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. Kai-Tai 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 well-chosen 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 Navier-Stokes equations in the primitive variables formulation.
The main difficulty in developing this moving mesh scheme is how to keep the
divergence-free 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 multi-grid 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
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