Thu 06 Jul 15:00: Randomized sketching of Krylov methods in numerical linear algebra
Many large-scale computations in numerical linear algebra are powered by Krylov methods, including the solution of linear systems of equations, least squares problems, linear and nonlinear eigenvalue problems, matrix functions and matrix equations, etc. We will discuss some recent ideas to speed up Krylov methods for these tasks using randomized sketching, and highlight some of the key challenges for future research.
- Speaker: Stefan Guettel (Manchester)
- Thursday 06 July 2023, 15:00-16:00
- Venue: Centre for Mathematical Sciences, MR14.
- Series: Applied and Computational Analysis; organiser: Matthew Colbrook.
Thu 22 Jun 15:30: Computing eigenvalues of the Laplacian on rough domains
In this talk, I shall present a joint work with Frank Rösler in which we consider the computability of eigenvalues of the Dirichlet Laplacian on bounded domains with rough, possibly fractal, boundaries. We work within the framework of Solvability Complexity Indices, which allows us to formulate the problem rigorously. On one hand, we construct an algorithm that provably converges for any domain satisfying a collection of mild topological hypotheses, on the other, we prove that there does not exist an algorithm of the same type which converges for an arbitrary bounded domain. Along the way, we develop new spectral convergence results for the Dirichlet Laplacian on rough domains, as well as a novel Poincaré-type inequality.
- Speaker: Alexei Stepanenko
- Thursday 22 June 2023, 15:30-16:30
- Venue: Centre for Mathematical Sciences, MR14.
- Series: Applied and Computational Analysis; organiser: Matthew Colbrook.
Thu 08 Jun 15:00: Integral equations for wave scattering by fractals
Integral equations are a powerful and popular tool for the numerical solution of linear PDEs for which a fundamental solution is available. They are of particular importance in the study of acoustic, electromagnetic and elastic wave propagation, where wave scattering problems posed in unbounded domains can often be formulated as an integral equation over the (typically bounded) scatterer or its boundary. For scatterers with smooth boundaries this is classical, but many real-life scatterers (e.g. trees/vegetation, snowflakes/ice crystal aggregates) are highly irregular. The case where the scatterer (or its boundary) is fractal poses particularly interesting challenges, and our recent investigations into this topic have led to new results in function spaces, variational problems, numerical quadrature and integral equations, which I will survey in this talk. Computationally, we have studied two main approaches: (1) approximate the fractal by a smoother “prefractal” shape, and (2) work with integral equations formulated directly on the fractal, with respect to the appropriate fractal (Hausdorff) measure. The latter approach seems to provide a clearer pathway for rigorous convergence analysis, but for numerical implementation requires accurate quadrature rules for evaluating singular integrals with respect to fractal measures.
- Speaker: David Hewett (UCL)
- Thursday 08 June 2023, 15:00-16:00
- Venue: Centre for Mathematical Sciences, MR14.
- Series: Applied and Computational Analysis; organiser: Matthew Colbrook.