A multigrid perspective on the parallel full approximation scheme in space and time

For the numerical solution of time-dependent partial differential equations, time-parallel methods have recently shown to provide a promising way to extend prevailing strong-scaling limits of numerical codes. One of the most complex methods in this field is the "Parallel Full Approximation Scheme in Space and Time" (PFASST). PFASST already shows promising results for many use cases and many more is work in progress. However, a solid and reliable mathematical foundation is still missing. We show that under certain assumptions the PFASST algorithm can be conveniently and rigorously described as a multigrid-in-time method. Following this equivalence, first steps towards a comprehensive analysis of PFASST using block-wise local Fourier analysis are taken. The theoretical results are applied to examples of diffusive and advective type

pySDC - Prototyping spectral deferred corrections

In this paper we present the Python framework pySDC for solving collocation problems with spectral deferred correction methods (SDC) and their time-parallel variant PFASST, the parallel full approximation scheme in space and time. pySDC features many implementations of SDC and PFASST, from simple implicit time-stepping to high-order implicit-explicit or multi-implicit splitting and multi-level spectral deferred corrections. It comes with many different, pre-implemented examples and has seven tutorials to help new users with their first steps. Time-parallelism is implemented either in an emulated way for debugging and prototyping as well as using MPI for benchmarking. The code is fully documented and tested using continuous integration, including most results of previous publications. Here, we describe the structure of the code by taking two different perspectives: the user's and the developer's perspective. While the first sheds light on the front-end, the examples and the tutorials, the second is used to describe the underlying implementation and the data structures. We show three different examples to highlight various aspects of the implementation, the capabilities and the usage of pySDC. Also, couplings to the FEniCS framework and PETSc, the latter including spatial parallelism with MPI, are described

The Coteau du Missouri : A Regional Study

The purpose of this study is to provide a general data base for future studies of and planning for studies of the region. It will also provide the people of South Dakota with information needed to derive a better understanding of the geography of South Dakota. The Department of Geography at South Dakota State University has adopted as a major goal of its graduate program the completion of a series of master’s theses on the geography of South Dakota. Each of these theses will examine the geography of one of the thirteen physiographic divisions that exist within the state. By 1988 studies which have been completed for South Dakota include the Coteau des Prairies, James River Highlands, Lake Dakota Plain, Minnesota River Lowland, and South Dakota Sandhil1s. These studies can be found in the thesis section of the library at South Dakota State University. This thesis is conducted with the hope that it will provide useful information for residents of the Coteau du Missouri, the Department of Geography at South Dakota State University, and any other individuals who may have an interest in the region. This thesis is a systematic regional study of the Coteau du Missouri of eastern South Dakota. The Coteau du Missouri occupies an area located on the eastern side of the Missouri River. It extends southward from the South Dakota-North Dakota border to the northwest corner of Bon Homme county in southeastern South Dakota. At its southernmost edge in South Dakota, the Missouri River cuts through the escarpment that forms the eastern boundary of the Coteau. It is nearly 75 miles wide at the North Dakota border, but narrows to a width of about 25miles at its southern edge in Bon Homme and Charles Mix counties. The Coteau occupies a curving belt of territory 200 miles long in a north-south extent between the Missouri River and the James River Lowland which comprises its eastern boundary. The Coteau du Missouri includes parts of 19 counties of South Dakota. These counties are Campbell, McPherson, Walworth, Edmunds, Potter, Faulk, Sully, Hughes, Hyde, Hand, Beadle, Buffalo, Jerauld, Brule, Aurora, Charles Mix, Douglas, Hutchinson, and Bon Homme

PFASST-ER: Combining the Parallel Full Approximation Scheme in Space and Time with parallelization across the method

To extend prevailing scaling limits when solving time-dependent partial differential equations, the parallel full approximation scheme in space and time (PFASST) has been shown to be a promising parallel-in-time integrator. Similar to a space-time multigrid, PFASST is able to compute multiple time-steps simultaneously and is therefore in particular suitable for large-scale applications on high performance computing systems. In this work we couple PFASST with a parallel spectral deferred correction (SDC) method, forming an unprecedented doubly time-parallel integrator. While PFASST provides global, large-scale "parallelization across the step", the inner parallel SDC method allows to integrate each individual time-step "parallel across the method" using a diagonalized local Quasi-Newton solver. This new method, which we call "PFASST with Enhanced concuRrency" (PFASST-ER), therefore exposes even more temporal parallelism. For two challenging nonlinear reaction-diffusion problems, we show that PFASST-ER works more efficiently than the classical variants of PFASST and can be used to run parallel-in-time beyond the number of time-steps.Comment: 12 pages, 12 figures, CVS PinT Workshop Proceeding

Time-parallel simulation of the Schr\"odinger Equation

The numerical simulation of the time-dependent Schr\"odinger equation for quantum systems is a very active research topic. Yet, resolving the solution sufficiently in space and time is challenging and mandates the use of modern high-performance computing systems. While classical parallelization techniques in space can reduce the runtime per time-step, novel parallel-in-time integrators expose parallelism in the temporal domain. They work, however, not very well for wave-type problems such as the Schr\"odinger equation. One notable exception is the rational approximation of exponential integrators. In this paper we derive an efficient variant of this approach suitable for the complex-valued Schr\"odinger equation. Using the Faber-Carath\'eodory-Fej\'er approximation, this variant is already a fast serial and in particular an efficient time-parallel integrator. It can be used to augment classical parallelization in space and we show the efficiency and effectiveness of our method along the lines of two challenging, realistic examples.Comment: 29 pages, 4 figures, 7 table

A parallel implementation of a diagonalization-based parallel-in-time integrator

We present and analyze a parallel implementation of a parallel-in-time method based on $\alpha$-circulant preconditioned Richardson iterations. While there are a lot of papers exploring this new class of single-level, time-parallel integrators from many perspectives, performance results of actual parallel runs are still missing. This leaves a critical gap, because the efficiency and applicability heavily rely on the actual parallel performance, with only limited guidance from theoretical considerations. Also, many challenges like selecting good parameters, finding suitable communication strategies, and performing a fair comparison to sequential time-stepping methods can be easily missed. In this paper, we first extend the original idea by using a collocation method of arbitrary order, which adds another level of parallelization in time. We derive an adaptive strategy to select a new $\alpha$-circulant preconditioner for each iteration during runtime for balancing convergence rates, round-off errors and inexactness in the individual time-steps. After addressing these more theoretical challenges, we present an open-source space- and doubly-time-parallel implementation and evaluate its performance for two different test problems

The Parallel Full Approximation Scheme in Space and Time for a Parabolic Finite Element Problem

The parallel full approximation scheme in space and time (PFASST) is a parallel-in-time integrator that allows to integrate multiple time-steps simultaneously. It has been shown to extend scaling limits of spatial parallelization strategies when coupled with finite differences, spectral discretizations, or particle methods. In this paper we show how to use PFASST together with a finite element discretization in space. While seemingly straightforward, the appearance of the mass matrix and the need to restrict iterates as well as residuals in space makes this task slightly more intricate. We derive the PFASST algorithm with mass matrices and appropriate prolongation and restriction operators and show numerically that PFASST can, after some initial iterations, gain two orders of accuracy per iteration