Contact Details
| Room No. | LT1007 (L907) |
|---|---|
| Telephone | +44 (0)141 548 3817 |
| Fax | +44 (0)141 548 3345 |
| oleg.davydov@strath.ac.uk | |
| Home Page | Homepage for Oleg Davydov (external link) |
Research Interests
Scattered data fitting
Approximation of multivariate data, or data defined on a manifold often requires a two-stage approach, where a number of reliable local approximations is created in the first stage, and the information delivered by them is used to build a smooth global approximant in the second stage. In the first stage one can apply e.g. least squares polynomials with carefully chosen degrees, or radial basis interpolants. The second stage can be organised with the help of a spline quasi-interpolant, a smooth finite element, or a partition of unity. I am developing robust two-stage algorithms able to work successfully with difficult real word data.
Poster (pdf file)
Multivariate splines
The focus of my research on multivariate splines is construction, theoretical investigation and applications of stable local bases that resemble well known properties of the univariate B-splines. Stable local bases provide a foundation for many applications of multivariate splines, e.g. data fitting or numerical solution of partial differential equations. Of a special interest are refinable bases that allow multilevel techniques, like wavelets or hierarchical bases.
Numerical solution of partial differential equations
I am interested in spline-based methods for numerical PDEs, where a recent result is a construction of a hierarchical Riesz basis for Sobolev space H2, which can be used for the preconditioning of linear systems arising from Galerkin discretisation of plate bending problems and other 4th order elliptic problems. I am also working on spline type finite elements suitable as discretisation tool in the context of a numerical method for fully non-linear PDEs introduced recently by K. Bohmer, as well as on polynomial curved finite elements. On the other hand, recently I became interested in meshless methods.
More info
Selected Publications
..
- A tension approach to controlling the shape of cubic spline surfaces on FVS triangulations
- Davydov O and Manni C
Journal of Computational and Applied Mathematics, 2010, 233, 1674-1684. - C2 piecewise cubic quasi-interpolants on a 6-direction mesh
- Davydov O and Sablonniere P
Journal of Approximation Theory, 2010, 162, 528-544. - Error bounds for anisotropic RBF interpolation
- Davydov O, Beatson R and Levesley J
J. Approx. Theory, 2010, 162, 512-527.
Vacancies & Studentships
Adaptive Meshless Methods Using Moving Nodes
Reference No.: OD/JAM1
Supervisors:
Date Advertised: 6th February 2010
Please contact Dr Oleg Davydov for further information.
This project will look at the use of moving node techniques to enhance the accuracy of meshless methods to solve numerically partial differential equations (PDEs).
Meshless methods are a new class of techniques that have the potential to revolutionise computational science and engineering.
The main attractive property of these methods is that they do not need a computational mesh but use a distribution of nodes within the spatial domain.
Mesh generation is often seen as the most time consuming aspect of the simulation of real-world problems. However, for most interesting problems the nodes in a meshless method need to be placed intelligently to avoid under-sampling of important physical features and for time-dependent problems these nodes also need to move.
Here, theoretical a priori and a posteriori error bounds should be used as guidance. In particular, methods based on radial basis functions are very promising because these functions provide an effective tool for the approximation on arbitrary nodes without the need to generate a mesh.
The interest in meshless methods is rapidly growing internationally among both engineers and numerical analysts.
Dr John Mackenzie has considerable experience in the analysis and development of moving mesh methods to solve PDEs. He has also recently supervised four PhD students in this area of research.
For time dependent problems moving mesh methods have been shown to be successful at solving a wide range of problems including melting and solidification problems, the simulation of liquid crystals, pattern formation in mathematical biology and the evaluation of options in mathematical finance.
For some movies go to http://www.maths.strath.ac.uk/~caas61/movies.html
Dr Oleg Davydov has worked on the approximation properties of radial basis functions and in the analysis of finite element methods.
His recent work includes the development and implementation of efficient algorithms for scattered data fitting based on local approximation with radial basis functions (see http://www.maths.strath.ac.uk/~aas04108/scat_data.html), as well as theoretical error bounds for the interpolation with radial basis functions.
This project would suit a student with an interest in the computational solution of PDEs and numerical approximation.
There would be plenty of scope within the project for the student to concentrate on analysis and/or computational aspects.
For the latter, confidence with MATLAB programming is expected.
It is envisaged that the developed algorithms will be applied to a realistic problem in the area of mathematical finance, mathematical biology or computational mechanics.
Data fitting and compression with multivariate splines
Reference No.: OD1
Supervisor: Dr Oleg Davydov
Date Advertised: 6th February 2010
Please contact Dr Oleg Davydov for further information.
A fast and at the same time reliable method of approximation of scattered data depending on several variables is the two-stage approach, where the data are split into small subsets, each of which is fitted separately by polynomials or radial basis functions in the first stage, and then the local patches are "glued" together using multivariate spline functions.
See http://www.maths.strath.ac.uk/~aas04108/scat_data.html
There are many possible variations of the method both in the first (local) and the second (global) stages.
Interesting research topics include for example:
- development of adaptive algorithms that automatically adjust the parameters, such as the degree of the approximation or the spline mesh, to the data features
- development of anisotropic methods specifically adjusted to data possessing strong local directional features
- compression of the resulting spline functions using wavelet type approaches
- fitting data on manifolds, such as global topography or aircraft surface pressure
- surface denoising.
Potential PhD projects may cover various aspects, from the theoretical investigation of the properties of local approximation methods and spline spaces, to the development of new algorithms, their implementation and applications in such areas as:
- computer graphics
- terrain modelling
- surface compression
- medical imaging
- computer aided design.