Associate Professor Kenneth Harrison

Room: S&C/3.008

Research

Operator theory in Hilbert space, computational complexity, signal processing.

Projects

Computational Complexity

There are algorithms for solving many practical mathematical problems. However the effectiveness of an algorithm depends very much on its complexity. If the number of computations grows too fast as size of the problem increases, then it is of very limited use, even when implemented on the fastest computers. In such cases alternative methods must be sought. There are a number of possibilities for honours projects in this area. Some of these involve looking at some interesting approximate algorithms which have recently been developed. They could involve a theoretical examination or an assessment of their performances based on real applications, or ideally, a combination of these approaches.

Graphs and Matrix Algebra

Finite-dimensional linear systems are usually modelled by matrices, and this is one important practical reason for studying their behaviour. Recently a number of interesting properties of some types of matrices have been shown to related in simple ways to properties of associated graphs. A project in this area would involve a survey of some of these connections between graphs and matrices and the opportunity to be involved in a search for others.

Sample Complexity of Linear System Identification

One important practical aspect of linear systems theory is the problem of determining the coefficients of the transfer function of an input/output model. This can be done by a observing the outputs resulting from a carefully chosen set of 'test' inputs. How large does this test set need to be in order to identify the system to a given level of accuracy, in the presence of noise or measurement error? Basic questions such as this have been shown to be related to some elementary properties of convolution products, and a project in this area could explore these connections further.

Linear System Representations.

The two standard models used in linear systems theory are the input/output models and the state-variable models. How are they related, and when should one be preferred over the other? These questions could be investigated in the first instance for discrete time, shift-invariant systems.

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