Eike Mueller

Eike Mueller, Xu Guo, Robert Scheichl et al.

Many problems in geophysical and atmospheric modelling require the fast solution of elliptic partial differential equations (PDEs) in "flat" three dimensional geometries. In particular, an anisotropic elliptic PDE for the pressure correction has to be solved at every time step in the dynamical core of many numerical weather prediction models, and equations of a very similar structure arise in global ocean models, subsurface flow simulations and gas and oil reservoir modelling. The elliptic solve is often the bottleneck of the forecast, and an algorithmically optimal method has to be used and implemented efficiently. Graphics Processing Units have been shown to be highly efficient for a wide range of applications in scientific computing, and recently iterative solvers have been parallelised on these architectures. We describe the GPU implementation and optimisation of a Preconditioned Conjugate Gradient (PCG) algorithm for the solution of a three dimensional anisotropic elliptic PDE for the pressure correction in NWP. Our implementation exploits the strong vertical anisotropy of the elliptic operator in the construction of a suitable preconditioner. As the algorithm is memory bound, performance can be improved significantly by reducing the amount of global memory access. We achieve this by using a matrix-free implementation which does not require explicit storage of the matrix and instead recalculates the local stencil. Global memory access can also be reduced by rewriting the algorithm using loop fusion and we show that this further reduces the runtime on the GPU. We demonstrate the performance of our matrix-free GPU code by comparing it to a sequential CPU implementation and to a matrix-explicit GPU code which uses existing libraries. The absolute performance of the algorithm for different problem sizes is quantified in terms of floating point throughput and global memory bandwidth.

Teo Deveney, Eike Mueller, Tony Shardlow

We introduce a deep learning accelerated methodology to solve PDE-based Bayesian inverse problems with guaranteed accuracy. This is motivated by the ill-posed problem of inferring a spatio-temporal heat-flux parameter known as the Biot number given temperature data, however the methodology is generalisable to other settings. To accelerate Bayesian inference, we develop a novel training scheme that uses data to adaptively train a neural-network surrogate simulating the parametric forward model. By simultaneously identifying an approximate posterior distribution over the Biot number, and weighting a physics-informed training loss according to this, our approach approximates forward and inverse solution together without any need for external solves. Using a random Chebyshev series, we outline how to approximate a Gaussian process prior, and using the surrogate we apply Hamiltonian Monte Carlo (HMC) to sample from the posterior distribution. We derive convergence of the surrogate posterior to the true posterior distribution in the Hellinger metric as our adaptive loss approaches zero. Additionally, we describe how this surrogate-accelerated HMC approach can be combined with traditional PDE solvers in a delayed-acceptance scheme to a-priori control the posterior accuracy. This overcomes a major limitation of deep learning-based surrogate approaches, which do not achieve guaranteed accuracy a-priori due to their non-convex training. Biot number calculations are involved in turbo-machinery design, which is safety critical and highly regulated, therefore it is important that our results have such mathematical guarantees. Our approach achieves fast mixing in high dimensions whilst retaining the convergence guarantees of a traditional PDE solver, and without the burden of evaluating this solver for proposals that are likely to be rejected. Numerical results are given using real and simulated data.