Daniel Ruprecht

Robert Speck, Daniel Ruprecht, Matthew Emmett et al.

The paper presents a combination of the time-parallel "parallel full approximation scheme in space and time" (PFASST) with a parallel multigrid method (PMG) in space, resulting in a mesh-based solver for the three-dimensional heat equation with a uniquely high degree of efficient concurrency. Parallel scaling tests are reported on the Cray XE6 machine "Monte Rosa" on up to 16,384 cores and on the IBM Blue Gene/Q system "JUQUEEN" on up to 65,536 cores. The efficacy of the combined spatial- and temporal parallelization is shown by demonstrating that using PFASST in addition to PMG significantly extends the strong-scaling limit. Implications of using spatial coarsening strategies in PFASST's multi-level hierarchy in large-scale parallel simulations are discussed.

Eugen Bronasco, Joscha Fregin, Daniel Ruprecht et al.

We interpret a wide range of flavors of Spectral Deferred Corrections (SDC) as Runge-Kutta methods (RKM). Using Butcher series, we show that the considered class of SDC methods achieve at least order p after p iterations compared to the underlying RKM, independently of the error discretisation chosen and the choice of nodes. For all collocation RKM, we analyse the phenomenon of order jumps in SDC iterations, where the order is increased by two at each iteration. We prove that it can be obtained by using appropriate inconsistent, implicit, parallelisable error discretisations. We also investigate the stability properties of the new SDC methods which can in general reduce to that of explicit RKM, but it can be improved by suitable combinations of error discretisations. We confirm the convergence analysis with numerical experiments and we apply relaxation RKM to derive SDC variants that conserve quadratic invariants.

Finn Sommer, Vamika Rathi, Sebastian Goetschel et al.

The Maxey-Riley-Gatignol equations (MaRGE) model the motion of spherical inertial particles in a fluid. They contain the Basset force, an integral term which models history effects due to the formation of wakes and boundary layer effects. This causes the force that acts on a particle to depend on its past trajectory and complicates the numerical solution of MaRGE. Therefore, the Basset force is often neglected, despite substantial evidence that it has both quantitative and qualitative impact on the movement patterns of modelled particles. Using the concept of universal differential equations, we propose an approximation of the history term via neural networks which approximates MaRGE by a system of ordinary differential equations that can be solved with standard numerical solvers like Runge-Kutta methods.

Lars Stietz, Sebastian Götschel, Peter Schleper et al.

Bathymetry reconstruction is an important problem in various fields, including oceanography and environmental monitoring. This paper presents a Bayesian inference approach to reconstructing bathymetries from point measurements of the water height. We test the method for parameterized and discretized bathymetries with synthetic data to evaluate its performance and limitations. Our results indicate that the Bayesian framework provides a robust approach to bathymetry reconstruction. Finally, we use the framework to reconstruct a real-world bathymetry in a wave flume from experimental measurements and compare its performance to an adjoint optimization method. The Bayesian approach improves the normalized root mean squared error (NRMSE) of the reconstruction and provides better qualitative features, while also quantifying uncertainty.

Maximilian Witte, Fabricio Rodrigues Lapolli, Philip Freese et al.

Using the nonlinear shallow water equations as benchmark, we demonstrate that a simulation with the ICON-O ocean model with a 20km resolution that is frequently corrected by a U-net-type neural network can achieve discretization errors of a simulation with 10km resolution. The network, originally developed for image-based super-resolution in post-processing, is trained to compute the difference between solutions on both meshes and is used to correct the coarse mesh every 12h. Our setup is the Galewsky test case, modeling transition of a barotropic instability into turbulent flow. We show that the ML-corrected coarse resolution run correctly maintains a balance flow and captures the transition to turbulence in line with the higher resolution simulation. After 8 day of simulation, the $L_2$-error of the corrected run is similar to a simulation run on the finer mesh. While mass is conserved in the corrected runs, we observe some spurious generation of kinetic energy.

Wisdom C. Agboh, Daniel Ruprecht, Mehmet R. Dogar

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

Johannes Steiner, Daniel Ruprecht, Robert Speck et al.

The paper presents first a linear stability analysis for the time-parallel Parareal method, using an IMEX Euler as coarse and a Runge-Kutta-3 method as fine propagator, confirming that dominant imaginary eigenvalues negatively affect Parareal's convergence. This suggests that when Parareal is applied to the nonlinear Navier-Stokes equations, problems for small viscosities could arise. Numerical results for a driven cavity benchmark are presented, confirming that Parareal's convergence can indeed deteriorate as viscosity decreases and the flow becomes increasingly dominated by convection. The effect is found to strongly depend on the spatial resolution.