Resiliency in numerical algorithm design for extreme scale simulations

This work is based on the seminar titled ‘Resiliency in Numerical Algorithm Design for Extreme Scale Simulations’ held March 1–6, 2020, at Schloss Dagstuhl, that was attended by all the authors. Advanced supercomputing is characterized by very high computation speeds at the cost of involving an enormous amount of resources and costs. A typical large-scale computation running for 48 h on a system consuming 20 MW, as predicted for exascale systems, would consume a million kWh, corresponding to about 100k Euro in energy cost for executing 1023 floating-point operations. It is clearly unacceptable to lose the whole computation if any of the several million parallel processes fails during the execution. Moreover, if a single operation suffers from a bit-flip error, should the whole computation be declared invalid? What about the notion of reproducibility itself: should this core paradigm of science be revised and refined for results that are obtained by large-scale simulation? Naive versions of conventional resilience techniques will not scale to the exascale regime: with a main memory footprint of tens of Petabytes, synchronously writing checkpoint data all the way to background storage at frequent intervals will create intolerable overheads in runtime and energy consumption. Forecasts show that the mean time between failures could be lower than the time to recover from such a checkpoint, so that large calculations at scale might not make any progress if robust alternatives are not investigated. More advanced resilience techniques must be devised. The key may lie in exploiting both advanced system features as well as specific application knowledge. Research will face two essential questions: (1) what are the reliability requirements for a particular computation and (2) how do we best design the algorithms and software to meet these requirements? While the analysis of use cases can help understand the particular reliability requirements, the construction of remedies is currently wide open. One avenue would be to refine and improve on system- or application-level checkpointing and rollback strategies in the case an error is detected. Developers might use fault notification interfaces and flexible runtime systems to respond to node failures in an application-dependent fashion. Novel numerical algorithms or more stochastic computational approaches may be required to meet accuracy requirements in the face of undetectable soft errors. These ideas constituted an essential topic of the seminar. The goal of this Dagstuhl Seminar was to bring together a diverse group of scientists with expertise in exascale computing to discuss novel ways to make applications resilient against detected and undetected faults. In particular, participants explored the role that algorithms and applications play in the holistic approach needed to tackle this challenge. This article gathers a broad range of perspectives on the role of algorithms, applications and systems in achieving resilience for extreme scale simulations. The ultimate goal is to spark novel ideas and encourage the development of concrete solutions for achieving such resilience holistically.Peer Reviewed"Article signat per 36 autors/es: Emmanuel Agullo, Mirco Altenbernd, Hartwig Anzt, Leonardo Bautista-Gomez, Tommaso Benacchio, Luca Bonaventura, Hans-Joachim Bungartz, Sanjay Chatterjee, Florina M. Ciorba, Nathan DeBardeleben, Daniel Drzisga, Sebastian Eibl, Christian Engelmann, Wilfried N. Gansterer, Luc Giraud, Dominik G ̈oddeke, Marco Heisig, Fabienne Jezequel, Nils Kohl, Xiaoye Sherry Li, Romain Lion, Miriam Mehl, Paul Mycek, Michael Obersteiner, Enrique S. Quintana-Ortiz, Francesco Rizzi, Ulrich Rude, Martin Schulz, Fred Fung, Robert Speck, Linda Stals, Keita Teranishi, Samuel Thibault, Dominik Thonnes, Andreas Wagner and Barbara Wohlmuth"Postprint (author's final draft

Software for Exascale Computing - SPPEXA 2016-2019

This open access book summarizes the research done and results obtained in the second funding phase of the Priority Program 1648 "Software for Exascale Computing" (SPPEXA) of the German Research Foundation (DFG) presented at the SPPEXA Symposium in Dresden during October 21-23, 2019. In that respect, it both represents a continuation of Vol. 113 in Springer’s series Lecture Notes in Computational Science and Engineering, the corresponding report of SPPEXA’s first funding phase, and provides an overview of SPPEXA’s contributions towards exascale computing in today's sumpercomputer technology. The individual chapters address one or more of the research directions (1) computational algorithms, (2) system software, (3) application software, (4) data management and exploration, (5) programming, and (6) software tools. The book has an interdisciplinary appeal: scholars from computational sub-fields in computer science, mathematics, physics, or engineering will find it of particular interest

Toward Performance-Portable PETSc for GPU-based Exascale Systems

The Portable Extensible Toolkit for Scientific computation (PETSc) library delivers scalable solvers for nonlinear time-dependent differential and algebraic equations and for numerical optimization.The PETSc design for performance portability addresses fundamental GPU accelerator challenges and stresses flexibility and extensibility by separating the programming model used by the application from that used by the library, and it enables application developers to use their preferred programming model, such as Kokkos, RAJA, SYCL, HIP, CUDA, or OpenCL, on upcoming exascale systems. A blueprint for using GPUs from PETSc-based codes is provided, and case studies emphasize the flexibility and high performance achieved on current GPU-based systems.Comment: 15 pages, 10 figures, 2 table

esys-Escript User’s Guide: Solving Partial Differential Equations with Escript and Finley Release - 4.0 (r5402)

esys.escript is a python-based environment for implementing mathematical models, in particular those based on coupled, non-linear, time-dependent partial differential equations. It consists of five major components: • esys.escript core library • finite element solvers esys.finley, esys.dudley, esys.ripley, and esys.speckley (which use fast vendor-supplied solvers or the included PASO linear solver library) • the meshing interface esys.pycad • a model library • an inversion module. All esys.escript modules should work under both python 2 and python 3, see Appendix E. The current version supports parallelization through MPI for distributed memory, OpenMP for shared memory on CPUs, as well as CUDA for some GPU-based solvers. This release comes with some significant changes and new features. Please see Appendix B for a detailed list. If you use this software in your research, then we would appreciate (but do not require) a citation. Some relevant references can be found in Appendix D

Parallel unstructured solvers for linear partial differential equations

This thesis presents the development of a parallel algorithm to solve symmetric systems of linear equations and the computational implementation of a parallel partial differential equations solver for unstructured meshes. The proposed method, called distributive conjugate gradient - DCG, is based on a single-level domain decomposition method and the conjugate gradient method to obtain a highly scalable parallel algorithm. An overview on methods for the discretization of domains and partial differential equations is given. The partition and refinement of meshes is discussed and the formulation of the weighted residual method for two- and three-dimensions presented. Some of the methods to solve systems of linear equations are introduced, highlighting the conjugate gradient method and domain decomposition methods. A parallel unstructured PDE solver is proposed and its actual implementation presented. Emphasis is given to the data partition adopted and the scheme used for communication among adjacent subdomains is explained. A series of experiments in processor scalability is also reported. The derivation and parallelization of DCG are presented and the method validated throughout numerical experiments. The method capabilities and limitations were investigated by the solution of the Poisson equation with various source terms. The experimental results obtained using the parallel solver developed as part of this work show that the algorithm presented is accurate and highly scalable, achieving roughly linear parallel speed-up in many of the cases tested

Automated cache optimisations of stencil computations for partial differential equations

This thesis focuses on numerical methods that solve partial differential equations. Our focal point is the finite difference method, which solves partial differential equations by approximating derivatives with explicit finite differences. These partial differential equation solvers consist of stencil computations on structured grids. Stencils for computing real-world practical applications are patterns often characterised by many memory accesses and non-trivial arithmetic expressions that lead to high computational costs compared to simple stencils used in much prior proof-of-concept work. In addition, the loop nests to express stencils on structured grids may often be complicated. This work is highly motivated by a specific domain of stencil computations where one of the challenges is non-aligned to the structured grid ("off-the-grid") operations. These operations update neighbouring grid points through scatter and gather operations via non-affine memory accesses, such as {A[B[i]]}. In addition to this challenge, these practical stencils often include many computation fields (need to store multiple grid copies), complex data dependencies and imperfect loop nests. In this work, we aim to increase the performance of stencil kernel execution. We study automated cache-memory-dependent optimisations for stencil computations. This work consists of two core parts with their respective contributions.The first part of our work tries to reduce the data movement in stencil computations of practical interest. Data movement is a dominant factor affecting the performance of high-performance computing applications. It has long been a target of optimisations due to its impact on execution time and energy consumption. This thesis tries to relieve this cost by applying temporal blocking optimisations, also known as time-tiling, to stencil computations. Temporal blocking is a well-known technique to enhance data reuse in stencil computations. However, it is rarely used in practical applications but rather in theoretical examples to prove its efficacy. Applying temporal blocking to scientific simulations is more complex. More specifically, in this work, we focus on the application context of seismic and medical imaging. In this area, we often encounter scatter and gather operations due to signal sources and receivers at arbitrary locations in the computational domain. These operations make the application of temporal blocking challenging. We present an approach to overcome this challenge and successfully apply temporal blocking.In the second part of our work, we extend the first part as an automated approach targeting a wide range of simulations modelled with partial differential equations. Since temporal blocking is error-prone, tedious to apply by hand and highly complex to assimilate theoretically and practically, we are motivated to automate its application and automatically generate code that benefits from it. We discuss algorithmic approaches and present a generalised compiler pipeline to automate the application of temporal blocking. These passes are written in the Devito compiler. They are used to accelerate the computation of stencil kernels in areas such as seismic and medical imaging, computational fluid dynamics and machine learning. \href{www.devitoproject.org}{Devito} is a Python package to implement optimised stencil computation (e.g., finite differences, image processing, machine learning) from high-level symbolic problem definitions. Devito builds on \href{www.sympy.org}{SymPy} and employs automated code generation and just-in-time compilation to execute optimised computational kernels on several computer platforms, including CPUs, GPUs, and clusters thereof. We show how we automate temporal blocking code generation without user intervention and often achieve better time-to-solution. We enable domain-specific optimisation through compiler passes and offer temporal blocking gains from a high-level symbolic abstraction. These automated optimisations benefit various computational kernels for solving real-world application problems.Open Acces

Productive and efficient computational science through domain-specific abstractions

In an ideal world, scientific applications are computationally efficient, maintainable and composable and allow scientists to work very productively. We argue that these goals are achievable for a specific application field by choosing suitable domain-specific abstractions that encapsulate domain knowledge with a high degree of expressiveness. This thesis demonstrates the design and composition of domain-specific abstractions by abstracting the stages a scientist goes through in formulating a problem of numerically solving a partial differential equation. Domain knowledge is used to transform this problem into a different, lower level representation and decompose it into parts which can be solved using existing tools. A system for the portable solution of partial differential equations using the finite element method on unstructured meshes is formulated, in which contributions from different scientific communities are composed to solve sophisticated problems. The concrete implementations of these domain-specific abstractions are Firedrake and PyOP2. Firedrake allows scientists to describe variational forms and discretisations for linear and non-linear finite element problems symbolically, in a notation very close to their mathematical models. PyOP2 abstracts the performance-portable parallel execution of local computations over the mesh on a range of hardware architectures, targeting multi-core CPUs, GPUs and accelerators. Thereby, a separation of concerns is achieved, in which Firedrake encapsulates domain knowledge about the finite element method separately from its efficient parallel execution in PyOP2, which in turn is completely agnostic to the higher abstraction layer. As a consequence of the composability of those abstractions, optimised implementations for different hardware architectures can be automatically generated without any changes to a single high-level source. Performance matches or exceeds what is realistically attainable by hand-written code. Firedrake and PyOP2 are combined to form a tool chain that is demonstrated to be competitive with or faster than available alternatives on a wide range of different finite element problems.Open Acces

Task-based adaptive multiresolution for time-space multi-scale reaction-diffusion systems on multi-core architectures

A new solver featuring time-space adaptation and error control has been recently introduced to tackle the numerical solution of stiff reaction-diffusion systems. Based on operator splitting, finite volume adaptive multiresolution and high order time integrators with specific stability properties for each operator, this strategy yields high computational efficiency for large multidimensional computations on standard architectures such as powerful workstations. However, the data structure of the original implementation, based on trees of pointers, provides limited opportunities for efficiency enhancements, while posing serious challenges in terms of parallel programming and load balancing. The present contribution proposes a new implementation of the whole set of numerical methods including Radau5 and ROCK4, relying on a fully different data structure together with the use of a specific library, TBB, for shared-memory, task-based parallelism with work-stealing. The performance of our implementation is assessed in a series of test-cases of increasing difficulty in two and three dimensions on multi-core and many-core architectures, demonstrating high scalability