Applied Mathematics Seminar
University of Leeds
Welcome to the Leeds Applied Mathematics Seminar
Autumn 2018
Monday 22 October 2018
Andrew Archer (University of Loughborough)
Introduction to classical density functional theory (DFT) and dynamical density functional theory (DDFT)
In this talk I will give an introduction to classical^** density functional theory (DFT) and dynamical density functional theory (DDFT). DFT is a statistical-mechanical theory for the average density distribution of a system of classical particles in the presence of an external field. It is a microscopic theory, able to describe the structure even down to the scale of the individual particles. A typical problem that can be solved using DFT is to determine the density distribution of molecules in a liquid in the presence of a container wall. It can also describe solid phases of matter and the density distribution at the interface between a crystal coexisting with the liquid phase. In addition to yielding particle density distributions, DFT can also be used to obtain thermodynamic quantities such as free energies and interfacial tensions. DDFT is a generalisation of DFT, able to describe the non-equilibrium time evolution of the density profile. Originally developed for systems of Brownian particles with stochastic equations of motion, the formalism has been extended to describe many body systems evolving under Newton's equations of motion and also underdamped stochastic equations of motion.
^** Classical DFT should not be confused with quantum DFT, a related theory for the density distribution of electrons in matter. I will not discuss quantum DFT.
Monday 12 November 2018
David Ham (Imperial College London)
Firedrake: combining symbolic and computational mathematics for high performance, high productivity simulation
There is a productivity problem in simulation science. Creating simulation code which deploys advanced numerical methods on complex PDEs coupled to sophisticated solvers and preconditioners is laborious, tedious, and error-prone. Far too many PhD students and researchers spend most of their time re-implementing and debugging known techniques rather than advancing their research goals, and the high cost of implementation impedes innovation: the first approach attempted, however poor, is frequently the only attempt as the resources required to try again are unavailable. As with many productivity problems across the economy, the root problem here is that processes which could and should be automated are instead being done by hand.
Numerically solving a PDE has two components: mathematically deriving the discrete computation, and then the actual calculation. The prevailing approach in scientific computing is to derive by hand and then code the calculation. However, this approach neglects the fact that the tools required to have a computed conduct symbolic mathematics are well-established. Firedrake, in contrast, brings together computer algebra with scientific computing to deliver a system which automates the numerical solution of PDEs by the finite element method. The user with a PDE to solve specifies the weak form of that equation in high level mathematical notation, including the choices of function spaces, boundary and initial conditions. Firedrake generates, compiles, and executes high performance parallel C code implementations of the discrete operators. The resulting linear and nonlinear systems are solved with the user-programmable and symbolically composable PETSc system. The symbolic interface to the solver system enables sophisticated operator preconditioners to be generated symbolically on the fly. The code generation layer employs advanced compiler technology which generates algorithmically optimal vectorised implementations which would be prohibitively complex to code by hand. Firedrake also composes with higher level tools which can exploit its symbolic reasoning capabilities to, for example, automatically execute the adjoint to a simulation.
In this talk I will present the core automated mathematical approach of the Firedrake and discuss in some detail recent advances in algorithmically optimal operator generation and hybridising solver capabilities.
Monday 19 November 2018
Suzanne Fielding (University of Durham)
Complex flows of complex fluids
Following a pedagogical introduction to complex fluids and to rheology (the study of deformation and flow), I shall discuss recent progress modelling instabilities that arise at the free surface where a flowing complex fluid meets the outside air, both in extensional flow and also (time permitting) in shear flow.