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

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

Performance Modeling of Inline Compression With Software Caching for Reducing the Memory Footprint in PYSDC

Modern HPC applications compute and analyze massive amounts of data. The data volume is growing faster than memory capabilities and storage improvements leading to performance bottlenecks. An example of this is pySDC, a framework for solving collocation problems iteratively using parallel-in-time methods. These methods require storing and exchanging 3D volume data for each parallel point in time. If a simulation consists of M parallel-in-time stages, where the full spatial problem has to be stored for the next iteration, the memory demand for a single state variable is M ×Nx ×Ny ×Nz per time-step. For an application simulation with many state variables or stages, the memory requirement is considerable. Data compression helps alleviate the overhead in memory by reducing the size of data and keeping it in compressed format. Inline compression compresses and decompresses the application’s working set as it moves in and out of main memory. Thus, it provides the system with the appearance of more main memory. Naive compressed arrays require a compression or decompression operation for each store or load and therefore hurt the performance of the application. By incorporating a software cache and storing decompressed values of the array, we limit the number of compression and decompression operations for the stores and loads, thereby improving performance overall. In this thesis, we build a compression manager and software cache manager for the pySDC framework to reduce the memory requirements and computational overhead. The compression manager wraps around LibPressio, a C++ compression library that abstracts all compressors. We utilize blosc, a lossless compressor for our compression manager, and build a software cache manager with various cache configurations and cache policies to work in cohesion with the compression manager. We build a performance model which evaluates the compression manager and cache manager’s performance on different metrics such as compression ratio and compression/decompression time. We test our framework on two different pySDC applications — e.g., Allen-Cahn and Heat-diffusion. ii Results show that incorporating compression and increasing the cache size for our applications inflates the total compressed size in bytes for the arrays and therefore reduces the compression ratio, in contrast to our expectations. However, incorporating the cache and a greater cache size reduces the number of compression/decompression calls to LibPressio as well as cache evictions, significantly reducing the computational overhead for pySDC. Thus, overall, our compression and cache manager help reduce the memory footprint in pySDC. Future work involves looking at improving the compression ratio and using lossy compression to achieve significant reduction in memory footprint

An arbitrary order time-stepping algorithm for tracking particles in inhomogeneous magnetic fields

The Lorentz equations describe the motion of electrically charged particles in electric and magnetic fields and are used widely in plasma physics. The most popular numerical algorithm for solving them is the Boris method, a variant of the Störmer-Verlet algorithm. Boris method is phase space volume conserving and simulated particles typically remain near the correct trajectory. However, it is only second order accurate. Therefore, in scenarios where it is not enough to know that a particle stays on the right trajectory but one needs to know where on the trajectory the particle is at a given time, Boris method requires very small time steps to deliver accurate phase information, making it computationally expensive. We derive an improved version of the high-order Boris spectral deferred correction algorithm (Boris-SDC) by adopting a convergence acceleration strategy for second order problems based on the Generalised Minimum Residual (GMRES) method. Our new algorithm is easy to implement as it still relies on the standard Boris method. Like Boris-SDC it can deliver arbitrary order of accuracy through simple changes of runtime parameter but possesses better long-term energy stability. We demonstrate for two examples, a magnetic mirror trap and the Solev'ev equilibrium, that the new method can deliver better accuracy at lower computational cost compared to the standard Boris method. While our examples are motivated by tracking ions in the magnetic field of a nuclear fusion reactor, the introduced algorithm can potentially deliver similar improvements in efficiency for other applications

Parallelizing spectral deferred corrections across the method

In this paper we present two strategies to enable “parallelization across the method” for spectral deferred corrections (SDC). Using standard low-order time-stepping methods in an iterative fashion, SDC can be seen as preconditioned Picard iteration for the collocation problem. Typically, a serial Gauß–Seidel-like preconditioner is used, computing updates for each collocation node one by one. The goal of this paper is to show how this process can be parallelized, so that all collocation nodes are updated simultaneously. The first strategy aims at finding parallel preconditioners for the Picard iteration and we test three choices using four different test problems. For the second strategy we diagonalize the quadrature matrix of the collocation problem directly. In order to integrate non-linear problems we employ simplified and inexact Newton methods. Here, we estimate the speed of convergence depending on the time-step size and verify our results using a non-linear diffusion problem