arXiv subjects

Robert Scheichl

Publications and source records attributed to Robert Scheichl.

66 records · Page 4Linked to original sources

arXiv2017

dune-composites -- A New Framework for High-Performance Finite Element Modelling of Laminates

Finite element (FE) analysis has the potential to offset much of the expensive experimental testing currently required to certify aerospace laminates. However, large numbers of degrees of freedom are necessary to model entire aircraft components whilst accurately resolving micro-scale defects. The new module dune-composites, implemented within DUNE by the authors, provides a tool to efficiently solve large-scale problems using novel iterative solvers. The key innovation is a preconditioner that guarantees a constant number of iterations regardless of the problem size. Its robustness has been shown rigorously in Spillane et al. (Numer. Math. 126, 2014) for isotropic problems. For anisotropic problems in composites it is verified numerically for the first time in this paper. The parallel implementation in DUNE scales almost optimally over thousands of cores. To demonstrate this, we present an original numerical study, varying the shape of a localised wrinkle and the effect this has on the strength of a curved laminate. This requires a high-fidelity mesh containing at least four layers of quadratic elements across each ply and interface layer, underlining the need for dune-composites, which can achieve run times of just over 2 minutes on 2048 cores for realistic composites problems with 173 million degrees of freedom.

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arXiv2016

Multilevel Quasi-Monte Carlo Methods for Lognormal Diffusion Problems

In this paper we present a rigorous cost and error analysis of a multilevel estimator based on randomly shifted Quasi-Monte Carlo (QMC) lattice rules for lognormal diffusion problems. These problems are motivated by uncertainty quantification problems in subsurface flow. We extend the convergence analysis in [Graham et al., Numer. Math. 2014] to multilevel Quasi-Monte Carlo finite element discretizations and give a constructive proof of the dimension-independent convergence of the QMC rules. More precisely, we provide suitable parameters for the construction of such rules that yield the required variance reduction for the multilevel scheme to achieve an $\varepsilon$-error with a cost of $\mathcal{O}(\varepsilon^{-θ})$ with $θ< 2$, and in practice even $θ\approx 1$, for sufficiently fast decaying covariance kernels of the underlying Gaussian random field inputs. This confirms that the computational gains due to the application of multilevel sampling methods and the gains due to the application of QMC methods, both demonstrated in earlier works for the same model problem, are complementary. A series of numerical experiments confirms these gains. The results show that in practice the multilevel QMC method consistently outperforms both the multilevel MC method and the single-level variants even for non-smooth problems.

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arXiv2016

Scheduling massively parallel multigrid for multilevel Monte Carlo methods

The computational complexity of naive, sampling-based uncertainty quantification for 3D partial differential equations is extremely high. Multilevel approaches, such as multilevel Monte Carlo (MLMC), can reduce the complexity significantly, but to exploit them fully in a parallel environment, sophisticated scheduling strategies are needed. Often fast algorithms that are executed in parallel are essential to compute fine level samples in 3D, whereas to compute individual coarse level samples only moderate numbers of processors can be employed efficiently. We make use of multiple instances of a parallel multigrid solver combined with advanced load balancing techniques. In particular, we optimize the concurrent execution across the three layers of the MLMC method: parallelization across levels, across samples, and across the spatial grid. The overall efficiency and performance of these methods will be analyzed. Here the scalability window of the multigrid solver is revealed as being essential, i.e., the property that the solution can be computed with a range of process numbers while maintaining good parallel efficiency. We evaluate the new scheduling strategies in a series of numerical tests, and conclude the paper demonstrating large 3D scaling experiments.

cs.CE↗

arXiv2016

Robust Numerical Upscaling of Elliptic Multiscale Problems at High Contrast

We present a new approach to the numerical upscaling for elliptic problems with rough diffusion coefficient at high contrast. It is based on the localizable orthogonal decomposition of $H^1$ into the image and the kernel of some novel stable quasi-interpolation operators with local $L^2$-approximation properties, independent of the contrast. We identify a set of sufficient assumptions on these quasi-interpolation operators that guarantee in principle optimal convergence without pre-asymptotic effects for high-contrast coefficients. We then give an example of a suitable operator and establish the assumptions for a particular class of high-contrast coefficients. So far this is not possible without any pre-asymptotic effects, but the optimal convergence is independent of the contrast and the asymptotic range is largely improved over other discretisation schemes. The new framework is sufficiently flexible to allow also for other choices of quasi-interpolation operators and the potential for fully robust numerical upscaling at high contrast.

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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.

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arXiv2013

Mixed Finite Element Analysis of Lognormal Diffusion and Multilevel Monte Carlo Methods

This work is motivated by the need to develop efficient tools for uncertainty quantification in subsurface flows associated with radioactive waste disposal studies. We consider single phase flow problems in random porous media described by correlated lognormal distributions. We are interested in the error introduced by a finite element discretisation of these problems. In contrast to several recent works on the analysis of standard nodal finite element discretisations, we consider here mass-conservative lowest order Raviart-Thomas mixed finite elements. This is very important since local mass conservation is highly desirable in realistic groundwater flow problems. Due to the limited spatial regularity and the lack of uniform ellipticity and boundedness of the operator the analysis is non-trivial in the presence of lognormal random fields. We establish finite element error bounds for Darcy velocity and pressure, as well as for a more accurate recovered pressure approximation. We then apply the error bounds to prove convergence of the multilevel Monte Carlo algorithm for estimating statistics of these quantities. Moreover, we prove convergence for a class of bounded, linear functionals of the Darcy velocity. An important special case is the approximation of the effective permeability in a 2D flow cell. We perform numerical experiments to confirm the convergence results.

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arXiv2013

Regularity for Maxwell eigenproblems in photonic crystal fibre modelling

The convergence behaviour and the design of numerical methods for modelling the flow of light in photonic crystal fibres depend critically on an understanding of the regularity of solutions to time-harmonic Maxwell equations in a three-dimensional, periodic, translationally invariant, heterogeneous medium. In this paper we determine the strength of the dominant singularities that occur at the interface between materials. By modifying earlier regularity theory for polygonal interfaces we find that on each subdomain, where the material in the fibre is constant, the regularity of in-plane components of the magnetic field are $H^{2-η}$ for all $η> 0$. This estimate is sharp in the sense that these components do not belong to $H^2$, in general. However, global regularity is restricted by the presence of an interface between these subdomains and the interface conditions imply only $H^{3/2-η}$ regularity across the interface. The results are useful to anyone applying a numerical method such as a finite element method or a planewave expansion method to model photonic crystal fibres or similar materials.

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arXiv2013

An Euler-Poisson Scheme for Lévy driven SDEs

We describe an Euler scheme to approximate solutions of Lévy driven Stochastic Differential Equations (SDE) where the grid points are random and given by the arrival times of a Poisson process. This result extends a previous work of the authors in Ferreiro-Castilla et al. (2012). We provide a complete numerical analysis of the algorithm to approximate the terminal value of the SDE and proof that the approximation converges in mean square error with rate $\mathcal{O}(n^{-1/2})$. The only requirement of the methodology is to have exact samples from the resolvent of the Lévy process driving the SDE; classic examples such as stable processes, subclasses of spectrally one sided Lévy processes and new families such as meromorphic Lévy processes (cf. Kuznetsov et al. (2011)) are some examples for which the implementation of our algorithm is straightforward.

math.PR↗

arXiv2013

Massively parallel solvers for elliptic PDEs in Numerical Weather- and Climate Prediction

The demand for substantial increases in the spatial resolution of global weather- and climate- prediction models makes it necessary to use numerically efficient and highly scalable algorithms to solve the equations of large scale atmospheric fluid dynamics. For stability and efficiency reasons several of the operational forecasting centres, in particular the Met Office and the ECMWF in the UK, use semi-implicit semi-Lagrangian time stepping in the dynamical core of the model. The additional burden with this approach is that a three dimensional elliptic partial differential equation (PDE) for the pressure correction has to be solved at every model time step and this often constitutes a significant proportion of the time spent in the dynamical core. To run within tight operational time scales the solver has to be parallelised and there seems to be a (perceived) misconception that elliptic solvers do not scale to large processor counts and hence implicit time stepping can not be used in very high resolution global models. After reviewing several methods for solving the elliptic PDE for the pressure correction and their application in atmospheric models we demonstrate the performance and very good scalability of Krylov subspace solvers and multigrid algorithms for a representative model equation with more than $10^{10}$ unknowns on 65536 cores on HECToR, the UK's national supercomputer. For this we tested and optimised solvers from two existing numerical libraries (DUNE and hypre) and implemented both a Conjugate Gradient solver and a geometric multigrid algorithm based on a tensor-product approach which exploits the strong vertical anisotropy of the discretised equation. We study both weak and strong scalability and compare the absolute solution times for all methods; in contrast to one-level methods the multigrid solver is robust with respect to parameter variations.

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arXiv2013

Multilevel Monte Carlo simulation for Levy processes based on the Wiener-Hopf factorisation

In Kuznetsov et al. (2011) a new Monte Carlo simulation technique was introduced for a large family of Levy processes that is based on the Wiener-Hopf decomposition. We pursue this idea further by combining their technique with the recently introduced multilevel Monte Carlo methodology. Moreover, we provide here for the first time a theoretical analysis of the new Monte Carlo simulation technique in Kuznetsov et al. (2011) and of its multilevel variant for computing expectations of functions depending on the historical trajectory of a Levy process. We derive rates of convergence for both methods and show that they are uniform with respect to the "jump activity" (e.g. characterised by the Blumenthal-Getoor index). We also present a modified version of the algorithm in Kuznetsov et al. (2011) which combined with the multilevel methodology obtains the optimal rate of convergence for general Levy processes and Lipschitz functionals. This final result is only a theoretical one at present, since it requires independent sampling from a triple of distributions which is currently only possible for a limited number of processes.

math.PR↗

arXiv2013

Matrix-free GPU implementation of a preconditioned conjugate gradient solver for anisotropic elliptic PDEs

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.

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