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

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

Algorithm 997: pySDC - Prototyping Spectral Deferred Corrections

In this article, we present the Python framework pySDC for solving collocation problems with spectral deferred correction (SDC) methods 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 timestepping to high-order implicit-explicit or multi-implicit splitting and multilevel SDCs. The software package comes with many different, preimplemented 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 or 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: those of the user and those of the developer. The first sheds light on the front-end, the examples, and the tutorials, and 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. In addition, couplings to the FEniCS framework and PETSc, the latter including spatial parallelism with MPI, are described