Parallel-in-time approaches for time-dependent PDEs have attracted a great deal of attention in the context of massively parallel environments. One such approach, multigrid reduction in time (MGRIT), extracts temporal parallelism by applying the reduction-based multigrid method for the temporal domain. To make efficient use of this parallelism, some processors need to be idle on a temporal coarse level of MGRIT. This paper proposes an optimization of the implementation that uses idle processors by assigning the spatial redistributed domain on coarse levels. It accelerates coarse-level spatial solvers and promotes early switching to parallelization-in-time by reducing its overhead. Although the spatial redistribution is contrary to the motivation of parallel-in-time approaches, it is expected to be effective when we need to balance spatial and temporal parallelism. This is because parallelization-in-time is already usually switched on when parallelization-in-space starts to saturate, so one can still benefit from additional parallelism. Numerical experiments demonstrate an improvement in runtime, at most, about 16.6% compared with pure MGRIT, which assigns best spatial and temporal parallelism at specific parallelism.

In this article, we introduce the Python framework PyMGRIT, which implements the multigrid-reduction-in-time (MGRIT) algorithm for solving (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, using a time-grid hierarchy. The PyMGRIT framework includes many different variants of the MGRIT algorithm, ranging from different multigrid cycle types and relaxation schemes, various coarsening strategies, including time-only and space-time coarsening, and the ability to utilize different time integrators on different levels in the multigrid hierachy. The comprehensive documentation with tutorials and many examples and the fully documented code allow an easy start into the work with the package. The functionality of the code is ensured by automated serial and parallel tests using continuous integration. PyMGRIT supports serial runs suitable for prototyping and testing of new approaches, as well as parallel runs using the Message Passing Interface (MPI). In this manuscript, we describe the implementation of the MGRIT algorithm in PyMGRIT and present the usage from both a user and a developer point of view. Three examples illustrate different aspects of the package itself, especially running tests with pure time parallelism, as well as space-time parallelism through the coupling of PyMGRIT with PETSc or Firedrake.

In this paper, we propose a multiplicative Schwartz-type block multi-color Gauss-Seidel (MS-BMC-GS) smoother for algebraic multigrid (AMG) methods. AMG is an excellent solver and one of the most effective preconditioners for Krylov subspace methods such as the conjugate gradient method. The achievable degree of parallelism, convergence ratio, and computational cost of AMG strongly depend on the chosen smoother. As multiple unknowns are relaxed simultaneously, the MS-BMC-GS smoother realizes higher convergence than the existing parallel Gauss-Seidel smoother. Although this increases the amount of computation, the increase in the computational time is mitigated by the high cache hit ratio owing to the novel blocking technique. Numerical experiments demonstrate that MS-BMC-GS outperforms the block multi-color GS smoother by 18%.

Summary SPIRAL is an autotuning, program generation, and code synthesis system that offers a fully automatic generation of highly optimized target codes, customized for the specific execution platform at hand. Initially, SPIRAL was targeted at problem domains in digital signal processing, later also at basic linear algebra. We open SPIRAL up to a new, practically relevant and challenging domain: multigrid solvers. SPIRAL is driven by algebraic transformation rules. We specify a set of such rules for a simple multigrid solver with a Richardson smoother for a discretized square 2D Poisson equation with Dirichlet boundary conditions. We present the target code that SPIRAL generates in static single‐assignment form and discuss its performance. While this example required no changes of or extensions to the SPIRAL system, more complex multigrid solvers may require small adaptations.

Summary Many problems formulated in partial differential equations lead to stencil‐type structures after applying an appropriate structured discretization. On one hand, exploiting these stencil structures in simulations can lead to massive performance improvements, compared to forming a sparse matrix. On the other hand, the generality of the simulation is restricted, depending on the exact definition of the stencils. In this article, we discuss the variability of stencils in the domain of porous‐media applications and present a family of models that grows in complexity. To demonstrate the relation between equation and discretization on the resulting stencil used to simulate the equation, we consider 4 models from the porous media domain. This way, we describe the influence of design decisions made during the discretization on the shape of stencils, to give application engineers' information on the variability they have to consider. This leads us to 2 variability models that shall help application engineers to understand the complexity and choices of stencil computations in the porous media domain.

Here, stencil codes are compute-intensive algorithms, in which data points arranged in a large grid are being recomputed repeatedly from the values of data points in a predefined neighborhood. This fixed neighborhood pattern is called a stencil. Stencils codes see wide-spread use in computing the discrete solutions of partial differential equations and systems composed of such equations.

We developed and optimized a multigrid method on the QPACE cluster. The QPACE cluster is an acclerator-based cluster using the Power Cell 8i CPU that is built by the special research field SFB TR 55 for Lattice Quantum Chromo dynamics computations. The cluster uses a custom 3D to rus network build using FPGAs. Our goal was to evaluate the QPACE architecture for a type of algorithm that uses a communication pattern not limited to nearest neighbor communication. We provide a model of the communication network taking into account the specific characteristics of the network and the network processor. For the implementation we chose to use an accelerator-centric programming model by using the SPUs, only.