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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, post-Doctoral Fellows, and good graduate students who are interested in our research areas.

# Associated Faculty

 Prof. Xiao-Wen Chang
 Prof. Chris Paige

# 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 over-determined systems or systems arising from ill-posed 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 ill-posed 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 MGS-GMRES 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 NP-hard. They may arise from many applications such as communications, cryptography, lattice design, Monte Carlo second-moment estimators, radar imaging, and global positioning systems. Computational efficiency is crucial for real-time applications. The goal of this project is to design fast and numerically reliable algorithms for solving over-determined or under-determined 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 M-estimation 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 Titley-Peloquin, Bidiagonalization, Core Problems, and Regularization of Discrete Ill-Posed Problems.
• Xiaohua Yang, Solving Integer Least Squares Problems.
Present M.Sc students:
• Mr. Ahmed Abu Safia (co-supervised with Prof. David Bryant)
• Mr. Xia Gao
• Ms. Qing Han
• Mr. Tianyang Zhou