Oxford

When approximating PDEs with the finite element method, large sparse linear systems must be solved. The ideal preconditioner yields convergence that is  algorithmically optimal and parameter robust, i.e. the number of Krylov iterations required to solve the linear system to a given accuracy does not grow substantially as the mesh or problem parameters are changed.

Achieving this for the stationary Navier-Stokes has proven challenging: LU factorisation is Reynolds-robust but scales poorly with degree of freedom count, while Schur complement approximations such as PCD and LSC degrade as the Reynolds number is increased.

Building on the work of Schöberl, Olshanskii and Benzi, in this talk we present the first preconditioner for the Newton linearisation of the stationary Navier--Stokes equations in three dimensions that achieves both optimal complexity and Reynolds-robustness. The scheme combines a novel tailored finite element discretisation, discrete augmented Lagrangian stabilisation, a custom prolongation operator involving local solves on coarse cells, and an additive patchwise relaxation on each
level. We present 3D simulations with over one billion degrees of freedom with robust performance from Reynolds number 10 to 5000.

Employing the usual multilevel Monte Carlo estimator, we introduce a framework for estimating the solutions of SDEs by an Euler-Maruyama scheme. By considering the expected value of such solutions, we produce simulations using approximately normal random variables, and recover the estimate from the exact normal distribution by use of a multilevel correction, leading to faster simulations without loss of accuracy. We will also highlight this concept in the framework of reduced precision and vectorised computations.

In this talk I will introduce a new mathematical model for the computational modelling of the active contraction of cardiac tissue using stress-assisted conductivity as the main mechanism for mechanoelectrical feedback. The coupling variable is the Kirchhoff stress and so the equations of hyperelasticity are written in mixed form and a suitable finite element formulation is proposed. Next I will introduce a simplified version of the coupled system, focusing on its analysis in terms of solvability and stability of continuous and discrete mixed-primal formulations, and the convergence of these methods will be illustrated through two numerical tests.

In petroleum reservoir simulation, the standard preconditioner is a two-stage process which involves solving a restricted pressure system with AMG. Initially designed for isothermal models, this approach is often used in the thermal case. However, it does not incorporate heat diffusion or the effects of temperature changes on fluid flow through viscosity and density. We seek to develop preconditioners which consider this cross-coupling between pressure and temperature. In order to study the effects of both pressure and temperature on fluid and heat flow, we first consider a model of non-isothermal single phase flow through porous media. For this model, we develop a block preconditioner with an efficient Schur complement approximation. Then, we extend this method for multiphase flow as a two-stage preconditioner.

The talk introduces the problem of completing a partially observed matrix whose columns obey a nonlinear structure. This is an extension of classical low-rank matrix completion where the structure is linear. Such matrices are in general full rank, but it is often possible to exhibit a low rank structure when the data is lifted to a higher dimensional space of features. The presence of a nonlinear lifting makes it impossible to write the problem using common low-rank matrix completion formulations. We investigate formulations as a nonconvex optimisation problem and optimisation on Riemannian manifolds.

This work determines an aim point selection strategy for players in order to improve their chances of winning at the classic darts game of 501. Although many previous studies have considered the problem of aim point selection in order to maximise the expected score a player can achieve, few have considered the more general strategical question of minimising the expected number of turns required for a player to finish. By casting the problem as a Markov decision process, a framework is derived for the identification of the optimal aim point for a player in an arbitrary game scenario.

There is no more classical problem of numerical PDE than the Laplace equation in a polygon, but Abi Gopal and I think we are on to a big step forward. The traditional approaches would be finite elements, giving a 2D representation of the solution, or integral equations, giving a 1D representation. The new approach, inspired by an approximation theory result of Donald Newman in 1964, leads to a "0D representation" -- the solution is the real part of a rational function with poles clustered exponentially near the corners of the polygon. The speed and accuracy of this approach are remarkable. For typical polygons of up to 8 vertices, we can solve the problem in less than a second on a laptop and evaluate the result in a few microseconds per point, with 6-digit accuracy all the way up to the corner singularities. We don't think existing methods come close to such performance. Next step: Helmholtz?