9 results
Filters
Settings
Now showing1 - 9 of 9
- Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Resilience against soft faults through adaptivity in spectral deferred correction(Springer, 2026-08-06); ; ; ; As supercomputers grow in hardware complexity, their susceptibility to faults increases, and measures need to be taken to ensure the correctness of results. Some numerical algorithms have certain characteristics that allow them to recover from some types of faults. It has been demonstrated that adaptive Runge–Kutta methods provide resilience against transient faults without adding computational cost. Using recent advances in adaptive step size selection for spectral deferred correction (SDC), an iterative numerical time stepping scheme that can achieve arbitrary order, we show that adaptive SDC can also detect and correct transient faults. Its performance is found to be comparable to that of the dedicated resilience strategy Hot Rod.Publicationtype: Conference Paper42 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Efficient massively space-time-parallel simulations with adaptive spectral deferred correction(2026); Spectral Deferred Correction (SDC) is a method for numerically integrating initial value problems. The method iteratively generates solutions to fully implicit Runge-Kutta methods with forward substitution using low order solves. This allows great flexibility, for instance in terms of splitting techniques or inexact solves. Furthermore, various parallel-in-time extensions exist that parallelize the solution of a single time-step or solve multiple steps concurrently. We propose two adaptive step size selection algorithms that tailor the ideas behind embedded Runge-Kutta methods to SDC. Both are completely generic and work only on intermediate values within the time-integration process. We show, with a range of experiments, that computational efficiency can be boosted significantly by employing these algorithms compared to standard SDC. Furthermore, we show that parallel-in-time adaptive SDC is competitive with state-of-the-art Runge-Kutta methods for stiff partial differential equations. We also show that adaptivity increases the resilience against soft faults in SDC. Soft faults are unanticipated alterations of the data stored in memory, brought about, for instance, by environmental radiation. Iterative or adaptive methods inherently provide an elevated level of resilience, which is well known also in the context of the embedded Runge-Kutta methods that the adaptive step size selection is based on. We then move on to port implementations of partial differential equations within the prototyping library pySDC to GPUs and make extensive space-time-parallel scaling tests. We find that the parallel-in-time extension diagonal SDC can help extend the scaling capabilities and allowed to run a Gray-Scott example on 3584 GPUs at decent parallel efficiency. Finally, we demonstrate that findings from the previous experiments translate to practical use via space-time-parallel production runs of Gray-Scott and Rayleigh-Benard convection using adaptive SDC.Publicationtype: Doctoral Thesis88 61 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Improving efficiency of parallel across the method spectral deferred corrections(Society for Industrial & Applied Mathematics (SIAM), 2025-02-11); ; Parallel across the method time integration can provide small scale parallelism when solving initial value problems. Spectral deferred corrections (SDCs) with a diagonal sweeper, closely related to iterated Runge-Kutta methods proposed by Van der Houwen and Sommeijer, can use a number of threads equal to the number of quadrature nodes in the underlying collocation method. However, convergence speed, efficiency, and stability depend critically on the coefficients of the used SDC preconditioner. Previous approaches used numerical optimization to find good diagonal coefficients. Instead, we propose an approach that allows one to find optimal diagonal coefficients analytically. We show that the resulting parallel SDC methods provide stability domains and convergence order very similar to those of well established serial SDC variants. Using a model for computational cost that assumes 80% efficiency of an implementation of parallel SDCs, we show that our variants are competitive with serial SDC, previously published parallel SDC coefficients, Picard iteration, and a fourth-order explicit as well as a fourth-order implicit diagonally implicit Runge-Kutta method.Publicationtype: Journal Article21 26 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Adaptive time step selection for spectral deferred correction(Springer, 2024-10-28); ; ; ; Spectral Deferred Correction (SDC) is an iterative method for the numerical solution of ordinary differential equations. It works by refining the numerical solution for an initial value problem by approximately solving differential equations for the error, and can be interpreted as a preconditioned fixed-point iteration for solving the fully implicit collocation problem. We adopt techniques from embedded Runge-Kutta Methods (RKM) to SDC in order to provide a mechanism for adaptive time step size selection and thus increase computational efficiency of SDC. We propose two SDC-specific estimates of the local error that are generic and do not rely on problem specific quantities. We demonstrate a gain in efficiency over standard SDC with fixed step size and compare efficiency favorably against state-of-the-art adaptive RKM.Publicationtype: Journal Article39 87 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Bathymetry reconstruction from experimental data using PDE-constrained optimisation(2024-06-30); ; ; ; Knowledge of the bottom topography, also called bathymetry, of rivers, seas or the ocean is important for many areas of maritime science and civil engineering. While direct measurements are possible, they are time consuming, expensive and inaccurate. Therefore, many approaches have been proposed how to infer the bathymetry from measurements of surface waves. Mathematically, this is an inverse problem where an unknown system state needs to be reconstructed from observations with a suitable model for the flow as constraint. In many cases, the shallow water equations can be used to describe the flow. While theoretical studies of the efficacy of such a PDE-constrained optimisation approach for bathymetry reconstruction exist, there seem to be few publications that study its application to data obtained from real-world measurements. This paper shows that the approach can, at least qualitatively, reconstruct a Gaussian-shaped bathymetry in a wave flume from measurements of the free surface level at up to three points. Achieved normalised root mean square errors (NRMSE) are in line with other approaches.Publicationtype: Journal Article52 89 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Impact of spatial coarsening on Parareal convergence for the linear advection equationThe Parareal parallel-in-time integration method often performs poorly when applied to hyperbolic partial differential equations. This effect is even more pronounced when the coarse propagator uses a reduced spatial resolution. However, some combinations of spatial discretization and numerical time stepping nevertheless allow for Parareal to converge with monotonically decreasing errors. This raises the question how these configurations can be distinguished theoretically from those where the error initially increases, sometimes over many orders of magnitude. For linear problems, we prove a theorem that implies that the 2-norm of the Parareal iteration matrix is not a suitable tool to predict convergence for hyperbolic problems when spatial coarsening is used. We then show numerical results that suggest that the pseudo-spectral radius can reliably indicate if a given configuration of Parareal will show transient growth or monotonic convergence. For the studied examples, it also provides a good quantitative estimate of the convergence rate in the first few Parareal iterations.Publicationtype: Conference Paper12 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , A unified analysis framework for iterative parallel-in-time algorithms(2023-10); ; ; Parallel-in-time integration has been the focus of intensive research efforts over the past two decades due to the advent of massively parallel computer architectures and the scaling limits of purely spatial parallelization. Various iterative parallel-in-time (PinT) algorithms have been proposed, like Parareal, PFASST, MGRIT, and Space-Time Multi-Grid (STMG). These methods have been described using different notations and the convergence estimates that are available for some of them are difficult to compare. We describe Parareal, PFASST, MGRIT and STMG for the Dahlquist model problem using a common notation and give precise convergence estimates using generating functions. This allows us, for the first time, to directly compare their convergence. We prove that all four methods eventually converge super-linearly and compare them directly numerically. Our framework also allows us to find new methods.Publicationtype: Journal ArticleScopus© Citations 2 234 207 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Parareal with a physics-informed neural network as coarse propagatorParallel-in-time algorithms provide an additional layer of concurrency for the numerical integration of models based on time-dependent differential equations. Methods like Parareal, which parallelize across multiple time steps, rely on a computationally cheap and coarse integrator to propagate information forward in time, while a parallelizable expensive fine propagator provides accuracy. Typically, the coarse method is a numerical integrator using lower resolution, reduced order or a simplified model. Our paper proposes to use a physics-informed neural network (PINN) instead. We demonstrate for the Black-Scholes equation, a partial differential equation from computational finance, that Parareal with a PINN coarse propagator provides better speedup than a numerical coarse propagator. Training and evaluating a neural network are both tasks whose computing patterns are well suited for GPUs. By contrast, mesh-based algorithms with their low computational intensity struggle to perform well. We show that moving the coarse propagator PINN to a GPU while running the numerical fine propagator on the CPU further improves Parareal’s single-node performance. This suggests that integrating machine learning techniques into parallel-in-time integration methods and exploiting their differences in computing patterns might offer a way to better utilize heterogeneous architectures.Publicationtype: Conference Paper43 73 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Impact of spatial coarsening on Parareal convergence(2021-11-19); ; We study the impact of spatial coarsening on the convergence of the Parareal algorithm, both theoretically and numerically. For initial value problems with a normal system matrix, we prove a lower bound for the Euclidean norm of the iteration matrix. When there is no physical or numerical diffusion, an immediate consequence is that the norm of the iteration matrix cannot be smaller than unoty as soon as the coarse problem has fewer degrees-of-freedom than the fine. This prevents a theoretical guarantee for monotonic convergence, which is necessary to obtain meaningful speedups. For diffusive problems, in the worst-case where the iteration error contracts only as fast as the powers of the iteration matrix norm, making Parareal as accurate as the fine method will take about as many iterations as there are processors, making meaningful speedup impossible. Numerical examples with a non-normal system matrix show that for diffusive problems good speedup is possible, but that for non-diffusive problems the negative impact of spatial coarsening on convergence is big.Publicationtype: Preprint144 177