A new paradigm for spectral localisation of operator pencils and analytic operator-valued functions

The natural frequencies of vibration of an elastic structure (bridge; airframe; bicycle frame) determine its mechanical behaviour. Frequencies which are likely to be excited by external loads should be designed out, in order to avoid mechanical failures. The yellow light emitted by sodium street-lamps (now mostly replaced by LCDs) was also an example of a natural frequency, determined by the law E = h f, where h is Planck's constant and E is determined by finding the allowed energy levels of electrons in a sodium atom: effectively, by solving something called an eigenvalue problem, for the Schroedinger equation.

The importance of eigenvalue problems in so many different applications has resulted in large amounts of public and industrial money being spent on the development of appropriate software, such as ARPACK and LAPACK. These deal with problems in which the eigen-parameter appears linearly. Similarly, huge efforts have been made to develop theoretical tools to understand, at the level of pure mathematics, where eigenvalues lie. Are they real or complex? Do they lie inside the unit disc, or outside? Do they have positive or negative real part? The numerical range is one of the most widely used tools for making such estimates.

However, eigen-parameters often appear polynomially or rationally. When this happens, a common approach has been to apply clever transformations to linearise the problem. In this proposal, we intend to show that this is generally not the best approach, by proving theorems and constructing estimates which treat the problem in its original form and get better estimates. In fact, we even propose that in most cases one should do the opposite of the usual approach: transform the original linear problem into a family of rational problems and, by considering estimates for the whole family together, obtain better estimates than can be obtained by direct treatment of the original. In fact, we shall show that this approach can, in theory, yield *all* information about the eigenvalues and, more generally, the spectrum, of the original problem.

The pathways from this project lead to two categories of impact: Knowledge and People, and Economy and Society.

Knowledge and People

We aim to train a postdoc with skills at the interface between analysis, applied mathematics and computational mathematics, opening for them a wide panorama
of career options by equipping them with skills which are valued both in industry and academia. Our project will link areas in which the UK is already an international leader, such as computational mathematics and numerical linear algebra, with areas in which we are internationally competitive and in which IRM 2004 and IRM 2010 recommended investment, including mathematical analysis. EPSRC has made what are arguably its most strategically significant investments in
doctoral education at this interface, funding CCA in Cambridge and the PDE CTD in Oxford. Funding our proposal would target the next level up, postdoctoral training. Many of the skills which would be acquired by the PDRA would also be applicable in other rapidly growing areas such as inverse problems and imaging, particularly in view of the fact that the PI has held continuous EPSRC funding on inverse-problem projects since January 2013.

Economy and Society

Ideas from spectral analysis of large matrices are now ubiquitous in daily life and data analysis. In Cardiff, our colleague Bertrand Gauthier of the
Statistics Group is working on a new technique for low-dimensional approximation of spectra of operators in reproducing kernel Hilbert spaces, which has shown great promise in the development of techniques in machine learning. Gauthier is actively discussing this work with the PI and other members of the PI's group. The
impact of these techniques will be felt by society and by our economy, even if those who experience the impact will be unaware of its origin. Cardiff School of
Mathematics also provides an outstanding environment in which to conduct this work as we have invested in new lectureships, both in mathematics and computer
science, to underpin these sorts of collaborations with a critical mass of resident expertise. As Director of Research for the School, the PI has had a key role in setting these strategic directions and choosing new appointees.

An important part of the fourth work package in this proposal will be to seek out the new applications such as these and provide the tools which allow techniques
initially developed by practitioners to be widened in their applicability. One enticing possible outcome here is a proper understanding of just why certain techniques in machine learning function well in certain situations and not in others. If achieved this would be comparable to the work of Hansen et al. in Cambridge
on compressed sensing, which provided the first properly rigorous analysis explaining the hidden limitations behind methods which, a priori, seemed too good to be
true.