arXiv subjects

Eike Hermann Müller

Publications and source records attributed to Eike Hermann Müller.

11 recordsLinked to original sources

arXiv2024

A Higher-order Hybridisable Discontinuous Galerkin IMEX method for the incompressible Euler equations

The incompressible Euler equations are an important model system in computational fluid dynamics. Fast high-order methods for the solution of this time-dependent system of partial differential equations are of particular interest: due to their exponential convergence in the polynomial degree they can make efficient use of computational resources. To address this challenge we describe a novel timestepping method which combines a hybridised Discontinuous Galerkin method for the spatial discretisation with IMEX timestepping schemes, thus achieving high-order accuracy in both space and time. The computational bottleneck is the solution of a (block-) sparse linear system to compute updates to pressure and velocity at each stage of the IMEX integrator. Following Chorin's projection approach, this update of the velocity and pressure fields is split into two stages. As a result, the hybridised equation for the implicit pressure-velocity problem is reduced to the well-known system which arises in hybridised mixed formulations of the Poisson- or diffusion problem and for which efficient multigrid preconditioners have been developed. Splitting errors can be reduced systematically by embedding this update into a preconditioned Richardson iteration. The accuracy and efficiency of the new method is demonstrated numerically for two time-dependent testcases that have been previously studied in the literature.

math.NA↗

arXiv2023

Exact conservation laws for neural network integrators of dynamical systems

The solution of time dependent differential equations with neural networks has attracted a lot of attention recently. The central idea is to learn the laws that govern the evolution of the solution from data, which might be polluted with random noise. However, in contrast to other machine learning applications, usually a lot is known about the system at hand. For example, for many dynamical systems physical quantities such as energy or (angular) momentum are exactly conserved. Hence, the neural network has to learn these conservation laws from data and they will only be satisfied approximately due to finite training time and random noise. In this paper we present an alternative approach which uses Noether's Theorem to inherently incorporate conservation laws into the architecture of the neural network. We demonstrate that this leads to better predictions for three model systems: the motion of a non-relativistic particle in a three-dimensional Newtonian gravitational potential, the motion of a massive relativistic particle in the Schwarzschild metric and a system of two interacting particles in four dimensions.

math.DS↗

arXiv2021

A new algorithm for electrostatic interactions in Monte Carlo simulations of charged particles

To minimise systematic errors in Monte Carlo simulations of charged particles, long range electrostatic interactions have to be calculated accurately and efficiently. Standard approaches, such as Ewald summation or the naive application of the classical Fast Multipole Method, result in a cost per Metropolis-Hastings step which grows in proportion to some positive power of the number of particles $N$ in the system. This prohibitively large cost prevents accurate simulations of systems with a sizeable number of particles. Currently, large systems are often simulated by truncating the Coulomb potential which introduces uncontrollable systematic errors. In this paper we present a new multilevel method which reduces the computational complexity to $\mathcal{O}(\log(N))$ per Metropolis-Hastings step, while maintaining errors which are comparable to direct Ewald summation. We show that compared to related previous work, our approach reduces the overall cost by better balancing time spent in the proposal- and acceptance- stages of each Metropolis-Hastings step. By simulating large systems with up to $N=10^5$ particles we demonstrate that our implementation is competitive with state-of-the-art MC packages and allows the simulation of very large systems of charged particles with accurate electrostatics.

physics.comp-ph↗

arXiv2020

Multilevel Monte Carlo for quantum mechanics on a lattice

Monte Carlo simulations of quantum field theories on a lattice become increasingly expensive as the continuum limit is approached since the cost per independent sample grows with a high power of the inverse lattice spacing. Simulations on fine lattices suffer from critical slowdown, the rapid growth of autocorrelations in the Markov chain. This causes a strong increase in the number of lattice configurations that have to be generated to obtain statistically significant results. This paper discusses hierarchical sampling methods to tame the growth in autocorrelations. Combined with multilevel variance reduction, this significantly reduces the computational cost of simulations for given tolerances $ε_{\text{disc}}$ on the discretisation error and $ε_{\text{stat}}$ on the statistical error. For observables with lattice errors of order $α$ and integrated autocorrelation times that grow like $τ_{\mathrm{int}}\propto a^{-z}$, multilevel Monte Carlo (MLMC) reduces the cost from $\mathcal{O}(ε_{\text{stat}}^{-2}ε_{\text{disc}}^{-(1+z)/α})$ to $\mathcal{O}(ε_{\text{stat}}^{-2}\vert\log ε_{\text{disc}} \vert^2+ε_{\text{disc}}^{-1/α})$ or $\mathcal{O}(ε_{\text{stat}}^{-2}+ε_{\text{disc}}^{-1/α})$. Higher gains are expected for simulations of quantum field theories in $D$ dimensions. The efficiency of the approach is demonstrated on two model systems, including a topological oscillator that is badly affected by critical slowdown from topological charge freezing. On fine lattices, the new methods are orders of magnitude faster than standard Hybrid Monte Carlo sampling. For high resolutions, MLMC can be used to accelerate even the cluster algorithm for the topological oscillator. Performance is further improved through perturbative matching which guarantees efficient coupling of theories on the multilevel hierarchy.

hep-lat↗

arXiv2020

Multigrid preconditioners for the mixed finite element dynamical core of the LFRic atmospheric model

Due to the wide separation of time scales in geophysical fluid dynamics, semi-implicit time integrators are commonly used in operational atmospheric forecast models. They guarantee the stable treatment of fast (acoustic and gravity) waves, while not suffering from severe restrictions on the timestep size. To propagate the state of the atmosphere forward in time, a non-linear equation for the prognostic variables has to be solved at every timestep. Since the nonlinearity is typically weak, this is done with a small number of Newton- or Picard- iterations, which in turn require the efficient solution of a large system on linear equations with $\mathcal{O}(10^6-10^9)$ unknowns. This linear solve is often the computationally most costly part of the model. In this paper an efficient linear solver for the LFRic next-generation model, currently developed by the Met Office, is described. The model uses an advanced mimetic finite element discretisation which makes the construction of efficient solvers challenging compared to models using standard finite-difference and finite-volume methods. The linear solver hinges on a bespoke multigrid preconditioner of the Schur-complement system for the pressure correction. By comparing to Krylov-subspace methods, the superior performance and robustness of the multigrid algorithm is demonstrated for standard test cases and realistic model setups. In production mode, the model will have to run in parallel on 100,000s of processing elements. As confirmed by numerical experiments, one particular advantage of the multigrid solver is its excellent parallel scalability due to avoiding expensive global reduction operations.

physics.comp-ph↗

arXiv2020

Parallel Performance of ARM ThunderX2 for Atomistic Simulation Algorithms

Atomistic simulation drives scientific advances in modern material science and accounts for a significant proportion of wall time on High Performance Computing facilities. It is important that algorithms are efficient and implementations are performant in a continuously diversifying hardware landscape. Furthermore, they have to be portable to make best use of the available computing resource. In this paper we assess the parallel performance of some key algorithms implemented in a performance portable framework developed by us. We consider Molecular Dynamics with short range interactions, the Fast Multipole Method and Kinetic Monte Carlo. To assess the performance of emerging architectures, we compare the Marvell ThunderX2 (ARM) architecture to traditional x86_64 hardware made available through the Azure cloud computing service.

cs.DC↗

arXiv2020

Fast electrostatic solvers for kinetic Monte Carlo simulations

Kinetic Monte Carlo (KMC) is an important computational tool in physics and chemistry. In contrast to standard Monte Carlo, KMC permits the description of time dependent dynamical processes and is not restricted to systems in equilibrium. Recently KMC has been applied successfully in modelling of novel energy materials such as Lithium-ion batteries and solar cells. We consider general solid state systems which contain free, interacting particles which can hop between localised sites in the material. The KMC transition rates for those hops depend on the change in total potential energy of the system. For charged particles this requires the frequent calculation of electrostatic interactions, which is usually the bottleneck of the simulation. To avoid this issue and obtain results in reasonable times, many studies replace the long-range potential by a short range approximation. This, however, leads to systematic errors and unphysical results. On the other hand standard electrostatic solvers such as Ewald summation or fast Poisson solvers are highly inefficient or introduce uncontrollable systematic errors at high resolution. In this paper we describe how the Fast Multipole Method by Greengard and Rokhlin can be adapted to overcome this issue by dramatically reducing computational costs. We exploit the fact that each update in the transition rate calculation corresponds to a single particle move and changes the configuration only by a small amount. This allows us to construct an algorithm which scales linearly in the number of charges for each KMC step, something which had not been deemed to be possible before. We demonstrate the performance and parallel scalability of the method by implementing it in a performance portable software library. We describe the high-level Python interface of the code which makes it easy to adapt to specific cases.

physics.comp-ph↗

arXiv2019

Matrix-free multigrid block-preconditioners for higher order Discontinuous Galerkin discretisations

Efficient and suitably preconditioned iterative solvers for elliptic partial differential equations (PDEs) of the convection-diffusion type are used in all fields of science and engineering. To achieve optimal performance, solvers have to exhibit high arithmetic intensity and need to exploit every form of parallelism available in modern manycore CPUs. The computationally most expensive components of the solver are the repeated applications of the linear operator and the preconditioner. For discretisations based on higher-order Discontinuous Galerkin methods, sum-factorisation results in a dramatic reduction of the computational complexity of the operator application while, at the same time, the matrix-free implementation can run at a significant fraction of the theoretical peak floating point performance. Multigrid methods for high order methods often rely on block-smoothers to reduce high-frequency error components within one grid cell. Traditionally, this requires the assembly and expensive dense matrix solve in each grid cell, which counteracts any improvements achieved in the fast matrix-free operator application. To overcome this issue, we present a new matrix-free implementation of block-smoothers. Inverting the block matrices iteratively avoids storage and factorisation of the matrix and makes it is possible to harness the full power of the CPU. We implemented a hybrid multigrid algorithm with matrix-free block-smoothers in the high order DG space combined with a low order coarse grid correction using algebraic multigrid where only low order components are explicitly assembled. The effectiveness of this approach is demonstrated by solving a set of representative elliptic PDEs of increasing complexity, including a convection dominated problem and the stationary SPE10 benchmark.

math.NA↗

arXiv2016

High level implementation of geometric multigrid solvers for finite element problems: applications in atmospheric modelling

The implementation of efficient multigrid preconditioners for elliptic partial differential equations (PDEs) is a challenge due to the complexity of the resulting algorithms and corresponding computer code. For sophisticated finite element discretisations on unstructured grids an efficient implementation can be very time consuming and requires the programmer to have in-depth knowledge of the mathematical theory, parallel computing and optimisation techniques on manycore CPUs. In this paper we show how the development of bespoke multigrid preconditioners can be simplified significantly by using a framework which allows the expression of the each component of the algorithm at the correct abstraction level. Our approach (1) allows the expression of the finite element problem in a language which is close to the mathematical formulation of the problem, (2) guarantees the automatic generation and efficient execution of parallel optimised low-level computer code and (3) is flexible enough to support different abstraction levels and give the programmer control over details of the preconditioner. We use the composable abstractions of the Firedrake/PyOP2 package to demonstrate the efficiency of this approach for the solution of strongly anisotropic PDEs in atmospheric modelling. The weak formulation of the PDE is expressed in Unified Form Language (UFL) and the lower PyOP2 abstraction layer allows the manual design of computational kernels for a bespoke geometric multigrid preconditioner. We compare the performance of this preconditioner to a single-level method and hypre's BoomerAMG algorithm. The Firedrake/PyOP2 code is inherently parallel and we present a detailed performance analysis for a single node (24 cores) on the ARCHER supercomputer. Our implementation utilises a significant fraction of the available memory bandwidth and shows very good weak scaling on up to 6,144 compute cores.

cs.MS↗

arXiv2015

Petascale elliptic solvers for anisotropic PDEs on GPU clusters

Memory bound applications such as solvers for large sparse systems of equations remain a challenge for GPUs. Fast solvers should be based on numerically efficient algorithms and implemented such that global memory access is minimised. To solve systems with up to one trillion ($10^{12}$) unknowns the code has to make efficient use of several million individual processor cores on large GPU clusters. We describe the multi-GPU implementation of two algorithmically optimal iterative solvers for anisotropic elliptic PDEs which are encountered in atmospheric modelling. In this application the condition number is large but independent of the grid resolution and both methods are asymptotically optimal, albeit with different absolute performance. We parallelise the solvers and adapt them to the specific features of GPU architectures, paying particular attention to efficient global memory access. We achieve a performance of up to 0.78 PFLOPs when solving an equation with $0.55\cdot 10^{12}$ unknowns on 16384 GPUs; this corresponds to about $3\%$ of the theoretical peak performance of the machine and we use more than $40\%$ of the peak memory bandwidth with a Conjugate Gradient (CG) solver. Although the other solver, a geometric multigrid algorithm, has a slightly worse performance in terms of FLOPs per second, overall it is faster as it needs less iterations to converge; the multigrid algorithm can solve a linear PDE with half a trillion unknowns in about one second.

cs.DC↗

arXiv2015

Efficient Multigrid Preconditioners for Atmospheric Flow Simulations at High Aspect Ratio

Many problems in fluid modelling require the efficient solution of highly anisotropic elliptic partial differential equations (PDEs) in "flat" domains. For example, in numerical weather- and climate-prediction an elliptic PDE for the pressure correction has to be solved at every time step in a thin spherical shell representing the global atmosphere. This elliptic solve can be one of the computationally most demanding components in semi-implicit semi-Lagrangian time stepping methods which are very popular as they allow for larger model time steps and better overall performance. With increasing model resolution, algorithmically efficient and scalable algorithms are essential to run the code under tight operational time constraints. We discuss the theory and practical application of bespoke geometric multigrid preconditioners for equations of this type. The algorithms deal with the strong anisotropy in the vertical direction by using the tensor-product approach originally analysed by Börm and Hiptmair [Numer. Algorithms, 26/3 (2001), pp. 219-234]. We extend the analysis to three dimensions under slightly weakened assumptions, and numerically demonstrate its efficiency for the solution of the elliptic PDE for the global pressure correction in atmospheric forecast models. For this we compare the performance of different multigrid preconditioners on a tensor-product grid with a semi-structured and quasi-uniform horizontal mesh and a one dimensional vertical grid. The code is implemented in the Distributed and Unified Numerics Environment (DUNE), which provides an easy-to-use and scalable environment for algorithms operating on tensor-product grids. Parallel scalability of our solvers on up to 20,480 cores is demonstrated on the HECToR supercomputer.

math.NA↗