Research
Members, visitors, postdoctoral fellows and students affiliated with the JRIAM
are engaged in frontier research in a variety of areas. These include:
 adaptive and spectral methods for numerical computation
 applied probability
 computational finance
 computational fluid dynamics
 data mining
 experimental design
 graph theory
 high dimensional integration and approximation
 Monte Carlo methods
 parallel computing
 optimization
Most of these research areas involve more than one member of
the JRIAM, and many members of the JRIAM are active in more than one area. For
more details check the websites of the members of the JRIAM.
Featured Research Programmes
The JRIAM conducts research in a number of fields in applied
mathematics. Some of the ongoing research programmes are described below. We
have aimed to build cohesive research teams. Each research area involves two
or more members of the JRIAM plus visitors, postdoctoral fellows, and students.
The references for the cited publications appear in section Publications.
Experimental design
In order to obtain as much information as possible from an
experiment it must be well designed. That is, for each experiment the levels
of the factors one can control must be chosen carefully. The uniform design,
of which Prof. KaiTai Fang is one of the pioneers, spreads design points as
evenly as possible over the experimental domain. Recent research by the JRIAM
in this area has included demonstrating the efficiency and robustness of the
uniform design and showing how uniform design improve upon traditional fractional
factorial designs by increasing the number of levels while maintaining an even
spread of design points [Fan02, FMM02,
HL02, MFL03].
New algorithms have been developed to construct uniform designs [FQ02,
FLQ03]. Researchers involved: K.T.
Fang, F. J. Hickernell, M. Y. Ai, D. K. J. Lin, R. Mukerjee, A. Zhang, Y. Zhang
High dimensional integration and approximation
Integrals of multivariate functions arise in a number of applications
in finance, physics, statistics and other fields. Pricing financial derivatives
is one example. Typically these integrals are approximated by weighted averages
of integrand values at certain wellchosen points. A related problem is how
to pproximate a function of many variables based on function values at certain
points. This research effort involves discovering new algorithms to solve such
problems, analyzing the error of algorithms, and determining when the convergence
rates can be made nearly independent of the dimension of the problem. Recent
research results have extended the error analysis to the important case where
the integration domain is unbounded (as in finance) [HSW03,
HSW04a, HSW04b,
HSW04c, HSW04d].
Work has also been done on expanding the spaces of functions for which a given
convergence rate holds [FH03a, FH03b,
HF03]. Researchers involved: F. J.
H ickernell, G. Fang, F. Huang, G. Wasilkowski, G. Wei, X. Y. Zang
Moving mesh methods for partial differential equations
Adaptive mesh methods have important applications for a variety
of physical and engineering areas such as solid and fluid dynamics, combustion,
heat transfer, material science etc. The physical phenomena in these areas develop
dynamically singular or nearly singular solutions in fairly localized regions,
such as shock waves, boundary layers, detonation waves etc. Successful implementation
of the adaptive strategy can increase the accuracy of the numerical approximations
and also decrease the computational cost. Recent results include solving incompressible
flow problems with moving finite element methods [DLTZ03].
This work presents the first effort in designing moving mesh algorithm to solve
the incompressible NavierStokes equations in the primitive variables formulation.
The main difficulty in developing this moving mesh scheme is how to keep the
divergencefree for the velocity field at each time level. The proposed numerical
scheme extends a recent adaptive grid method based on harmonic mapping [LTZ01,
LTZ02]. Researchers involved: T. Tang.
P.W. Zhang, R. Li, Y.N. Di, Z.R. Zhang, W.L. Lee
Numerical solutions for PDEs in large solution domains
Many physical problems are posed in large solution domains.
To make numerical computations possible, the solution domain has to be truncated
into a finite region, and therefore artificial boundary conditions have to be
designed. Moreover, even with successful domain truncation, fast solution solvers
are extremely important for the underlying PDE problems in large solution domains.
Recent results in this area include new algorithms for problems in solid mechanics
[WX03] and for problems in computational
finance [HW03]. Researchers involved:
T. Tang, X.N. Wu, H.D. Han, Z.H. Teng, J.C. Jin
High accuracy methods
Spectral methods are highly accurate methods for solving partial
differential equations and similar problems. Radial basis function methods,
allow more freedom in the placement of design points, but have recently been
shown to also obtain the high accuracy of spectral methods. Research by JRIAM
members includes developing Hermite spectral method for partial differential
equations in unbounded domains [FGT02]
and a multigrid method for elliptical equations [HSTX03].
There has also been recent success in extending spectral methods to high dimensional
problems by sampling on lattices [LH03a,LH03b].
Researchers involved: F. J. Hickernell, T. Tang, W. M. Xue, B. Fornberg,
D. Li
Statistical inference
An important part of statistics is the inference of parameter
values from data. For more complicated problems finding estimators that are
consistent, efficient and robust is a difficult problem. Recent results by [HYFW02]
develop consistent estimates for the intensity of droplets from a spray flame
when the recording instrument is subject to dead time. Other recent work in
statistical inference includes [MHF04,
PF02]. Researchers involved: K.T.
Fang, S.Y. He, C. L. Mei
External Grants:
 Liao, LiZhi, HKBU12319816, Sept. 1, 2016  Aug. 31, 2019, HK$326,811, ``Neurodynamical approach for linearly constrained convex programming’’
 Liao, LiZhi, HKBU12302019, Sept. 1, 2019  Aug. 31, 2022, HK$332,261, ``Interior point polynomialtime algorithms for linearly constrained convex programming’’
 Liao, LiZhi, HKBU12300920, Jan. 1, 2021  June30, 2023, HK$396,967, ``Neurodynamical approach for optimization problems with partial orthogonality constraints in supervised learning’’
 Cheng, MingYen, Essence codings in functional linear regression: identification, interpretation and application (HKBU12304120), RGC General Research Fund, HK$ 599,861, January 2021December 2023.
 Ling, Leevan, GRF/12301520 [11.23.4551.173072] $576,337, Kernelbased approximation methods for curvatures estimations and applications on raw point clouds, 1/202112/2023.
 Ling, Leevan, GRF/12301419 [11.23.4551.172940] $332,261, On the Asymptotics of Leastsquares Kernel Collocation Methods, 1/202012/2022.
 Ling, Leevan, GRF/12303818 [11.23.4551.172846] $456,452, Meshfree Methods for Coupled BulkSurface Problems, 1/201912/2021.
 Ling, Leevan, GRF/12301917 [11.23.4551.172746] $314,900, Meshfree methods for parabolic equations based on weighted leastsquares, 1/201812/2020.
 Ling, Leevan, GRF/12304114 [3214341] $614,810, Erroroptimal methods for PDEs arise from meshless theories, 01/201512/2017.
 Liu, Hongyu, Mathematical and computational studies of geomagnetic anomaly detections, HKBU12302919, HK$502,444, 01 January 2020  31 December 2022
 Liu, Hongyu, Mathematical analysis on scattering from corner singularities, inverse shape problems and geometric structures of transmission eigenfunctions, HKBU12301218, HK$304,301, 01 September 2018  03 March 2020
 Ng, Michael K. P., Preconditioning Methods for Inverse Source Problem in Fractional Diffusion Equations, HKBU12300519, HK$332,261, 01 January 2020  31 December 2022
 Ng, Michael K. P., Orthogonal Nonnegative Matrix and Tensor Factorizations: Analysis, Algorithms and Applications, HKBU12300218, 172824, HK$456,452, 01 January 2019  11 September 2019
 Ng, Michael K. P., Nonlinear Toeplitzlike Matrix Systems in Image Restoration, HKBU12200317, 172732, HK$472,351, 01 January 2018  11 September 2019
 Ng, Michael K. P., Sparse Multilinear PageRank: Formulation, Algorithm and Analysis, HKBU12306616 3216066, HK$727,647, 1 January 2017  31 December 2019
 Tai, XueCheng, RG(R) RC/1718/02MATH, Research on Scientific Computing and Imaging Science, HKBU, HK$500,000, 1 Aug 2017 (24 months).
 Tai, XueCheng, HKBU12300819, Efficient Splitting And Preconditioning Algorithms For Total Variation And Elastica Energy Minimization For Variational Image Processing And Computer Vision, HK$502,444, 01 January 2020  31 December 2022
 Tai, XueCheng, Mathematical modelling and analysis of deep neural networks for solving structured differential and integral models, NSFC/RGC Joint Research Scheme, HK$1,177,667, 1 Jan 2020 (36 months).
 Zhu, Lixing, HKBU12302720, Estimation of conditional Mquantile treatment effect, HKD 529,289, 1/1/202131/12/2023
 Zhu, Lixing, HKBU12303419, Order determination for large dimensional matrices and its application, HKD 502,444, 1/1/202031/12/2022
 Zhu, Lixing, HKBU123028/18, Model checking for regressions: local smoothingbased tests with global smoothing features, HKD 651,273, 1/1/201931/12/2021
 Zhu, Lixing, HKBU123017/17, Thresholding double ridge ratio criteria for order determination and their applications, HKD 472,351, 1/1/201831/12/2020
 Zhu, Lixing, HKBU123289/16P, Nonparametric generalized likelihood ratio and significant testing for regressions: dimension reduction approaches, HK$ 326,811, 1/1/201731/12/2019
 Zhu, Lixing, HKBU123031/15P, Sufficient dimension reduction with mixture multivariate skewelliptical distributions, HK$ 451,255, 1/1/201630/06/2019
 Chiu, Sung Nok, HKBU12317616, Analysis of multivariate spatial point patterns, January 2017  December 2019, HK$410,371.
 Chiu, Sung Nok, HKBU12301215, Asymptotic methods for spatial point processes, January 2016  December 2018, HK$451,255.
 Fan, Jun, Mathematical analysis of kernel based modal regression schemes. Early Career Scheme (ECS), Hong Kong, HKD 446,678, September 2018  August 2021.
 Fan, Jun, Learning theory of Kaczmarz Methods for Phase Retrieval. National Natural Science Foundation of China (NSFC), China, RMB 220,000, January 2019 December 2021.
 Fan, Jun, Weighted tubal decompositions for tensors with applications in image processing. GRF HKBU12301619, HKD 502,444, September 2019  August 2022.
 Fan, Jun, Analysis of kernelbased learning and testing schemes for largescale data. GRF HKBU12302819, HKD 332,261, January 2020  December 2022.
 Fan, Jun, Learning theory of outcome weighted schemes and related topics. GRF HKBU12303220 , HKD 599,861, January 2021  December 2023.
 Hon, Sean Y. S., ECS 22300921, $951,823, Preconditioning for Toeplitzlike Systems in Evolutionary Differential Equations, 20212024
 Hon, Sean Y. S., Startup Allowance for Croucher Study Awards, $500,000, A Fast Iterative Method for Illconditioned Toeplitz Systems, 20212024
 Kwok, Felix, Domain decomposition based nonlinear preconditioning for optimal control problems, HKBU12300720, HK$599,861
 Kwok, Felix, Adaptive Pipelining Methods for ParallelinTime Integration, HKBU12301018, 172828, HK$456,452, 01 January 2019  31 December 2021
 Kwok, Felix, Spacetime Bmethods for the integration of nonlinear parabolic PDEs with blowup, HKBU12301817, 172744, HK$472,351, 01 January 2018  31 December 2020
 Lam, Andrew K. F., Modelling and analysis of diffuse interface models for twophase micropolar fluid flows. GRF 14303420, HKD 555,754, Jan 2021 to Dec 2023
 Lam, Andrew K. F., On Cahn–Hilliard models with singular potentials and source terms. GRF 14302319, HKD 502,444, Sep 2019 to Aug 2022
 Lam, Andrew K. F., Mathematical studies of a phase field approach to shape optimization. GRF 14302218, HKD 456,452, Jan 2019 to Dec 2021
 Peng, Heng, HKBU 12302615, Revisiting the Generalized Likelihood Ratio Test and Its Related Nonparametric Regression Inference Problems, 1/201612/2018, HKD 451,255
 Peng, Heng, HKBU 12303618, Tradeoﬀs between Statistical Eﬃciency and Complexity of Computation by Discretization and Partial consistency, 1/201912/2021, HKD 456,452
 Tong, Tiejun, HKBU12303918, Some new developments in hypothesis testing for highdimensional data, HK$ 456,452, December 2019May 2022.
 Tong, Tiejun, Differencebased methods in statistics and their wide applications. National Natural Science Foundation of China, CNY 480,000, January 2017  December 2020.
 Tong, Tiejun, Data transformation for metaanalysis with application to evidencebased medicine. Health and Medica Add to l Research Fund, Hong Kong, HK$ 376,960, July 2017  June 2019.
