arXiv subjects

C. C. Pain

Publications and source records attributed to C. C. Pain.

17 recordsLinked to original sources

arXiv2025

Solving advection equations with reduction multigrids on GPUs

Methods for solving hyperbolic systems typically depend on unknown ordering (e.g., Gauss-Seidel, or sweep/wavefront/marching methods) to achieve good convergence. For many discretisations, mesh types or decompositions these methods do not scale well in parallel. In this work we demonstrate that the combination of AIRG (a reduction multigrid which uses GMRES polynomials) and PMISR DDC (a CF splitting algorithm which gives diagonally dominant submatrices) can be used to solve linear advection equations in parallel on GPUs with good weak scaling. We find that GMRES polynomials are well suited to GPUs when applied matrix-free, either as smoothers (at low order) or as an approximate coarse grid solver (at high order). To improve the parallel performance we automatically truncate the multigrid hierarchy given the quality of the polynomials as coarse grid solvers. Solving time-independent advection equations in 2D on structured grids, we find 66-101% weak scaling efficiency in the solve and 47-63% in the setup with AIRG, across the majority of Lumi-G, a pre-exascale GPU machine.

math.NA↗

arXiv2024

Coarsening and parallelism with reduction multigrids for hyperbolic Boltzmann transport

Reduction multigrids have recently shown good performance in hyperbolic problems without the need for Gauss-Seidel smoothers. When applied to the hyperbolic limit of the Boltzmann Transport Equation (BTE), these methods result in very close to $\mathcal{O}(n)$ growth in work with problem size on unstructured grids. This scalability relies on the CF splitting producing an $A_\textrm{ff}$ block that is easy to invert. We introduce a parallel two-pass CF splitting designed to give diagonally dominant $A_\textrm{ff}$. The first pass computes a maximal independent set in the symmetrized strong connections. The second pass converts F-points to C-points based on the row-wise diagonal dominance of $A_\textrm{ff}$. We find this two-pass CF splitting outperforms common CF splittings available in hypre. Furthermore, parallelisation of reduction multigrids in hyperbolic problems is difficult as we require both long-range grid-transfer operators and slow coarsenings (with rates of $\sim$1/2 in both 2D and 3D). We find that good parallel performance in the setup and solve is dependent on several factors: repartitioning the coarse grids, reducing the number of active MPI ranks as we coarsen, truncating the multigrid hierarchy and applying a GMRES polynomial as a coarse-grid solver. We compare the performance of two different reduction multigrids, AIRG (that we developed previously) and the hypre implementation of $\ell$AIR. In the streaming limit with AIRG, we demonstrate 81\% weak scaling efficiency in the solve from 2 to 64 nodes (256 to 8196 cores) with only 8.8k unknowns per core, with solve times up to 5.9$\times$ smaller than the $\ell$AIR implementation in hypre.

math.NA↗

arXiv2023

AIR multigrid with GMRES polynomials (AIRG) and additive preconditioners for Boltzmann transport

We develop a reduction multigrid based on approximate ideal restriction (AIR) for use with asymmetric linear systems. We use fixed-order GMRES polynomials to approximate $A_\textrm{ff}^{-1}$ and we use these polynomials to build grid transfer operators and perform F-point smoothing. We can also apply a fixed sparsity to these polynomials to prevent fill-in. When applied in the streaming limit of the Boltzmann Transport Equation (BTE), with a P$^0$ angular discretisation and a low-memory spatial discretisation on unstructured grids, this "AIRG" multigrid used as a preconditioner to an outer GMRES iteration outperforms the lAIR implementation in hypre, with two to three times less work. AIRG is very close to scalable; we find either fixed work in the solve with slight growth in the setup, or slight growth in the solve with fixed work in the setup when using fixed sparsity. Using fixed sparsity we see less than 20% growth in the work of the solve with either 6 levels of spatial refinement or 3 levels of angular refinement. In problems with scattering AIRG performs as well as lAIR, but using the full matrix with scattering is not scalable. We then present an iterative method designed for use with scattering which uses the additive combination of two fixed-sparsity preconditioners applied to the angular flux; a single AIRG V-cycle on the streaming/removal operator and a DSA method with a CG FEM. We find with space or angle refinement our iterative method is very close to scalable with fixed memory use.

physics.comp-ph↗

arXiv2023

Solving the Discretised Boltzmann Transport Equations using Neural Networks: Applications in Neutron Transport

In this paper we solve the Boltzmann transport equation using AI libraries. The reason why this is attractive is because it enables one to use the highly optimised software within AI libraries, enabling one to run on different computer architectures and enables one to tap into the vast quantity of community based software that has been developed for AI and ML applications e.g. mixed arithmetic precision or model parallelism. Here we take the first steps towards developing this approach for the Boltzmann transport equation and develop the necessary methods in order to do that effectively. This includes: 1) A space-angle multigrid solution method that can extract the level of parallelism necessary to run efficiently on GPUs or new AI computers. 2) A new Convolutional Finite Element Method (ConvFEM) that greatly simplifies the implementation of high order finite elements (quadratic to quintic, say). 3) A new non-linear Petrov-Galerkin method that introduces dissipation anisotropically.

cs.CE↗

arXiv2023

Solving the Discretised Neutron Diffusion Equations using Neural Networks

This paper presents a new approach which uses the tools within Artificial Intelligence (AI) software libraries as an alternative way of solving partial differential equations (PDEs) that have been discretised using standard numerical methods. In particular, we describe how to represent numerical discretisations arising from the finite volume and finite element methods by pre-determining the weights of convolutional layers within a neural network. As the weights are defined by the discretisation scheme, no training of the network is required and the solutions obtained are identical (accounting for solver tolerances) to those obtained with standard codes often written in Fortran or C++. We also explain how to implement the Jacobi method and a multigrid solver using the functions available in AI libraries. For the latter, we use a U-Net architecture which is able to represent a sawtooth multigrid method. A benefit of using AI libraries in this way is that one can exploit their power and their built-in technologies. For example, their executions are already optimised for different computer architectures, whether it be CPUs, GPUs or new-generation AI processors. In this article, we apply the proposed approach to eigenvalue problems in reactor physics where neutron transport is described by diffusion theory. For a fuel assembly benchmark, we demonstrate that the solution obtained from our new approach is the same (accounting for solver tolerances) as that obtained from the same discretisation coded in a standard way using Fortran. We then proceed to solve a reactor core benchmark using the new approach.

cs.CE↗

arXiv2023

Angular adaptivity in P0 space and reduced tolerance solves for Boltzmann transport

Previously we developed an adaptive method in angle, based on solving in Haar wavelet space with a matrix-free multigrid for Boltzmann transport problems. This method scalably mapped to the underlying P$^0$ space during every matrix-free matrix-vector product, however the multigrid method itself was not scalable in the streaming limit. To tackle this we recently built an iterative method based on using an ideal restriction multigrid with frozen GMRES polynomials (AIRG) for Boltzmann transport that showed scalable work with uniform P$^0$ angle in the streaming and scattering limits. This paper details the practical requirements of using this new iterative method with angular adaptivity. Hence we modify our angular adaptivity to occur directly in P$^0$ space, rather than the Haar space. We then develop a modified stabilisation term for our FEM method that results in scalable growth in the number of non-zeros in the streaming operator with P$^0$ adaptivity. We can therefore combine the use of this iterative method with P$^0$ angular adaptivity to solve problems in both the scattering and streaming limits, with close to fixed work and memory use. We also present a CF splitting for multigrid methods based on element agglomeration combined with angular adaptivity, that can produce a semi-coarsening in the streaming limit without access to the matrix entries. The equivalence between our adapted P$^0$ and Haar wavelet spaces also allows us to introduce a robust convergence test for our iterative method when using regular adaptivity. This allows the early termination of the solve in each adapt step, reducing the cost of producing an adapted angular discretisation.

physics.comp-ph↗

arXiv2021

Numerical study of COVID-19 spatial-temporal spreading in London

Recent study reported that an aerosolised virus (COVID-19) can survive in the air for a few hours. It is highly possible that people get infected with the disease by breathing and contact with items contaminated by the aerosolised virus. However, the aerosolised virus transmission and trajectories in various meteorological environments remain unclear. This paper has investigated the movement of aerosolised viruses from a high concentration source across a dense urban area. The case study looks at the highly air polluted areas of London: University College Hospital (UCH) and King Cross and St Pancras International Station (KCSPI). We explored the spread and decay of COVID-19 released from the hospital and railway stations with the prescribed meteorological conditions. The study has three key findings: the primary result is that it is possible for the virus to travel from meters up to hundred meters from the source location. The secondary finding shows viruses released into the atmosphere from entry and exit points at KCSPI remain trapped within a small radial distance of < 50m. This strengthens the case for the use of face coverings to reduce the infection rate. The final finding shows that there are different levels of risk at various door locations for UCH, depending on which door is used there can be a higher concentration of COVID-19. Although our results are based on London, since the fundamental knowledge processes are the same, our study can be further extended to other locations (especially the highly air polluted areas) in the world.

physics.med-ph↗

arXiv2020

A comparison of element agglomeration algorithms for unstructured geometric multigrid

This paper compares the performance of seven different element agglomeration algorithms on unstructured triangular/tetrahedral meshes when used as part of a geometric multigrid. Five of these algorithms come from the literature on AMGe multigrid and mesh partitioning methods. The resulting multigrid schemes are tested matrix-free on two problems in 2D and 3D taken from radiation transport applications; one of which is in the diffusion limit. In two dimensions all coarsening algorithms result in multigrid methods which perform similarly, but in three dimensions aggressive element agglomeration performed by METIS produces the shortest runtimes and multigrid setup times.

math.NA↗

arXiv2020

An advanced hybrid deep adversarial autoencoder for parameterized nonlinear fluid flow modelling

Considering the high computation cost produced in conventional computation fluid dynamic simulations, machine learning methods have been introduced to flow dynamic simulations in recent years. However, most of studies focus mainly on existing fluid fields learning, the prediction of spatio-temporal nonlinear fluid flows in varying parameterized space has been neglected. In this work, we propose a hybrid deep adversarial autoencoder (DAA) to integrate generative adversarial network (GAN) and variational autoencoder (VAE) for predicting parameterized nonlinear fluid flows in spatial and temporal space. High-dimensional inputs are compressed into the low-representation representations by nonlinear functions in a convolutional encoder. In this way, the predictive fluid flows reconstructed in a convolutional decoder contain the dynamic flow physics of high nonlinearity and chaotic nature. In addition, the low-representation representations are applied into the adversarial network for model training and parameter optimization, which enables a fast computation process. The capability of the hybrid DAA is demonstrated by varying inputs on a water collapse example. Numerical results show that this hybrid DAA has successfully captured the spatio-temporal flow features with CPU speed-up of three orders of magnitude. Promising results suggests that the hybrid DAA can play a critical role in efficiently and accurately predicting complex flows in future.

physics.flu-dyn↗

arXiv2020

Data-driven modelling of nonlinear spatio-temporal fluid flows using a deep convolutional generative adversarial network

Deep learning techniques for improving fluid flow modelling have gained significant attention in recent years. Advanced deep learning techniques achieve great progress in rapidly predicting fluid flows without prior knowledge of the underlying physical relationships. Advanced deep learning techniques achieve great progress in rapidly predicting fluid flows without prior knowledge of the underlying physical relationships. However, most of existing researches focused mainly on either sequence learning or spatial learning, rarely on both spatial and temporal dynamics of fluid flows (Reichstein et al., 2019). In this work, an Artificial Intelligence (AI) fluid model based on a general deep convolutional generative adversarial network (DCGAN) has been developed for predicting spatio-temporal flow distributions. In deep convolutional networks, the high-dimensional flows can be converted into the low-dimensional "latent" representations. The complex features of flow dynamics can be captured by the adversarial networks. The above DCGAN fluid model enables us to provide reasonable predictive accuracy of flow fields while maintaining a high computational efficiency. The performance of the DCGAN is illustrated for two test cases of Hokkaido tsunami with different incoming waves along the coastal line. It is demonstrated that the results from the DCGAN are comparable with those from the original high fidelity model (Fluidity). The spatio-temporal flow features have been represented as the flow evolves, especially, the wave phases and flow peaks can be captured accurately. In addition, the results illustrate that the online CPU cost is reduced by five orders of magnitude compared to the original high fidelity model simulations. The promising results show that the DCGAN can provide rapid and reliable spatio-temporal prediction for nonlinear fluid flows.

physics.ao-ph↗

arXiv2019

Goal-based angular adaptivity for Boltzmann transport in the presence of ray-effects

Boltzmann transport problems often involve heavy streaming, where particles propagate long distance due to the dominance of advection over particle interaction. If an insufficiently refined non-rotationally invariant angular discretisation is used, there are areas of the problem where no particles will propogate. These "ray-effects" are problematic for goal-based error metrics with angular adaptivty, as the metrics in the pre-asymptotic region will be zero/incorrect and angular adaptivity will not occur. In this work we use low-order filtered spherical harmonics, which is rotationally invariant and hence not subject to ray-effects, to "bootstrap" our error metric and enable highly refined anisotropic angular adaptivity with a Haar wavelet angular discretisation. We test this on three simple problems with pure streaming where we know a priori where refinement should occur. We show our method is robust and produces adapted angular discretisations that match the results produced by fixed refinement with either reduced runtime or a constant additional cost with angular refinement.

math.NA↗

arXiv2019

Angular adaptivity with spherical harmonics for Boltzmann transport

This paper describes an angular adaptivity algorithm for Boltzmann transport applications which uses Pn and filtered Pn expansions, allowing for different expansion orders across space/energy. Our spatial discretisation is specifically designed to use less memory than competing DG schemes and also gives us direct access to the amount of stabilisation applied at each node. For filtered Pn expansions, we then use our adaptive process in combination with this net amount of stabilisation to compute a spatially dependent filter strength that does not depend on a priori spatial information. This applies heavy filtering only where discontinuities are present, allowing the filtered Pn expansion to retain high-order convergence where possible. Regular and goal-based error metrics are shown and both the adapted Pn and adapted filtered Pn methods show significant reductions in DOFs and runtime. The adapted filtered Pn with our spatially dependent filter shows close to fixed iteration counts and up to high-order is even competitive with P0 discretisations in problems with heavy advection.

physics.comp-ph↗

arXiv2019

Scalable angular adaptivity for Boltzmann transport

This paper describes an angular adaptivity algorithm for Boltzmann transport applications which for the first time shows evidence of $\mathcal{O}(n)$ scaling in both runtime and memory usage, where $n$ is the number of adapted angles. This adaptivity uses Haar wavelets, which perform structured $h$-adaptivity built on top of a hierarchical P$_0$ FEM discretisation of a 2D angular domain, allowing different anisotropic angular resolution to be applied across space/energy. Fixed angular refinement, along with regular and goal-based error metrics are shown in three example problems taken from neutronics/radiative transfer applications. We use a spatial discretisation designed to use less memory than competing alternatives in general applications and gives us the flexibility to use a matrix-free multgrid method as our iterative method. This relies on scalable matrix-vector products using Fast Wavelet Transforms and allows the use of traditional sweep algorithms if desired.

physics.comp-ph↗

arXiv2018

Goal-based sensitivity maps using time windows and ensemble perturbations

We present an approach for forming sensitivity maps (or sensitivites) using ensembles. The method is an alternative to using an adjoint, which can be very challenging to formulate and also computationally expensive to solve. The main novelties of the presented approach are: 1) the use of goals, weighting the perturbation to help resolve the most important sensitivities, 2) the use of time windows, which enable the perturbations to be optimised independently for each window and 3) re-orthogonalisation of the solution through time, which helps optimise each perturbation when calculating sensitivity maps. These novel methods greatly reduce the number of ensembles required to form the sensitivity maps as demonstrated in this paper. As the presented method relies solely on ensembles obtained from the forward model, it can therefore be applied directly to forward models of arbitrary complexity arising from, for example, multi-physics coupling, legacy codes or model chains. It can also be applied to compute sensitivities for optimisation of sensor placement, optimisation for design or control, goal-based mesh adaptivity, assessment of goals (e.g. hazard assessment and mitigation in the natural environment), determining the worth of current data and data assimilation. We analyse and demonstrate the efficiency of the approach by applying the method to advection problems and also a non-linear heterogeneous multi-phase porous media problem, showing, in all cases, that the number of ensembles required to obtain accurate sensitivity maps is relatively low, in the order of 10s.

cs.CE↗

arXiv2009

Solving the Poisson equation on small aspect ratio domains using unstructured meshes

We discuss the ill conditioning of the matrix for the discretised Poisson equation in the small aspect ratio limit, and motivate this problem in the context of nonhydrostatic ocean modelling. Efficient iterative solvers for the Poisson equation in small aspect ratio domains are crucial for the successful development of nonhydrostatic ocean models on unstructured meshes. We introduce a new multigrid preconditioner for the Poisson problem which can be used with finite element discretisations on general unstructured meshes; this preconditioner is motivated by the fact that the Poisson problem has a condition number which is independent of aspect ratio when Dirichlet boundary conditions are imposed on the top surface of the domain. This leads to the first level in an algebraic multigrid solver (which can be extended by further conventional algebraic multigrid stages), and an additive smoother. We illustrate the method with numerical tests on unstructured meshes, which show that the preconditioner makes a dramatic improvement on a more standard multigrid preconditioner approach, and also show that the additive smoother produces better results than standard SOR smoothing. This new solver method makes it feasible to run nonhydrostatic unstructured mesh ocean models in small aspect ratio domains.

math.NA↗

arXiv2008

A mixed discontinuous/continuous finite element pair for shallow-water ocean modelling

We introduce a mixed discontinuous/continuous finite element pair for ocean modelling, with continuous quadratic pressure/layer depth and discontinuous velocity. We investigate the finite element pair applied to the linear shallow-water equations on an f-plane. The element pair has the property that all geostrophically balanced states which strongly satisfy the boundary conditions have discrete divergence equal to exactly zero and hence are exactly steady states of the discretised equations. This means that the finite element pair has excellent geostrophic balance properties. We illustrate these properties using numerical tests and provide convergence calculations which show that the discretisation has quadratic errors, indicating that the element pair is stable.

math.NA↗

arXiv2007

LBB Stability of a Mixed Discontinuous/Continuous Galerkin Finite Element Pair

We introduce a new mixed discontinuous/continuous Galerkin finite element for solving the 2- and 3-dimensional wave equations and equations of incompressible flow. The element, which we refer to as P1dg-P2, uses discontinuous piecewise linear functions for velocity and continuous piecewise quadratic functions for pressure. The aim of introducing the mixed formulation is to produce a new flexible element choice for triangular and tetrahedral meshes which satisfies the LBB stability condition and hence has no spurious zero-energy modes. We illustrate this property with numerical integrations of the wave equation in two dimensions, an analysis of the resultant discrete Laplace operator in two and three dimensions, and a normal mode analysis of the semi-discrete wave equation in one dimension.

math.NA↗