We consider optimal-scaling multigrid solvers for the linear systems that arise from the discretization of problems with evolutionary behavior. Typically, solution algorithms for evolution equations are based on a time-marching approach, solving sequentially for one time step after the other. Parallelism in these traditional time-integration techniques is limited to spatial parallelism. However, current trends in computer architectures are leading towards systems with more, but not faster, processors. Therefore, faster compute speeds must come from greater parallelism. One approach to achieve parallelism in time is with multigrid, but extending classical multigrid methods for elliptic operators to this setting is not straightforward. In this paper, we present a non-intrusive, optimal-scaling time-parallel method based on multigrid reduction (MGR). We demonstrate optimal-ity of our multigrid-reduction-in-time algorithm (MGRIT) for solving diffusion equations in two and three space dimensions in numerical experiments. Furthermore, through both parallel performance models and actual parallel numerical results, we show that we can achieve significant speedup in comparison to sequential time marching on modern architectures. 1. Introduction. One of the major challenges facing the computational science community with future architectures is that faster compute speeds must come from increased concurrency, since clock speeds are no longer increasing but core counts are going up sharply. As a consequence, traditional time marching is becoming a huge sequential bottleneck in time integration simulations in the following way: improving simulation accuracy by scaling up the spatial resolution requires a similar (or greater) increase in the temporal resolution, which is also required to maintain stability in explicit methods. As a result, numerical time integration involves many more time steps leading to long overall compute times, since parallelizing only in space limits concurrency. Solving for multiple time steps in parallel and, therefore, increasing concurrency would remove this time integration bottleneck. Because time is sequential in nature, the idea of simultaneously solving for multiple time steps is not intuitive. Yet it is possible, with work on this topic going back to as early as 1964 [33]. However, most research on this subject has been done within the past 30 years including [2, 7-10, 14-22, 25, 28, 31, 32, 38, 40-44]. One approach to achieve parallelism in time is with multigrid methods. The parareal in time method, introduced by Lions, Maday, and Turinici in [25], can be interpreted as a two-level multigrid method [16], even though the leading idea came from a spatial domain decomposition approach. The algorithm is optimal, but concurrency is limited since the coarse-grid solve is still sequential. Considering true multilevel (not two-level) schemes, only a few methods exhibit full multigrid optimality and concurrency such as [21, 42,43], and most are designed for specific problems or discretizations. Furt...

Abstract. We consider optimal-scaling multigrid solvers for the linear systems that arise from the discretization of problems with evolutionary behavior. Typically, solution algorithms for evolution equations are based on a time-marching approach, solving sequentially for one time step after the other. Parallelism in these traditional time-integration techniques is limited to spatial parallelism. However, current trends in computer architectures are leading towards systems with more, but not faster, processors. Therefore, faster compute speeds must come from greater parallelism. One approach to achieve parallelism in time is with multigrid, but extending classical multigrid methods for elliptic operators to this setting is not straightforward. In this paper, we present a non-intrusive, optimal-scaling time-parallel method based on multigrid reduction (MGR). We demonstrate optimality of our multigrid-reduction-in-time algorithm (MGRIT) for solving diffusion equations in two and three space dimensions in numerical experiments. Furthermore, through both parallel performance models and actual parallel numerical results, we show that we can achieve significant speedup in comparison to sequential time marching on modern architectures.Key words. parabolic problems, reduction-based multigrid, multigrid-in-time, parareal AMS subject classifications. 65F10, 65M22, 65M551. Introduction. One of the major challenges facing the computational science community with future architectures is that faster compute speeds must come from increased concurrency, since clock speeds are no longer increasing but core counts are going up sharply. As a consequence, traditional time marching is becoming a huge sequential bottleneck in time integration simulations in the following way: improving simulation accuracy by scaling up the spatial resolution requires a similar (or greater) increase in the temporal resolution, which is also required to maintain stability in explicit methods. As a result, numerical time integration involves many more time steps leading to long overall compute times, since parallelizing only in space limits concurrency. Solving for multiple time steps in parallel and, therefore, increasing concurrency would remove this time integration bottleneck.Because time is sequential in nature, the idea of simultaneously solving for multiple time steps is not intuitive. Yet it is possible, with work on this topic going back to as early as 1964 [33]. However, most research on this subject has been done within the past 30 years including [2, 7-10, 14-22, 25, 28, 31, 32, 38, 40-44]. One approach to achieve parallelism in time is with multigrid methods. The parareal in time method, introduced by Lions, Maday, and Turinici in [25], can be interpreted as a two-level multigrid method [16], even though the leading idea came from a spatial domain decomposition approach. The algorithm is optimal, but concurrency is limited since the coarse-grid solve is still sequential. Considering true multilevel (not two-level) schemes, only a few method...

Summary Multigrid and related multilevel methods are the approaches of choice for solving linear systems that result from discretization of a wide class of PDEs. A large gap, however, exists between the theoretical analysis of these algorithms and their actual performance. This paper focuses on the extension of the well‐known local mode (often local Fourier) analysis approach to a wider class of problems. The semi‐algebraic mode analysis (SAMA) proposed here couples standard local Fourier analysis approaches with algebraic computation to enable analysis of a wider class of problems, including those with strong advective character. The predictive nature of SAMA is demonstrated by applying it to the parabolic diffusion equation in one and two space dimensions, elliptic diffusion in layered media, as well as a two‐dimensional convection‐diffusion problem. These examples show that accounting for boundary conditions and heterogeneity enables accurate predictions of the short‐term and asymptotic convergence behavior for multigrid and related multilevel methods. Copyright © 2015 John Wiley & Sons, Ltd.

We consider numerical methods for generalized diffusion equations that are motivated by the transport problems arising in electron beam radiation therapy planning. While Monte Carlo methods are typically used for simulations of the forward-peaked scattering behavior of electron beams, rough calculations suggest that grid-based discretizations can provide more efficient simulations if the discretizations can be made sufficiently accurate, and optimal solvers can be found for the resulting linear systems. The multigrid method for model two-dimensional transport problems presented in [C. Börgers and S. MacLachlan, J. Comput. Phys., 229 (2010), pp. 2914-2931 shows the necessary optimal scaling with some dependence on the choice of scattering kernel. In order to understand this behavior, local Fourier analysis can be applied to the two-grid cycle. Using this approach, expressions for the error-propagation operators of the coarse-grid correction and relaxation steps, projected onto the fine-grid harmonic spaces, can be found. In this paper, we consider easier problems of the form of generalized diffusion problems in space-time that are analogous to model two-dimensional transport problems. We present local Fourier analysis results for these space-time model problems and compare with convergence factors of Börgers and MacLachlan. Since one of our model problems is the diffusion equation itself, we also compare to convergence factors for the diffusion equation of [S. Vandewalle and G. Horton, Computing, 54 (1995), pp. 317-330]. The results presented here show that local Fourier analysis does not offer its usual predictivity of the convergence behavior of the diffusion equation and the generalized diffusion equations until we consider unrealistically long time intervals.