Numerical analysis and high-accuracy algorithms for scattering, spectra, and interdisciplinary applications.

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Mathematics provides an elegant framework to understand physical and biological phenomena, and the tools with which to probe the detailed behaviour of these systems. In this research program, I will be investigating some specific physical and biological systems from a mathematical viewpoint. A key goal is to develop robust and efficient computational techniques to help further our understanding. My research interests are concentrated in two major areas: (A) numerical analysis for scattering problems (with a recent focus on associated spectral problems), and (B) applications of mathematics to interdisciplinary problems. There is a natural mathematical connection between these two areas, with novel problems in science and engineering requiring the development of new algorithms. Conversely, careful analysis and simulation can yield deep insight into basic science questions. Computational scattering is concerned with the development and analysis of algorithms for studying wave propagation. My proposed research program in (A) continues the investigation and creation of fast and accurate structure--preserving algorithms to study challenging problems in scattering. My research has evolved to include eigenvalue problems (EVP). A new direction is the related area of computational spectral geometry, which concerns numerically simulating the interplay between the shape of an object and its spectrum. The most intuitive example is that of the fundamental tones of drumheads: given a drum, can we tell what its tones will be? And conversely, if we wanted to design a drumhead with a given set of tones, how would we find its shape? Some of these questions are classical, and others are of immediate societal relevance. The development of these computational tools presents many interesting mathematical and computational challenges, which we address in the program. Amongst the avenues we will explore is the possibility of using ideas from Bayesian optimization to develop new strategies for spectral calculations. The proposed program involves investigation in several related topics, including the analysis and approximation of an unusual constrained EVP arising in fluid--structure interaction. and the development of high--accuracy computational strategies for mixed Dirichlet--Neumann and Steklov eigenvalue computations. We will provide approximation strategies for these EVP, and investigate their use for computational spectral geometry. In area (B), I bring mathematical tools to bear on interdisciplinary problems, in collaborations with researchers in science and industry. Some existing questions concern applications of ideas from (A). In others, I will continue efforts to understand at a fundamental level how muscles produce force: how does muscle architecture, mass and tissue properties change force output? Mathematics can be used to shed light on the behaviour of these and other systems.