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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