Wave propagation and scattering is an important area which despite its long history (the "wave equation'' was derived by d'Alembert in 1747) still provides significant challenges in analysis and computation. Wave propagation has many important applications such as telecommunications, non-destructive testing, geological exploration, tomography, radar, sonar, and other military uses. It is very important for the applications areas to use reliable and efficient computational techniques, and developing these typically involves a combination of applied and numerical analysis.

One of my main research interests is the numerical approximation of time domain wave scattering problems which are formulated as boundary integral equations.   These problems are computationally challenging: convolution quadrature methods which use global time basis functions are numerically stable (but involve dense system matrices and are hence computationally expensive), and it is known that methods which use local time basis functions are much less stable (any perturbations such as induced by numerical quadrature can result in an exponentially unstable approximation).  Recent collaborative work is focused on the development of methods which share some of the desirable characteristics of convolution quadrature but the underlying basis functions have narrow support, which means that these new approximation methods are computationally efficient.alysis.

Another area of interest is the development of mathematical models (using elasticity) for the mechanical properties of various materials, and associated problems in elastic stability.