Multilevel convergence analysis of multigrid-reduction-in-time

This paper presents a multilevel convergence framework for multigrid-reduction-in-time (MGRIT) as a generalization of previous two-grid estimates. The framework provides a priori upper bounds on the convergence of MGRIT V- and F-cycles, with different relaxation schemes, by deriving the respective residual and error propagation operators. The residual and error operators are functions of the time stepping operator, analyzed directly and bounded in norm, both numerically and analytically. We present various upper bounds of different computational cost and varying sharpness. These upper bounds are complemented by proposing analytic formulae for the approximate convergence factor of V-cycle algorithms that take the number of fine grid time points, the temporal coarsening factors, and the eigenvalues of the time stepping operator as parameters. The paper concludes with supporting numerical investigations of parabolic (anisotropic diffusion) and hyperbolic (wave equation) model problems. We assess the sharpness of the bounds and the quality of the approximate convergence factors. Observations from these numerical investigations demonstrate the value of the proposed multilevel convergence framework for estimating MGRIT convergence a priori and for the design of a convergent algorithm. We further highlight that observations in the literature are captured by the theory, including that two-level Parareal and multilevel MGRIT with F-relaxation do not yield scalable algorithms and the benefit of a stronger relaxation scheme. An important observation is that with increasing numbers of levels MGRIT convergence deteriorates for the hyperbolic model problem, while constant convergence factors can be achieved for the diffusion equation. The theory also indicates that L-stable Runge-Kutta schemes are more amendable to multilevel parallel-in-time integration with MGRIT than A-stable Runge-Kutta schemes.Comment: 26 pages; 17 pages Supplementary Material

The Multicomputer Toolbox - First-Generation Scalable Libraries

First-generation scalable parallel libraries have been achieved, and are maturing, within the Multicomputer Toolbox. The Toolbox includes sparse, dense, iterative linear algebra, a stiff ODE/DAE solver, and an open software technology for additional numerical algorithms, plus an inter-architecture Makefile mechanism for building applications. We have devised C-based strategies for useful classes of distributed data structures, including distributed matrices and vectors. The underlying Zipcodemessage passing system has enabled process-grid abstractions of multicomputers, communication contexts, and process groups, all characteristics needed for building scalable libraries, and scalable application software. We describe the data-distribution-independent approach to building scalable libraries, which is needed so that applications do not unnecessarily have to redistribute data at high expense. We discuss the strategy used for implementing data-distribution mappings. We also describe high-level message-passing constructs used to achieve flexibility in transmission of data structures (Zipcode invoices). We expect Zipcode and MPI message-passing interfaces (which will incorporate many features from Zipcode, mentioned above) to co-exist in the future. We discuss progress thus far in achieving uniform interfaces for different algorithms for the same operation, which are needed to create poly-algorithms. Poly-algorithms are needed to widen the potential for scalability; uniform interfaces make simpler the testing of alternative methods with an application (whether for parallelism or for convergence, or both). We indicate that data-distribution-independent algorithms are sometimes more efficient than fixed-data-distribution counterparts, because redistribution of data can be avoided, and that this question is strongly application dependent

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

An Introduction to Algebraic Multigrid An Introduction to Algebraic Multigrid

Abstract Algebraic multigrid (AMG) solves linear systems based on multigrid principles, but in a way that only depends on the coefficients in the underlying matrix. The author begins with a basic introduction to AMG methods, and then describes some more recent advances and theoretical developments

Multigrid on Massively Parallel Architectures

. The scalable implementation of multigrid methods for machines with several thousands of processors is investigated. Parallel performance models are presented for two dierent structured-grid multigrid algorithms. These performance models are then used in the discussion of two implementation topics: replicating computations to reduce communications, and mixed programming models for multigrid codes on clusters of SMPs. Special attention is paid to comparing moderatesized parallelism and large-scale parallelism. Results are given from existing multigrid codes to support the discussion. 1 Introduction Computer simulations play an increasingly important role in scientic investigations. As a result, codes are being developed to solve complex multi-physics problems at very high resolutions. Such large-scale simulations require massively parallel computing, but this is not sucient. One also needs scalable algorithms such as multigrid, and scalable implementations of these algorithms. The ..