arXiv subjects

Daniel Ruprecht

Publications and source records attributed to Daniel Ruprecht.

At least 37 records · Page 2Linked to original sources

arXiv2024

Spectral deferred correction methods for second-order problems

Spectral deferred corrections (SDC) are a class of iterative methods for the numerical solution of ordinary differential equations. SDC can be interpreted as a Picard iteration to solve a fully implicit collocation problem, preconditioned with a low-order method. It has been widely studied for first-order problems, using explicit, implicit or implicit-explicit Euler and other low-order methods as preconditioner. For first-order problems, SDC achieves arbitrary order of accuracy and possesses good stability properties. While numerical results for SDC applied to the second-order Lorentz equations exist, no theoretical results are available for SDC applied to second-order problems. We present an analysis of the convergence and stability properties of SDC using velocity-Verlet as the base method for general second-order initial value problems. Our analysis proves that the order of convergence depends on whether the force in the system depends on the velocity. We also demonstrate that the SDC iteration is stable under certain conditions. Finally, we show that SDC can be computationally more efficient than a simple Picard iteration or a fourth-order Runge-Kutta-Nyström method.

math.NA↗

arXiv2023

Parareal with a physics-informed neural network as coarse propagator

Parallel-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.

math.NA↗

arXiv2022

Parallel Computation of Inverse Compton Scattering Radiation Spectra based on Liénard-Wiechert Potentials

Inverse Compton Scattering (ICS) has gained much attention recently because of its promise for the development of table-top-size X-ray light sources. Precise and fast simulation is an indispensable tool for predicting the radiation property of a given machine design and to optimize its parameters. Instead of the conventional approach to compute radiation spectra which directly evaluates the discretized Fourier integral of the Liénard-Wiechert field given analytically (referred to as the frequency-domain method), this article focuses on an approach where the field is recorded along the observer time on a uniform time grid which is then used to compute the radiation spectra after completion of the simulation, referred to as the time-domain method. Besides the derivation and implementation details of the proposed method, we analyze possible parallelization schemes and compare the parallel performance of the proposed time-domain method with the frequency-domain method. We will characterize scenarios/conditions under which one method is expected to outperform the other.

physics.comp-ph↗

arXiv2021

New applications for the Boris Spectral Deferred Correction algorithm for plasma simulations

The paper investigates two new use cases for the Boris Spectral Deferred Corrections (Boris-SDC) time integrator for plasma simulations. First, we show that using Boris-SDC as a particle pusher in an electrostatic particle-in-cell (PIC) code can, at least in the linear regime, improve simulation accuracy compared with the standard second order Boris method. In some instances, the higher order of Boris-SDC even allows a much larger time step, leading to modest computational gains. Second, we propose a modification of Boris-SDC for the relativistic regime. Based on an implementation of Boris-SDC in the \textsc{runko} PIC code, we demonstrate for a relativistic Penning trap that Boris-SDC retains its high order of convergence for velocities ranging from $0.5c$ to $>0.99c$. We also show that for the force-free case where acceleration from electric and magnetic field cancel, Boris-SDC produces less numerical drift than Boris.

math.NA↗

arXiv2021

Twelve Ways To Fool The Masses When Giving Parallel-In-Time Results

Getting good speedup -- let alone high parallel efficiency -- for parallel-in-time (PinT) integration examples can be frustratingly difficult. The high complexity and large number of parameters in PinT methods can easily (and unintentionally) lead to numerical experiments that overestimate the algorithm's performance. In the tradition of Bailey's article "Twelve ways to fool the masses when giving performance results on parallel computers", we discuss and demonstrate pitfalls to avoid when evaluating performance of PinT methods. Despite being written in a light-hearted tone, this paper is intended to raise awareness that there are many ways to unintentionally fool yourself and others and that by avoiding these fallacies more meaningful PinT performance results can be obtained.

math.NA↗

arXiv2020

Performance of the BGSDC integrator for computing fast ion trajectories in nuclear fusion reactors

Modelling neutral beam injection (NBI) in fusion reactors requires computing the trajectories of large ensembles of particles. Slowing down times of up to one second combined with nanosecond time steps make these simulations computationally very costly. This paper explores the performance of BGSDC, a new numerical time stepping method, for tracking ions generated by NBI in the DIII-D and JET reactors. BGSDC is a high-order generalisation of the Boris method, combining it with spectral deferred corrections and the Generalized Minimal Residual method GMRES. Without collision modelling, where numerical drift can be quantified accurately, we find that BGSDC can deliver higher quality particle distributions than the standard Boris integrator at comparable cost or comparable distributions at lower cost. With collision models, quantifying accuracy is difficult but we show that BGSDC produces stable distributions at larger time steps than Boris.

physics.comp-ph↗

arXiv2020

Parareal with a Learned Coarse Model for Robotic Manipulation

A key component of many robotics model-based planning and control algorithms is physics predictions, that is, forecasting a sequence of states given an initial state and a sequence of controls. This process is slow and a major computational bottleneck for robotics planning algorithms. Parallel-in-time integration methods can help to leverage parallel computing to accelerate physics predictions and thus planning. The Parareal algorithm iterates between a coarse serial integrator and a fine parallel integrator. A key challenge is to devise a coarse model that is computationally cheap but accurate enough for Parareal to converge quickly. Here, we investigate the use of a deep neural network physics model as a coarse model for Parareal in the context of robotic manipulation. In simulated experiments using the physics engine Mujoco as fine propagator we show that the learned coarse model leads to faster Parareal convergence than a coarse physics-based model. We further show that the learned coarse model allows to apply Parareal to scenarios with multiple objects, where the physics-based coarse model is not applicable. Finally, we conduct experiments on a real robot and show that Parareal predictions are close to real-world physics predictions for robotic pushing of multiple objects. Videos are at https://youtu.be/wCh2o1rf-gA.

cs.RO↗

arXiv2020

Performance of parallel-in-time integration for Rayleigh Bénard Convection

Rayleigh-Bénard convection (RBC) is a fundamental problem of fluid dynamics, with many applications to geophysical, astrophysical, and industrial flows. Understanding RBC at parameter regimes of interest requires complex physical or numerical experiments. Numerical simulations require large amounts of computational resources; in order to more efficiently use the large numbers of processors now available in large high performance computing clusters, novel parallelisation strategies are required. To this end, we investigate the performance of the parallel-in-time algorithm Parareal when used in numerical simulations of RBC. We present the first parallel-in-time speedups for RBC simulations at finite Prandtl number. We also investigate the problem of convergence of Parareal with respect to to statistical numerical quantities, such as the Nusselt number, and discuss the importance of reliable online stopping criteria in these cases.

physics.comp-ph↗

arXiv2019

Shared Memory Pipelined Parareal

For the parallel-in-time integration method Parareal, pipelining can be used to hide some of the cost of the serial correction step and improve its efficiency. The paper introduces a basic OpenMP implementation of pipelined Parareal and compares it to a standard MPI-based variant. Both versions yield almost identical runtimes, but, depending on the compiler, the OpenMP variant consumes about 7% less energy and has a significantly smaller memory footprint. However, its higher implementation complexity might make it difficult to use in legacy codes and in combination with spatial parallelisation.

cs.MS↗

arXiv2019

Combining Coarse and Fine Physics for Manipulation using Parallel-in-Time Integration

We present a method for fast and accurate physics-based predictions during non-prehensile manipulation planning and control. Given an initial state and a sequence of controls, the problem of predicting the resulting sequence of states is a key component of a variety of model-based planning and control algorithms. We propose combining a coarse (i.e. computationally cheap but not very accurate) predictive physics model, with a fine (i.e. computationally expensive but accurate) predictive physics model, to generate a hybrid model that is at the required speed and accuracy for a given manipulation task. Our approach is based on the Parareal algorithm, a parallel-in-time integration method used for computing numerical solutions for general systems of ordinary differential equations. We adapt Parareal to combine a coarse pushing model with an off-the-shelf physics engine to deliver physics-based predictions that are as accurate as the physics engine but run in substantially less wall-clock time, thanks to parallelization across time. We use these physics-based predictions in a model-predictive-control framework based on trajectory optimization, to plan pushing actions that avoid an obstacle and reach a goal location. We show that with hybrid physics models, we can achieve the same success rates as the planner that uses the off-the-shelf physics engine directly, but significantly faster. We present experiments in simulation and on a real robotic setup. Videos are available here: https://youtu.be/5e9oTeu4JOU

cs.RO↗

arXiv2019

An arbitrary order time-stepping algorithm for tracking particles in inhomogeneous magnetic fields

The Lorentz equations describe the motion of electrically charged particles in electric and magnetic fields and are used widely in plasma physics. The most popular numerical algorithm for solving them is the Boris method, a variant of the Störmer-Verlet algorithm. Boris' method is phase space volume conserving and simulated particles typically remain near the correct trajectory. However, it is only second order accurate. Therefore, in scenarios where it is not enough to know that a particle stays on the right trajectory but one needs to know where on the trajectory the particle is at a given time, Boris method requires very small time steps to deliver accurate phase information, making it computationally expensive. We derive an improved version of the high-order Boris spectral deferred correction algorithm (Boris-SDC) by adopting a convergence acceleration strategy for second order problems based on the Generalised Minimum Residual (GMRES) method. Our new algorithm is easy to implement as it still relies on the standard Boris method. Like Boris-SDC it can deliver arbitrary order of accuracy through simple changes of runtime parameter but possesses better long-term energy stability. We demonstrate for two examples, a magnetic mirror trap and the Solev'ev equilibrium, that the new method can deliver better accuracy at lower computational cost compared to the standard Boris method. While our examples are motivated by tracking ions in the magnetic field of a nuclear fusion reactor, the introduced algorithm can potentially deliver similar improvements in efficiency for other applications.

math.NA↗

arXiv2019

Parallel-in-time integration of Kinematic Dynamos

The precise mechanisms responsible for the natural dynamos in the Earth and Sun are still not fully understood. Numerical simulations of natural dynamos are extremely computationally intensive, and are carried out in parameter regimes many orders of magnitude away from real conditions. Parallelization in space is a common strategy to speed up simulations on high performance computers, but eventually hits a scaling limit. Additional directions of parallelization are desirable to utilise the high number of processor cores now available. Parallel-in-time methods can deliver speed up in addition to that offered by spatial partitioning but have not yet been applied to dynamo simulations. This paper investigates the feasibility of using the parallel-in-time algorithm Parareal to speed up initial value problem simulations of the kinematic dynamo, using the open source Dedalus spectral solver. Both the time independent Roberts and time dependent Galloway-Proctor 2.5D dynamos are investigated over a range of magnetic Reynolds numbers. Speed ups beyond those possible from spatial parallelization are found in both cases. Results for the Galloway-Proctor flow are promising, with Parareal efficiency found to be close to 0.3. Roberts flow results are less efficient, but Parareal still shows some speed up over spatial parallelization alone. Parallel in space and time speed ups of $\sim300$ were found for 1600 cores for the Galloway-Proctor flow, with total parallel efficiency of $\sim0.16$.

physics.comp-ph↗

arXiv2017

Wave propagation characteristics of Parareal

The paper derives and analyses the (semi-)discrete dispersion relation of the Parareal parallel-in-time integration method. It investigates Parareal's wave propagation characteristics with the aim to better understand what causes the well documented stability problems for hyperbolic equations. The analysis shows that the instability is caused by convergence of the amplification factor to the exact value from above for medium to high wave numbers. Phase errors in the coarse propagator are identified as the culprit, which suggests that specifically tailored coarse level methods could provide a remedy.

math.NA↗

arXiv2017

Toward transient finite element simulation of thermal deformation of machine tools in real-time

Finite element models without simplifying assumptions can accurately describe the spatial and temporal distribution of heat in machine tools as well as the resulting deformation. In principle, this allows to correct for displacements of the Tool Centre Point and enables high precision manufacturing. However, the computational cost of FEM models and restriction to generic algorithms in commercial tools like ANSYS prevents their operational use since simulations have to run faster than real-time. For the case where heat diffusion is slow compared to machine movement, we introduce a tailored implicit-explicit multi-rate time stepping method of higher order based on spectral deferred corrections. Using the open-source FEM library DUNE, we show that fully coupled simulations of the temperature field are possible in real-time for a machine consisting of a stock sliding up and down on rails attached to a stand.

cs.CE↗

arXiv2017

Parallel-in-Space-and-Time Simulation of the Three-Dimensional, Unsteady Navier-Stokes Equations for Incompressible Flow

In this paper we combine the Parareal parallel-in-time method together with spatial parallelization and investigate this space-time parallel scheme by means of solving the three-dimensional incompressible Navier-Stokes equations. Parallelization of time stepping provides a new direction of parallelization and allows to employ additional cores to further speed up simulations after spatial parallelization has saturated. We report on numerical experiments performed on a Cray XE6, simulating a driven cavity flow with and without obstacles. Distributed memory parallelization is used in both space and time, featuring up to 2,048 cores in total. It is confirmed that the space-time-parallel method can provide speedup beyond the saturation of the spatial parallelization.

cs.CE↗

arXiv2016

Time parallel gravitational collapse simulation

This article demonstrates the applicability of the parallel-in-time method Parareal to the numerical solution of the Einstein gravity equations for the spherical collapse of a massless scalar field. To account for the shrinking of the spatial domain in time, a tailored load balancing scheme is proposed and compared to load balancing based on number of time steps alone. The performance of Parareal is studied for both the sub-critical and black hole case; our experiments show that Parareal generates substantial speedup and, in the super-critical regime, can reproduce Choptuik's black hole mass scaling law.

gr-qc↗

arXiv2016

Spectral deferred corrections with fast-wave slow-wave splitting

The paper investigates a variant of semi-implicit spectral deferred corrections (SISDC) in which the stiff, fast dynamics correspond to fast propagating waves ("fast-wave slow-wave problem"). We show that for a scalar test problem with two imaginary eigenvalues $i λ_{fast}$, $i λ_{slow}$, having $Δt \left(\left| λ_{fast} \right| + \left| λ_{slow} \right| \right) < 1$ is sufficient for the fast-wave slow-wave SDC (FWSW-SDC) iteration to converge and that in the limit of infinitely fast waves the convergence rate of the non-split version is retained. Stability function and discrete dispersion relation are derived and show that the method is stable for essentially arbitrary fast-wave CFL numbers as long as the slow dynamics are resolved. The method causes little numerical diffusion and its semi-discrete phase speed is accurate also for large wave number modes. Performance is studied for an acoustic-advection problem and for the linearised Boussinesq equations, describing compressible, stratified flow. FWSW-SDC is compared to a diagonally implicit Runge-Kutta (DIRK) and IMEX Runge-Kutta (IMEX) method and found to be competitive in terms of both accuracy and cost.

math.NA↗

arXiv2016

Toward fault-tolerant parallel-in-time integration with PFASST

We introduce and analyze different strategies for the parallel-in-time integration method PFASST to recover from hard faults and subsequent data loss. Since PFASST stores solutions at multiple time steps on different processors, information from adjacent steps can be used to recover after a processor has failed. PFASST's multi-level hierarchy allows to use the coarse level for correcting the reconstructed solution, which can help to minimize overhead. A theoretical model is devised linking overhead to the number of additional PFASST iterations required for convergence after a fault. The potential efficiency of different strategies is assessed in terms of required additional iterations for examples of diffusive and advective type.

cs.DC↗