We develop new error estimates for the one-dimensional advection equation, considering general space-time discretization schemes based on Runge–Kutta methods and finite difference discretizations. We then derive conditions on the number of points per wavelength for a given error tolerance from these new estimates. Our analysis also shows the existence of synergistic space-time discretization methods that permit to gain one order of accuracy at a given CFL number. Our new error estimates can be used to analyze the choice of space-time discretizations considered when testing Parallel-in-Time methods.
This review article serves to summarize the many advances in time-parallel computations since the excellent review article by Gander, "50 years of Time Parallel Integration" (Gander, in: 50 years of time parallel time integration. Multiple shooting and time domain decomposition, Springer, Berlin, 2015). We focus, when possible, on applications of time parallelism and the observed speedup and efficiency, highlighting the challenges and benefits of parallel time computations. The applications covered range from numerous PDE-based simulations (both hyperbolic and parabolic), to PDE-constrained optimization, powergrid simulations, and machine learning. The time-parallel methods covered range from various iterative schemes (multigrid, waveform, multiple shooting, domain decomposition) to direct time-parallel methods.
This paper describes a space-time parallel algorithm with space-time adaptive mesh refinement (AMR). AMR with subcycling is added to multigrid reduction-in-time (MGRIT) in order to provide solution efficient adaptive grids with a reduction in work performed on coarser grids. This algorithm is achieved by integrating two software libraries: XBraid (Parallel time integration with multigrid. https://computation.llnl.gov/projects/parallel-timeintegration-multigrid ) and Chombo (Chombo software package for AMR applications—design document, 2014). The former is a parallel time integration library using multigrid and the latter is a massively parallel structured AMR library. Employing this adaptive space-time parallel algorithm is Chord (Comput Fluids 123:202–217, 2015), a computational fluid dynamics (CFD) application code for solving compressible fluid dynamics problems. For the same solution accuracy, speedups are demonstrated from the use of space-time parallelization over the time-sequential integration on Couette flow and Stokes’ second problem. On a transient Couette flow case, at least a $$1.5\times $$ speedup is achieved, and with a time periodic problem, a speedup of up to $$13.7\times $$ over the time-sequential case is obtained. In both cases, the speedup is achieved by adding processors and exploring additional parallelization in time. The numerical experiments show the algorithm is promising for CFD applications that can take advantage of the time parallelism. Future work will focus on improving the parallel performance and providing more tests with complex fluid dynamics to demonstrate the full potential of the algorithm.
The code d3f++ developed under the leadership of GRS for simulating 3D flow and solute transport includes an advanced formulation for fracture flow and transport in lower-dimensional elements that has still been in need of qualification. A variety of in-situ experiments in the granitic rock at the Swedish Hard Rock Laboratory Äspö offers an excellent basis for this purpose. Particularly well suited is the work performed in the TRUE Block Scale Project where a hydrostructural model comprising 30 large-scale fractures in a cube-shaped domain with a side length of 200 m has been set up for simulating groundwater flow which in turn forms the basis for reproducing the results of the injection–extraction tracer test “C2”. The present paper deals with modelling 3D flow and solute transport with the code d3f++ as a qualification exercise. The modelling results reveal an excellent representation of the contrasts in flow velocity across the fractures. Also the jumps in tracer concentration along intersecting fractures where waters with different tracer concentrations mix are handled well by the code. Calibration has nevertheless been required in order to achieve a reasonable fit of measured and calculated breakthrough curves.
Starting from the general question, if there is a connection between the mathematical capabilities of a student and his native language, we aim at comparing natural languages with mathematical language quantitatively. In [ 20 ] we set up an approach to compare language structures using Natural Language Processors (NLP). However, difficulties arose with the quality of the structural analysis of the NLP used just comparing simple sentences in different but closely related natural languages. We now present a comparison of different available NLPs and discuss the results. The comparison confirms the results from [ 20 ], showing that current NLPs are not capable of analysing even simple sentences such that resulting structures between different natural languages can be compared.
The mathematical characterization of the sound of a musical instrument still follows Schumann’s laws (Schumann in Physik der klangfarben, Leipzig, 1929). According to this theory, the resonances of the instrument body, “the formants”, filter the oscillations of the sound generator (e.g., strings) and produce the characteristic “timbre” of an instrument. This is a strong simplification of the actual situation. It applies to a point source and can be easily performed by a loudspeaker, disregarding the three dimensional structure of music instruments. To describe the effect of geometry and material of the instruments, we set up a 3d model and simulate it using the simulation system UG4 (Vogel et al. in Comput Vis Sci 16(4):165–179, 2013; Reiter et al. in Comput Vis Sci 16(4):151–164, 2014). We aim to capture the oscillation behavior of eigenfrequencies of a harpsichord soundboard and investigate how well a model for the oscillation behavior of the soundboard approximates the oscillation behavior of the whole instrument. We resolve the complicated geometry by several unstructured 3d grids and take into account the anisotropy of wood. The oscillation behavior of the soundboard is modeled following the laws of linear orthotropic elasticity with homogenous boundary conditions. The associated eigenproblem is discretized using FEM and solved with the iterative PINVIT method using an efficient GMG preconditioner (Neymeyr in A hierarchy of preconditioned eigensolvers for elliptic differential operators. Habilitation dissertation, University of Tübingen, 2001). The latter allows us to resolve the harpsichord with a high resolution hybrid grid, which is required to capture fine modes of the simulated eigenfrequencies. We computed the first 16 eigenmodes and eigenfrequencies with a resolution of 1.8 billion unknowns each on Shaheen II supercomputer ( https://www.hpc.kaust.edu.sa/content/shaheen-ii ). To verify our results, we compare them with measurement data obtained from an experimental modal analysis of a real reference harpsichord.
Multigrid and domain-decomposition methods are now widely used in large-scale simulation studies. Meanwhile, many types of preconditioners are available for speeding the convergence. Users of analysis models are interested in the method or preconditioner that minimizes the computing time. In this paper, we analyze a thermal problem and a solid problem using two free, open-source software programs—the UG4 framework and the ADVENTURE system—based on geometric multigrid (GM) solvers and balancing domain-decomposition (BDD) solvers. We examine and report the computing times and iteration numbers to convergence of the two programs, and generate a common software interface between the UG4 framework and ADVENTURE system. Through this software interface, we can compare the solvers of the two software systems using the same mesh data. The results were presented on the CX400 system at the Information Technology Center of Nagoya University, Japan. The convergence rate of GM solver improved after suitable smoothing and scaling to finer grids. The BDD solver is suitable for large-scale analyses of structures with detailed and complex geometries.
The generation of detailed three dimensional meshes for the simulation of groundwater flow in thin layered domains is crucial to capture important properties of the underlying domains and to reach a satisfying accuracy. At the same time, this level of detail poses high demands both on suitable hardware and numerical solver efficiency. Parallel multigrid methods have been shown to exhibit near optimal weak scalability for massively parallel computations of density driven flow. A fully automated parameterized algorithm for prism based meshing of coarse grids from height data of individual layers is presented. Special structures like pinch outs of individual layers are preserved. The resulting grid is used as a starting point for parallel mesh and hierarchy creation through interweaved projected refinement and redistribution. Efficiency and applicability of the proposed approach are demonstrated for a parallel multigrid based simulation of a realistic sample problem.
We investigate algebraic multigrid (AMG) methods for the linear systems arising from the discretization of Navier–Stokes equations via the finite pointset method. In the segregated approach , three pressure systems and one velocity system need to be solved. In the coupled approach , one of the pressure systems is coupled with the velocity system, leading to a coupled velocity-pressure saddle point system. The discretization of the differential operators used in FPM leads to non-symmetric matrices that do not have the M-matrix property. Even though the theoretical framework for many AMG methods requires these properties, our AMG methods can be successfully applied to these matrices and show a robust and scalable convergence when compared to a BiCGStab(2) solver.
The Poisson-Boltzmann equation (PBE) is a nonlinear elliptic PDE that arises in biomolecular modeling and is a fundamental tool for structural biology. It is used to calculate electrostatic potentials around an ensemble of fixed charges immersed in an ionic solution. Efficient numerical computation of the PBE yields a high number of degrees of freedom in the resultant algebraic system of equations. Coupled with the fact that in most cases the PBE requires to be solved multiple times for a large number of system configurations, this poses great computational challenges to conventional numerical techniques. To accelerate such computations, we here present the reduced basis method (RBM) which greatly reduces this computational complexity by constructing a reduced order model of typically low dimension. In this study, we employ a simple version of the PBE for proof of concept and discretize the linearized PBE (LPBE) with a centered finite difference scheme. The resultant linear system is solved by the aggregation-based algebraic multigrid method at different samples of ionic strength on a three-dimensional Cartesian grid. The discretized LPBE, which we call the high-fidelity full order model (FOM), yields solution as accurate as other LPBE solvers. We then apply the RBM to FOM. The discrete empirical interpolation method (DEIM) is applied to the Dirichlet boundary conditions which are nonaffine with the parameter, to reduce the complexity of the reduced order model (ROM). From the numerical results, we notice that the RBM reduces the model order from $\mathcal{N} = 2\times 10^{6}$ to $N = 6$ at an accuracy of $10^{-9}$ and reduces computational time by a factor of approximately $7,600$. DEIM, on the other hand, is also used in the offline-online phase of solving the ROM for different values of parameters which provides a speed-up of $20$ for a single iteration of the greedy algorithm.
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 statistical numerical quantities, such as the Nusselt number, and discuss the importance of reliable online stopping criteria in these cases.
The parareal algorithm is known to allow for a significant reduction in wall clock time for accurate numerical solutions by parallelising across the time dimension. We present and test a micro-macro version of parareal, in which the fine propagator is based on a (high-dimensional, slow-fast) stochastic microscopic model, and the coarse propagator is based on a low-dimensional approximate effective dynamics at slow time scales. At the microscopic level, we use an ensemble of Monte Carlo particles, whereas the approximate coarse propagator uses the (deterministic) Fokker–Planck equation for the slow degrees of freedom. The required coupling between microscopic and macroscopic representations of the system introduces several design options, specifically on how to generate a microscopic probability distribution consistent with a required macroscopic probability distribution and how to perform the coarse-level updating of the macroscopic probability distribution in a meaningful manner. We numerically study how these design options affect the efficiency of the algorithm in a number of situations. The choice of the coarse-level updating operator strongly impacts the result, with a superior performance if addition and subtraction of the quantile function (inverse cumulative distribution) is used. How microscopic states are generated has a less pronounced impact, provided a suitable prior microscopic state is used.
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. Code (https://doi.org/10.5281/zenodo.3779085) and videos (https://youtu.be/wCh2o1rf-gA) are publicly available.
We apply the multigrid-reduction-in-time (MGRIT) algorithm to an eddy current simulation of a two-dimensional induction machine supplied by a pulse-width-modulation signal. To resolve the fast-switching excitations, small time steps are needed, such that parallelization in time becomes highly relevant for reducing the simulation time. The MGRIT algorithm is an iterative method that allows calculating multiple time steps simultaneously by using a time-grid hierarchy. It is particularly well suited for introducing time parallelism in the simulation of electrical machines using existing application codes, as MGRIT is a non-intrusive approach that essentially uses the same time integrator as a traditional time-stepping algorithm. However, the key difficulty when using time-stepping routines of existing application codes for the MGRIT algorithm is that the cost of the time integrator on coarse time grids must be less expensive than on the fine grid to allow for speedup over sequential time stepping on the fine grid. To overcome this difficulty, we consider reducing the costs of the coarse-level problems by adding spatial coarsening. We investigate effects of spatial coarsening on MGRIT convergence when applied to two numerical models of an induction machine, one with linear material laws and a full nonlinear model. Parallel results demonstrate significant speedup in the simulation time compared to sequential time stepping, even for moderate numbers of processors.
To extend prevailing scaling limits when solving time-dependent partial differential equations, the parallel full approximation scheme in space and time (PFASST) has been shown to be a promising parallel-in-time integrator. Similar to space–time multigrid, PFASST is able to compute multiple time-steps simultaneously and is therefore in particular suitable for large-scale applications on high performance computing systems. In this work we couple PFASST with a parallel spectral deferred correction (SDC) method, forming an unprecedented doubly time-parallel integrator. While PFASST provides global, large-scale “parallelization across the step”, the inner parallel SDC method allows integrating each individual time-step “parallel across the method” using a diagonalized local Quasi-Newton solver. This new method, which we call “PFASST with Enhanced concuRrency” (PFASST-ER), therefore exposes even more temporal concurrency. For two challenging nonlinear reaction-diffusion problems, we show that PFASST-ER works more efficiently than the classical variants of PFASST and can use more processors than time-steps.
Porous and heterogeneous materials are found in many applications from composites, membranes, chemical reactors, and other engineered materials to biological matter and natural subsurface structures. In this work we propose an integrated approach to generate, study and upscale transport equations in random and periodic porous structures. The geometry generation is based on random algorithms or ballistic deposition. In particular, a new algorithm is proposed to generate random packings of ellipsoids with random orientation and tunable porosity and connectivity. The porous structure is then meshed using locally refined Cartesian-based or unstructured strategies. Transport equations are thus solved in a finite-volume formulation with quasi-periodic boundary conditions to simplify the upscaling problem by solving simple closure problems consistent with the classical theory of homogenisation for linear advection-diffusion-reaction operators. Existing simulation codes are extended with novel developments and integrated to produce a fully open-source simulation pipeline. A showcase of a few interesting three-dimensional applications of these computational approaches is then presented. Firstly, convergence properties and the transport and dispersion properties of a periodic arrangement of spheres are studied. Then, heat transfer problems are considered in a pipe with layers of deposited particles of different heights, and in heterogeneous anisotropic materials.
We present a technique for proving convergence of h and hp adaptive finite element methods through comparison with certain reference refinement schemes based on interpolation error. We then construct a testing environment where properties of different adaptive approaches can be evaluated and improved.
The brain is a complex organ operating on multiple scales. From molecular events that inform electrical and biochemical cellular responses, the brain interconnects processes all the way up to the massive network size of billions of brain cells. This strongly coupled, nonlinear, system has been subject to research that has turned increasingly multidisciplinary. The seminal work of Hodgkin and Huxley in the 1950s made use of experimental data to derive a coherent physical model of electrical signaling in neurons, which can be solved using mathematical and computational methods, thus bringing together neuroscience, physics, mathematics, and computer science. Over the last decades numerous projects have been dedicated to modeling and simulation of specific parts of molecular dynamics, neuronal signaling, and neural network behavior. Simulators have been developed around a specific objective and scale, in order to cope with the underlying computational complexity. Often times a dimension reduction approach allows larger scale simulations, this however has the inherent drawback of losing insight into structure-function interplay at the cellular level. This paper gives an overview of the project NeuroBox that has the objective of integrating multiple brain scales and associated physical models into one unified framework. NeuroBox hosts geometry and anatomical reconstruction methods, such that detailed three-dimensional domains can be integrated into numerical simulations of models based on partial differential equations. The project further focusses on deriving numerical methods for handling complex computational domains, and to couple multiple spatial dimensions. The latter allows the user to specify in which parts of the biological problem high-dimensional representations are necessary and where low-dimensional approximations are acceptable. NeuroBox offers workflow user interfaces that are automatically generated with VRL-Studio and can be controlled by non-experts. The project further uses uG4 as the numerical backend, and therefore accesses highly advanced discretization methods as well as hierarchical and scalable numerical solvers for very large neurobiological problems.
