Scheme: Wolfson Research Merit Awards
Organisation: University of Manchester
Dates: May 2014-Apr 2019
Summary: The numerical solution of algebraic eigenvalue problems, both linear and nonlinear, is a key technology underpinning many areas of computational science and engineering, including acoustics, aeronautics, control theory and nanotechnology. The eigenvalues represent key physical quantities, such as vibrational frequencies in structural engineering problems. In all these areas, the need for fast and numerically reliable solution of eigenvalue problems arises. The problems can be large, so that time to solution can be unacceptably long, and they can be very ill conditioned, making it difficult to obtain accurate solutions. Both aspects are exacerbated by the trend to towards extreme designs, such as in high speed trains and "superjumbo" jets. I am interested in fundamental mathematical and computational challenges driven by this diverse range of applications.
One strand of my research is to provide new theory, algorithms and software that will allow solution of problems that could not previously be solved reliably, or for which there is no library software available. For general nonlinear eigenvalue problems, I believe that this can be achieved with a new technique based on a dynamic approximation of the nonlinear part by a linear eigenproblem. Making it work reliably will involve a theoretical analysis of the approximation process. I also intend to provide faster and more meaningful solution of problems for which available techniques do not fully respect the problem structure (such as different kinds of symmetry).
The second aspect of my work deals with the applications of tropical algebra to numerical linear algebra. The tropical (or max-plus) algebra consists of the set of real numbers, along with negative infinity, equipped with two binary operations, maximization and addition. Tropical methods have demonstrated applications in a wide range of fields including analysis of discrete event systems, control theory, combinatorial optimization and scheduling, and algebraic geometry. There is strong evidence that tropical algebra can also be useful to numerical linear algebra, in particular for matrix problems with large variation in the data and which are difficult to solve accurately. For such problems, tropical solutions tend to be good approximations to solutions in the classical algebra and they can be much cheaper to compute. My plan is to use tropical approximation of eigenvalues in the development of numerically stable nonlinear eigensolvers and preconditioners for highly nonsymmetric large sparse matrices.