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Details of Grant 

EPSRC Reference: EP/I005293/1
Title: Nonlinear Eigenvalue Problems: Theory and Numerics
Principal Investigator: Tisseur, Professor F
Other Investigators:
Researcher Co-investigators:
Project Partners:
Department: Mathematics
Organisation: The University of Manchester
Scheme: Leadership Fellowships
Starts: 01 March 2011 Ends: 29 February 2016 Value (£): 1,218,799
EPSRC Research Topic Classifications:
Numerical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
09 Jun 2010 EPSRC Fellowships 2010 Interview Panel B Announced
Summary on Grant Application Form
Nonlinear eigenvalue problems arise in a wide variety of science and engineering applications, such as the dynamic analysis of mechanical systems (where the eigenvalues represent vibrational frequencies), the linear stability of flows in fluid mechanics, the stability analysis of time-delay systems, and electronic band structure calculations for photonic crystals. These problems present many mathematical challenges. For some there is a lack of underlying theory. For others numerical methods struggle to provide any accuracy or to solve very large problems in a reasonable time.The trend towards extreme designs (such as in micro-electromechanical (MEMS) devices and superjumbo jets) means that these nonlinear eigenproblems are often poorly conditioned (hence difficult to solve accurately) while also having algebraic structure that should be exploited in a numerical method in order to ensure physical meaning of the computed results. As a specific example, in a project at TU Berlin modelling the sound and vibration levels in European high-speed trains it was found that standard finite element packages provided no correct figures in the computed solutions until linear algebra techniques of the type to be developed in this project were brought into play in the underlying quadratic (degree 2) eigenvalue problem (see the cover article in SIAM News, Nov. 2004).With the help of the funded research team I will develop theory and methods that enable the solution of new classes of emerging eigenproblems (e.g., rational) and more efficient and more accurate solution of existing problems. The project will exploit the new concept of structure preserving transformations for matrix polynomials and a new linearization-based approach for rational eigenproblems. For the general nonlinear eigenproblem, we will to devise good approximations to the nonlinear parts by rational or polynomial functions that will then be handled with techniques for the latter problems.The work will have significant impact through the provision of algorithms and software, either open source or distributed through numerical libraries, that enables efficient computer solution of these problems.
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Organisation Website: http://www.man.ac.uk