics, HKUST. Current: PhD, UBC.
  6. Meng Wang. PhD Program (2011-2015), Department of Mathematics, HKUST. Current: Credit Suisse.
  7. Yu-Sing Sean Hon. MPhil Program (2012-2014), Department of Mathematics, HKUST. Current: PhD, Oxford University.
  8. Wenbin Li. PhD Program (2010-2014), Department of Mathematics, HKUST. Current: Postdoc, Michigan State University.
  9. Guoqiao You. PhD Program (2010-2014), Department of Mathematics, HKUST. Current: Nanjing Audit University China.

Undergraduate Students Mentored

  1. Hugo Wai-Leung MAK, HKUST Department of Mathematics, UROP1100 2014 Summer.
  2. Wing-Fai KWAN, HKUST Department of Mathematics, UROP1100 2014 Summer.
  3. Geyue CAI, HKUST School of Science, UROP1100 2014 Summer.
  4. Fai-Tsing Ronson CHAU, HKUST Department of Mathematics, UROP1100 2013 Fall.
  5. Liu Zhao, Zhejiang University Department of Mathematics, HKUST School of Science Summer Research Program 2013.
  6. Yunshen Zhou, Zhejiang University Department of Mathematics, HKUST School of Science Summer Research Program 2013.
  7. Tsun-Tak LAI and Lung-Fai WONG, HKUST Department of Mathematics, UROP1100 2013 Summer.
  8. Xingyu HE and Kaiqi YANG, HKUST School of Science, UROP1100 2013 Summer.
  9. Zhaoming XIE, HKUST Department of Computer Engineering, UROP1100 2013 Summer.
  10. Tianyu WANG and Yueshen WU, HKUST Department of Mathematics and Department of Chemistry, UROP1100 2013 Spring.
  11. Yau-Chuen YAM and Yiu-Ting LAI, HKUST Department of Mathematics, UROP1100 2012 Fall.
  12. Ching-Yeung AU, HKUST Department of Physics, UROP1100 2012 Fall.
  13. Leung-Hong Cheng, HKUST Department of Mathematics, UROP1000 2012 Summer.
  14. Siyu Zhou, HKUST Department of Mathematics, UROP1000 2012 Summer.
  15. Ho-Yin Ma and Wai-Leung Yip, HKUST Department of Mathematics, UROP1000 2012 Summer.
  16. Jiezhong Luo, HKUST Department of Mathematics, UROP1000 2012 Summer.
  17. Ka-Ching Law, HKUST Department of Physics, UROP1000 2012 Summer.
  18. Ruofan Li, HKUST Department of Mathematics, UROP1000 2012 Summer.
  19. Chi-Wai Yuen, HKUST Department of Mathematics and Department of Physics, UROP1000 2012 Summer.
  20. Yat-Hei Lam, HKUST Department of Physics, UROP 2011 Summer.
  21. Weixiang Xu, HKUST Department of Mathematics, UROP 2011 Spring.
  22. Yu-Sing Sean Hon, HKUST Department of Mathematics, UROP 2010 Fall, 2011 Spring, 2012 Summer.
  23. Chun-Kong Danny Tsang, HKUST Department of Mathematics, UROP 2010 Fall.
  24. Ziaomeng Shine Zhai, HKUST Department of Computer Science and Engineering, UROP 2010 Summer.