arXiv subjects

Stephanie Friedhoff

Publications and source records attributed to Stephanie Friedhoff.

10 recordsLinked to original sources

arXiv2021

Asynchronous Truncated Multigrid-reduction-in-time (AT-MGRIT)

In this paper, we present the new "asynchronous truncated multigrid-reduction-in-time" (AT-MGRIT) algorithm for introducing time parallelism to the solution of discretized time-dependent problems. The new algorithm is based on the multigrid-reduction-in-time (MGRIT) approach, which, in certain settings, is equivalent to another common multilevel parallel-in-time method, Parareal. In contrast to Parareal and MGRIT that both consider a global temporal grid over the entire time interval on the coarsest level, the AT-MGRIT algorithm uses truncated local time grids on the coarsest level, each grid covering certain temporal subintervals. These local grids can be solved completely independent from each other, which reduces the sequential part of the algorithm and, thus, increases parallelism in the method. Here, we study the effect of using truncated local coarse grids on the convergence of the algorithm, both theoretically and numerically, and show, using challenging nonlinear problems, that the new algorithm consistently outperforms classical Parareal/MGRIT in terms of time to solution.

math.NA↗

arXiv2021

Weighted Relaxation for Multigrid Reduction in Time

Based on current trends in computer architectures, faster compute speeds must come from increased parallelism rather than increased clock speeds, which are currently stagnate. This situation has created the well-known bottleneck for sequential time-integration, where each individual time-value (i.e., time-step) is computed sequentially. One approach to alleviate this and achieve parallelism in time is with multigrid. In this work, we consider multigrid-reduction-in-time (MGRIT), a multilevel method applied to the time dimension that computes multiple time-steps in parallel. Like all multigrid methods, MGRIT relies on the complementary relationship between relaxation on a fine-grid and a correction from the coarse grid to solve the problem. All current MGRIT implementations are based on unweighted-Jacobi relaxation; here we introduce the concept of weighted relaxation to MGRIT. We derive new convergence bounds for weighted relaxation, and use this analysis to guide the selection of relaxation weights. Numerical results then demonstrate that non-unitary relaxation weights consistently yield faster convergence rates and lower iteration counts for MGRIT when compared with unweighted relaxation. In most cases, weighted relaxation yields a 10%-20% saving in iterations. For A-stable integration schemes, results also illustrate that under-relaxation can restore convergence in some cases where unweighted relaxation is not convergent.

math.NA↗

arXiv2020

PyMGRIT: A Python Package for the parallel-in-time method MGRIT

In this paper, we introduce the Python framework PyMGRIT, which implements the multigrid-reduction-in-time (MGRIT) algorithm for solving the (non-)linear systems arising from the discretization of time-dependent problems. The MGRIT algorithm is a reduction-based iterative method that allows parallel-in-time simulations, i. e., calculating multiple time steps simultaneously in a simulation, by using a time-grid hierarchy. The PyMGRIT framework features many different variants of the MGRIT algorithm, ranging from different multigrid cycle types and relaxation schemes, as well as various coarsening strategies, including time-only and space-time coarsening, to using different time integrators on different levels in the multigrid hierachy. PyMGRIT allows serial runs for prototyping and testing of new approaches, as well as parallel runs using the Message Passing Interface (MPI). Here, we describe the implementation of the MGRIT algorithm in PyMGRIT and present the usage from both user and developer point of views. Three examples illustrate different aspects of the package, including pure time parallelism as well as space-time parallelism by coupling PyMGRIT with PETSc or Firedrake, which enable spatial parallelism through MPI.

cs.MS↗

arXiv2020

On "Optimal" h-Independent Convergence of Parareal and MGRIT using Runge-Kutta Time Integration

Although convergence of the Parareal and multigrid-reduction-in-time (MGRIT) parallel-in-time algorithms is well studied, results on their optimality is limited. Appealling to recently derived tight bounds of two-level Parareal and MGRIT convergence, this paper proves (or disproves) $h_x$- and $h_t$-independent convergence of two-level Parareal and MGRIT, for linear problems of the form $\mathbf{u}'(t) + \mathcal{L}\mathbf{u}(t) = f(t)$, where $\mathcal{L}$ is symmetric positive definite and Runge-Kutta time integration is used. The theory presented in this paper also encompasses analysis of some modified Parareal algorithms, such as the $θ$-Parareal method, and shows that not all Runge-Kutta schemes are equal from the perspective of parallel-in-time. Some schemes, particularly L-stable methods, offer significantly better convergence than others as they are guaranteed to converge rapidly at both limits of small and large $h_tξ$, where $ξ$ denotes an eigenvalue of $\mathcal{L}$ and $h_t$ time-step size. On the other hand, some schemes do not obtain $h$-optimal convergence, and two-level convergence is restricted to certain regimes. In certain cases, an $\mathcal{O}(1)$ factor change in time step $h_t$ or coarsening factor $k$ can be the difference between convergence factors $ρ\approx0.02$ and divergence! The analysis is extended to skew symmetric operators as well, which cannot obtain $h$-independent convergence and, in fact, will generally not converge for a sufficiently large number of time steps. Numerical results confirm the analysis in practice and emphasize the importance of a priori analysis in choosing an effective coarse-grid scheme and coarsening factor. A Mathematica notebook to perform a priori two-grid analysis is available at https://github.com/XBraid/xbraid-convergence-est.

math.NA↗

arXiv2019

Parallel-in-Time Simulation of an Electrical Machine using MGRIT

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 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. 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.

math.NA↗

arXiv2019

On selecting coarse-grid operators for Parareal and MGRIT applied to linear advection

We consider the parallel time integration of the linear advection equation with the Parareal and two-level multigrid-reduction-in-time (MGRIT) algorithms. Our aim is to develop a better understanding of the convergence behaviour of these algorithms for this problem, which is known to be poor relative to the diffusion equation, its model parabolic counterpart. Using Fourier analysis, we derive new convergence estimates for these algorithms which, in conjunction with existing convergence theory, provide insight into the origins of this poor performance. We then use this theory to explore improved coarse-grid time-stepping operators. For several high-order discretizations of the advection equation, we demonstrate that there exist non-standard coarse-grid time stepping operators that yield significant improvements over the standard choice of rediscretization.

math.NA↗

arXiv2019

Optimizing multigrid reduction-in-time (MGRIT) and Parareal coarse-grid operators for linear advection

Parallel-in-time methods, such as multigrid reduction-in-time (MGRIT) and Parareal, provide an attractive option for increasing concurrency when simulating time-dependent PDEs in modern high-performance computing environments. While these techniques have been very successful for parabolic equations, it has often been observed that their performance suffers dramatically when applied to advection-dominated problems or purely hyperbolic PDEs using standard rediscretization approaches on coarse grids. In this paper, we apply MGRIT or Parareal to the constant-coefficient linear advection equation, appealing to existing convergence theory to provide insight into the typically non-scalable or even divergent behavior of these solvers for this problem. To overcome these failings, we replace rediscretization on coarse grids with improved coarse-grid operators that are computed by applying optimization techniques to approximately minimize error estimates from the convergence theory. One of our main findings is that, in order to obtain fast convergence as for parabolic problems, coarse-grid operators should take into account the behavior of the hyperbolic problem by tracking the characteristic curves. Our approach is tested for schemes of various orders using explicit or implicit Runge-Kutta methods combined with upwind-finite-difference spatial discretizations. In all cases, we obtain scalable convergence in just a handful of iterations, with parallel tests also showing significant speed-ups over sequential time-stepping. Our insight of tracking characteristics on coarse grids provides a key idea for solving the long-standing problem of efficient parallel-in-time integration for hyperbolic PDEs.

math.NA↗

arXiv2019

Convergence analysis for parallel-in-time solution of hyperbolic systems

Parallel-in-time algorithms have been successfully employed for reducing time-to-solution of a variety of partial differential equations, especially for diffusive (parabolic-type) equations. A major failing of parallel-in-time approaches to date, however, is that most methods show instabilities or poor convergence for hyperbolic problems. This paper focuses on the analysis of the convergence behavior of multigrid methods for the parallel-in-time solution of hyperbolic problems. Three analysis tools are considered that differ, in particular, in the treatment of the time dimension: (1) space-time local Fourier analysis, using a Fourier ansatz in space and time, (2) semi-algebraic mode analysis, coupling standard local Fourier analysis approaches in space with algebraic computation in time, and (3) a two-level reduction analysis, considering error propagation only on the coarse time grid. In this paper, we show how insights from reduction analysis can be used to improve feasibility of the semi-algebraic mode analysis, resulting in a tool that offers the best features of both analysis techniques. Following validating numerical results, we investigate what insights the combined analysis framework can offer for two model hyperbolic problems, the linear advection equation in one space dimension and linear elasticity in two space dimensions.

math.NA↗

arXiv2019

Multigrid-reduction-in-time for Eddy Current problems

Parallel-in-time methods have shown success for reducing the simulation time of many time-dependent problems. Here, we consider applying the multigrid-reduction-in-time (MGRIT) algorithm to a voltage-driven eddy current model problem.

math.NA↗

arXiv2018

Exploring Parallel-in-Time Approaches for Eddy Current Problems

We consider the usage of parallel-in-time algorithms of the Parareal and multigrid-reduction-in-time (MGRIT) methodologies for the parallel-in-time solution of the eddy current problem. Via application of these methods to a two-dimensional model problem for a coaxial cable model, we show that a significant speedup can be achieved in comparison to sequential time stepping.

math.NA↗