arXiv subjects

Rolf Krause

Publications and source records attributed to Rolf Krause.

88 records · Page 5Linked to original sources

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

Towards a multigrid method for the minimum-cost flow problem

We present a first step towards a multigrid method for solving the min-cost flow problem. Specifically, we present a strategy that takes advantage of existing black-box fast iterative linear solvers, i.e. algebraic multigrid methods. We show with standard benchmarks that, while less competitive than combinatorial techniques on small problems that run on a single core, our approach scales well with problem size, complexity, and number of processors, allowing for tackling large-scale problems on modern parallel architectures. Our approach is based on combining interior-point with multigrid methods for solving the nonlinear KKT equations via Newton's method. However, the Jacobian matrix arising in the Newton iteration is indefinite and its condition number cannot be expected to be bounded. In fact, the eigenvalues of the Jacobian can both vanish and blow up near the solution, leading to a significant slow-down of the convergence speed of iterative solvers - or to the loss of convergence at all. In order to allow for the application of multigrid methods, which have been originally designed for elliptic problems, we furthermore show that the occurring Jacobian can be interpreted as the stiffness matrix of a mixed formulation of the weighted graph Laplacian of the network, whose metric depends on the slack variables and the multipliers of the inequality constraints. Together with our regularization, this allows for the application of a black-box algebraic multigrid method on the Schur-complement of the system.

math.OC↗

arXiv2016

An Iterative Approach for Time Integration Based on Discontinuous Galerkin Methods

We present a new class of iterative schemes for solving initial value problems (IVP) based on discontinuous Galerkin (DG) methods. Starting from the weak DG formulation of an IVP, we derive a new iterative method based on a preconditioned Picard iteration. Using this approach, we can systematically construct explicit, implicit and semi-implicit schemes with arbitrary order of accuracy. We also show that the same schemes can be constructed by solving a series of correction equations based on the DG weak formulation. The accuracy of the schemes is proven to be $\min\{2p+1, K+1\}$ with $p$ the degree of the DG polynomial basis and $K$ the number of iterations. The stability is explored numerically; we show that the implicit schemes are $A$-stable at least for $0 \leq p \leq 9$. Furthermore, we combine the methods with a multilevel strategy to accelerate their convergence speed. The new multilevel scheme is intended to provide a flexible framework for high order space-time discretizations and to be coupled with space-time multigrid techniques for solving partial differential equations (PDEs). We present numerical examples for ODEs and PDEs to analyze the performance of the new methods. Moreover, the newly proposed class of methods, due to its structure, is also a competitive and promising candidate for parallel in time algorithms such as Parareal, PFASST, multigrid in time, etc.

math.NA↗

arXiv2015

Explicit Parallel-in-time Integration of a Linear Acoustic-Advection System

The applicability of the Parareal parallel-in-time integration scheme for the solution of a linear, two-dimensional hyperbolic acoustic-advection system, which is often used as a test case for integration schemes for numerical weather prediction (NWP), is addressed. Parallel-in-time schemes are a possible way to increase, on the algorithmic level, the amount of parallelism, a requirement arising from the rapidly growing number of CPUs in high performance computer systems. A recently introduced modification of the "parallel implicit time-integration algorithm" could successfully solve hyperbolic problems arising in structural dynamics. It has later been cast into the framework of Parareal. The present paper adapts this modified Parareal and employs it for the solution of a hyperbolic flow problem, where the initial value problem solved in parallel arises from the spatial discretization of a partial differential equation by a finite difference method. It is demonstrated that the modified Parareal is stable and can produce reasonably accurate solutions while allowing for a noticeable reduction of the time-to-solution. The implementation relies on integration schemes already widely used in NWP (RK-3, partially split forward Euler, forward-backward). It is demonstrated that using an explicit partially split scheme for the coarse integrator allows to avoid the use of an implicit scheme while still achieving speedup.

cs.CE↗

arXiv2015

Parareal convergence for 2D unsteady flow around a cylinder

In this technical report we study the convergence of Parareal for 2D incompressible flow around a cylinder for different viscosities. Two methods are used as fine integrator: backward Euler and a fractional step method. It is found that Parareal converges better for the implicit Euler, likely because it under-resolves the fine-scale dynamics as a result of numerical diffusion.

cs.CE↗

arXiv2015

Numerical simulation of skin transport using Parareal

In-silico investigation of skin permeation is an important but also computationally demanding problem. To resolve all scales involved in full detail will not only require exascale computing capacities but also suitable parallel algorithms. This article investigates the applicability of the time-parallel Parareal algorithm to a brick and mortar setup, a precursory problem to skin permeation. The C++ library Lib4PrM implementing Parareal is combined with the UG4 simulation framework, which provides the spatial discretization and parallelization. The combination's performance is studied with respect to convergence and speedup. It is confirmed that anisotropies in the domain and jumps in diffusion coefficients only have a minor impact on Parareal's convergence. The influence of load imbalances in time due to differences in number of iterations required by the spatial solver as well as spatio-temporal weak scaling is discussed.

cs.CE↗

arXiv2014

A stencil-based implementation of Parareal in the C++ domain specific embedded language STELLA

In view of the rapid rise of the number of cores in modern supercomputers, time-parallel methods that introduce concurrency along the temporal axis are becoming increasingly popular. For the solution of time-dependent partial differential equations, these methods can add another direction for concurrency on top of spatial parallelization. The paper presents an implementation of the time-parallel Parareal method in a C++ domain specific language for stencil computations (STELLA). STELLA provides both an OpenMP and a CUDA backend for a shared memory parallelization, using the CPU or GPU inside a node for the spatial stencils. Here, we intertwine this node-wise spatial parallelism with the time-parallel Parareal. This is done by adding an MPI-based implementation of Parareal, which allows us to parallelize in time across nodes. The performance of Parareal with both backends is analyzed in terms of speedup, parallel efficiency and energy-to-solution for an advection-diffusion problem with a time-dependent diffusion coefficient.

cs.DC↗

arXiv2014

Convergence of Parareal for the Navier-Stokes equations depending on the Reynolds number

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.

math.NA↗

arXiv2014

A multi-level spectral deferred correction method

The spectral deferred correction (SDC) method is an iterative scheme for computing a higher-order collocation solution to an ODE by performing a series of correction sweeps using a low-order timestepping method. This paper examines a variation of SDC for the temporal integration of PDEs called multi-level spectral deferred corrections (MLSDC), where sweeps are performed on a hierarchy of levels and an FAS correction term, as in nonlinear multigrid methods, couples solutions on different levels. Three different strategies to reduce the computational cost of correction sweeps on the coarser levels are examined: reducing the degrees of freedom, reducing the order of the spatial discretization, and reducing the accuracy when solving linear systems arising in implicit temporal integration. Several numerical examples demonstrate the effect of multi-level coarsening on the convergence and cost of SDC integration. In particular, MLSDC can provide significant savings in compute time compared to SDC for a three-dimensional problem.

math.NA↗

arXiv2014

A space-time parallel solver for the three-dimensional heat equation

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.

math.NA↗

arXiv2014

Inexact spectral deferred corrections

Spectral deferred correction (SDC) methods are an attractive approach to iteratively computing collocation solutions to an ODE by performing so-called sweeps with a low-order time stepping method. SDC allows to easily construct high order split methods where e.g. stiff terms of the ODE are treated implicitly. This requires the solution to full accuracy of multiple linear systems of equations during each sweep, e.g. with a multigrid method. In this paper, we present an inexact variant of SDC, where each solve of a linear system is replaced by a single multigrid V-cycle and thus significantly reduces the cost for each sweep. For the investigated examples, this strategy results only in a small increase of the number of required sweeps and we demonstrate that "inexact spectral deferred corrections" can provide a dramatic reduction of the overall number of multigrid V-cycles required to complete an SDC time step.

math.NA↗

arXiv2014

Parareal for diffusion problems with space- and time-dependent coefficients

For the time-parallel Parareal method, there exists both numerical and analytical proof that it converges very well for diffusive problems like the heat equation. Many applications, however, do not lead to simple homogeneous diffusive scenarios but feature strongly inhomogeneous and possibly anisotropic coefficients. The paper presents results from a numerical study of how space- and time-dependent coefficients in a diffusion setup affect Parareal's convergence behaviour. It is shown that, for the presented examples, non-constant diffusion coefficients have only marginal influence on how fast Parareal converges. Furthermore, an example is shown that illustrates how for linear problems the maximum singular value of the Parareal iteration matrix can be used to estimate convergence rates.

math.NA↗

arXiv2013

Probabilistic Analysis of LCF Crack Initiation Life of a Turbine Blade under Thermomechanical Loading

An accurate assessment for fatigue damage as a function of activation and deactivation cycles is vital for the design of many engineering parts. In this paper we extend the probabilistic and local approach to this problem proposed in [1,2] and [3] to the case of non-constant temperature fields and thermomechanical loading. The method has been implemented as a finite element postprocessor and applied to an example case of a gas-turbine blade which is made of a conventionally cast nickel base superalloy.

math.NA↗

arXiv2013

A Probabilistic Model for LCF

Fatigue life of components or test specimens often exhibit a significant scatter. Furthermore, size effects have a non-negligible influence on fatigue life of parts with different geometries. We present a new probabilistic model for low-cycle fatigue (LCF) in the context of polycrystalline metal. The model takes size effects and inhomogeneous strain fields into account by means of the Poisson point process (PPP). This approach is based on the assumption of independently occurring LCF cracks and the Coffin-Manson-Basquin (CMB) equation. Within the probabilistic model, we give a new and more physical interpretation of the CMB parameters which are in the original approach no material parameters in a strict sense, as they depend on the specimen geometry. Calibration and validation of the proposed model is performed using results of strain controlled LCF tests of specimens with different surface areas. The test specimens are made of the nickel base superalloy RENE 80.

stat.AP↗

arXiv2013

Risk estimation for LCF crack initiation

An accurate risk assessment for fatigue damage is of vital importance for the design and service of today's turbomachinery components. We present an approach for quantifying the probability of crack initiation due to surface driven low-cycle fatigue (LCF). This approach is based on the theory of failure-time processes and takes inhomogeneous stress fields and size effects into account. The method has been implemented as a finite-element postprocessor which uses quadrature formulae of higher order. Results of applying this new approach to an example case of a gas-turbine compressor disk are discussed.

math.NA↗