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Scientific Computing

Our research interests are in the field of scientific computing with particular emphasis on numerical linear algebra and its applications. Scientific computing is the field of study concerned with using computers to analyse and solve scientific and engineering problems. To that end it constructs mathematical models and develops numerical solution techniques. Numerical linear algebra is at the heart of most scientific computing. Many problems can be effectively represented, or approximated, and then solved on computers as matrix based problems, while others require the solution of a large number of intermediate such problems. We develop algorithms and theory for solving basic and general problems, as well as problems arising from specific applied areas.
We welcome visitors, postDoctoral Fellows, and good graduate students who are interested in our research areas.

Associated Faculty



Interesting Projects

 Core problems in linear systems.
The basic theory of core problems in the area of
estimation for linear systems has recently been developed.
Problems where the data is uncertain,
or where the precision of computation has a nontrivial effect,
such as overdetermined systems or systems arising from illposed problems,
can contain both redundant and irrelevant data.
This can create both mathematical and computational difficulties.
The theory of core problems simplifies the theory,
analysis and computations for such problems.
In applying these ideas it is important to examine the subtle problems
that arise in determining the dimensions of core problems.
Another project is to combine the understanding of core problem ideas
with those used to regularize illposed problems.
The difficulties are greatly magnified when the ideas are applied to
iterative methods for large sparse problems.
 Iterative algorithms for large sparse matrix problems.
The MGSGMRES method is a well known method for solving
large sparse unsymmetric linear systems of equations.
Recently it was shown that it is numerically backward stable,
and so it can be used with confidence.
But it requires storage for an increasing sequence of
supposedly orthogonal vectors, and uses all of these at each step,
often leading to storage problems and restarting.
There is another class of methods which only uses a small fixed number of
vectors at each step, yet in theory still has finite convergence.
The LSQR method is of this type,
and is widely used for large sparse least squares problems. But it, and related methods such as Craig's method, are often not considered for compatible unsymmetric systems of equations  even though they might be more efficient than GMRES for some problems.
To understand when these methods can be advantageous
it is important to develop an improved theory of convergence, and to delineate their practical behaviour more clearly. This knowledge could then be used to develop useful preconditioners to improve the convergence of such methods.
 Integer least squares (ILS) problems.
ILS problems, also referred to as closest point problems, are NPhard. They may arise from many applications such as communications, cryptography, lattice design, Monte Carlo secondmoment estimators, radar imaging, and global positioning systems. Computational efficiency is crucial for realtime applications. The goal of this project is to design fast and numerically reliable algorithms for
solving overdetermined or underdetermined ILS problems with various constraints, and to develop a software package for these problems.
 Global navigation satellite systems (GNSS).
GNSS is an important and computationally intensive area,
and has a vast number of applications. The goal of this project is to apply and develop various estimation methods such as Huber's Mestimation and partial linear model estimation for positioning. Computational efficiency and numerical reliability will be considered in developing algorithms.
Developing a MATLAB toolbox is part of this project.

Students

Present Ph.D. students:
 Ivo Panayotov (Dept. of Mathematics and Statistics),
Krylov Subspace Methods for Solving Linear Systems.

David TitleyPeloquin,
Bidiagonalization, Core Problems, and Regularization of Discrete IllPosed Problems.
 Xiaohua Yang,
Solving Integer Least Squares Problems.
Present M.Sc students:
 Mr. Ahmed Abu Safia (cosupervised with Prof. David Bryant)
 Mr. Xia Gao
 Ms. Qing Han
 Mr. Tianyang Zhou

