Preparing sparse solvers for exascale computing.

Sparse solvers provide essential functionality for a wide variety of scientific applications. Highly parallel sparse solvers are essential for continuing advances in high-fidelity, multi-physics and multi-scale simulations, especially as we target exascale platforms. This paper describes the challenges, strategies and progress of the US Department of Energy Exascale Computing project towards providing sparse solvers for exascale computing platforms. We address the demands of systems with thousands of high-performance node devices where exposing concurrency, hiding latency and creating alternative algorithms become essential. The efforts described here are works in progress, highlighting current success and upcoming challenges. This article is part of a discussion meeting issue 'Numerical algorithms for high-performance computational science'

Parallel Smoothers for Matrix-based Multigrid Methods on Unstructured Meshes Using Multicore CPUs and GPUs

Multigrid methods are efficient and fast solvers for problems typically modeled by partial differential equations of elliptic type. For problems with complex geometries and local singularities stencil-type discrete operators on equidistant Cartesian grids need to be replaced by more flexible concepts for unstructured meshes in order to properly resolve all problem-inherent specifics and for maintaining a moderate number of unknowns. However, flexibility in the meshes goes along with severe drawbacks with respect to parallel execution – especially with respect to the definition of adequate smoothers. This point becomes in particular pronounced in the framework of fine-grained parallelism on GPUs with hundreds of execution units. We use the approach of matrixbased multigrid that has high flexibility and adapts well to the exigences of modern computing platforms. In this work we investigate multi-colored Gauß-Seidel type smoothers, the power(q)-pattern enhanced multi-colored ILU(p) smoothers with fillins

Iteration-fusing conjugate gradient for sparse linear systems with MPI + OmpSs

In this paper, we target the parallel solution of sparse linear systems via iterative Krylov subspace-based method enhanced with a block-Jacobi preconditioner on a cluster of multicore processors. In order to tackle large-scale problems, we develop task-parallel implementations of the preconditioned conjugate gradient method that improve the interoperability between the message-passing interface and OmpSs programming models. Specifically, we progressively integrate several communication-reduction and iteration-fusing strategies into the initial code, obtaining more efficient versions of the method. For all these implementations, we analyze the communication patterns and perform a comparative analysis of their performance and scalability on a cluster consisting of 32 nodes with 24 cores each. The experimental analysis shows that the techniques described in the paper outperform the classical method by a margin that varies between 6 and 48%, depending on the evaluation.This research was partially supported by the H2020 EU FETHPC Project 671602 “INTERTWinE.” The researchers from Universidad Jaume I were sponsored by Project TIN2017-82972-R of the Spanish Ministerio de Economía y Competitividad. Maria Barreda was supported by the POSDOC-A/2017/11 project from the Universitat Jaume I.Peer ReviewedPostprint (author's final draft

Heterogeneous multicore systems for signal processing

This thesis explores the capabilities of heterogeneous multi-core systems, based on multiple Graphics Processing Units (GPUs) in a standard desktop framework. Multi-GPU accelerated desk side computers are an appealing alternative to other high performance computing (HPC) systems: being composed of commodity hardware components fabricated in large quantities, their price-performance ratio is unparalleled in the world of high performance computing. Essentially bringing “supercomputing to the masses”, this opens up new possibilities for application fields where investing in HPC resources had been considered unfeasible before. One of these is the field of bioelectrical imaging, a class of medical imaging technologies that occupy a low-cost niche next to million-dollar systems like functional Magnetic Resonance Imaging (fMRI). In the scope of this work, several computational challenges encountered in bioelectrical imaging are tackled with this new kind of computing resource, striving to help these methods approach their true potential. Specifically, the following main contributions were made: Firstly, a novel dual-GPU implementation of parallel triangular matrix inversion (TMI) is presented, addressing an crucial kernel in computation of multi-mesh head models of encephalographic (EEG) source localization. This includes not only a highly efficient implementation of the routine itself achieving excellent speedups versus an optimized CPU implementation, but also a novel GPU-friendly compressed storage scheme for triangular matrices. Secondly, a scalable multi-GPU solver for non-hermitian linear systems was implemented. It is integrated into a simulation environment for electrical impedance tomography (EIT) that requires frequent solution of complex systems with millions of unknowns, a task that this solution can perform within seconds. In terms of computational throughput, it outperforms not only an highly optimized multi-CPU reference, but related GPU-based work as well. Finally, a GPU-accelerated graphical EEG real-time source localization software was implemented. Thanks to acceleration, it can meet real-time requirements in unpreceeded anatomical detail running more complex localization algorithms. Additionally, a novel implementation to extract anatomical priors from static Magnetic Resonance (MR) scansions has been included

Multigrain Affinity for Heterogeneous Work Stealing

International audienceIn a parallel computing context, peak performance is hard to reach with irregular applications such as sparse linear algebra operations. It requires dynamic adjustments to automatically balance the workload between several processors. The problem becomes even more complicated when an architecture contains processing units with radically different computing capabilities. We present a hierarchical scheduling scheme designed to harness several CPUs and a GPU. It is built on a two-level work stealing mechanism tightly coupled to a software-managed cache. We show that our approach is well suited to dynamically control heterogeneous architectures, while taking advantage of a reduction of data transfers

Application of HPC in eddy current electromagnetic problem solution

As engineering problems are becoming more and more advanced, the size of an average model solved by partial differential equations is rapidly growing and, in order to keep simulation times within reasonable bounds, both faster computers and more efficient software implementations are needed. In the first part of this thesis, the full potential of simulation software has been exploited through high performance parallel computing techniques. In particular, the simulation of induction heating processes is accomplished within reasonable solution times, by implementing different parallel direct solvers for large sparse linear system, in the solution process of a commercial software. The performance of such library on shared memory systems has been remarkably improved by implementing a multithreaded version of MUMPS (MUltifrontal Massively Parallel Solver) library, which have been tested on benchmark matrices arising from typical induction heating process simulations. A new multithreading approach and a low rank approximation technique have been implemented and developed by MUMPS team in Lyon and Toulouse. In the context of a collaboration between MUMPS team and DII-University of Padova, a preliminary version of such functionalities could be tested on induction heating benchmark problems, and a substantial reduction of the computational cost and memory requirements could be achieved. In the second part of this thesis, some examples of design methodology by virtual prototyping have been described. Complex multiphysics simulations involving electromagnetic, circuital, thermal and mechanical problems have been performed by exploiting parallel solvers, as developed in the first part of this thesis. Finally, multiobjective stochastic optimization algorithms have been applied to multiphysics 3D model simulations in search of a set of improved induction heating device configurations

The fast multipole method at exascale

This thesis presents a top to bottom analysis on designing and implementing fast algorithms for current and future systems. We present new analysis, algorithmic techniques, and implementations of the Fast Multipole Method (FMM) for solving N- body problems. We target the FMM because it is broadly applicable to a variety of scientific particle simulations used to study electromagnetic, fluid, and gravitational phenomena, among others. Importantly, the FMM has asymptotically optimal time complexity with guaranteed approximation accuracy. As such, it is among the most attractive solutions for scalable particle simulation on future extreme scale systems. We specifically address two key challenges. The first challenge is how to engineer fast code for today’s platforms. We present the first in-depth study of multicore op- timizations and tuning for FMM, along with a systematic approach for transforming a conventionally-parallelized FMM into a highly-tuned one. We introduce novel opti- mizations that significantly improve the within-node scalability of the FMM, thereby enabling high-performance in the face of multicore and manycore systems. The second challenge is how to understand scalability on future systems. We present a new algorithmic complexity analysis of the FMM that considers both intra- and inter- node communication costs. Using these models, we present results for choosing the optimal algorithmic tuning parameter. This analysis also yields the surprising prediction that although the FMM is largely compute-bound today, and therefore highly scalable on current systems, the trajectory of processor architecture designs, if there are no significant changes could cause it to become communication-bound as early as the year 2015. This prediction suggests the utility of our analysis approach, which directly relates algorithmic and architectural characteristics, for enabling a new kind of highlevel algorithm-architecture co-design. To demonstrate the scientific significance of FMM, we present two applications namely, direct simulation of blood which is a multi-scale multi-physics problem and large-scale biomolecular electrostatics. MoBo (Moving Boundaries) is the infrastruc- ture for the direct numerical simulation of blood. It comprises of two key algorithmic components of which FMM is one. We were able to simulate blood flow using Stoke- sian dynamics on 200,000 cores of Jaguar, a peta-flop system and achieve a sustained performance of 0.7 Petaflop/s. The second application we propose as future work in this thesis is biomolecular electrostatics where we solve for the electrical potential using the boundary-integral formulation discretized with boundary element methods (BEM). The computational kernel in solving the large linear system is dense matrix vector multiply which we propose can be calculated using our scalable FMM. We propose to begin with the two dielectric problem where the electrostatic field is cal- culated using two continuum dielectric medium, the solvent and the molecule. This is only a first step to solving biologically challenging problems which have more than two dielectric medium, ion-exclusion layers, and solvent filled cavities. Finally, given the difficulty in producing high-performance scalable code, productivity is a key concern. Recently, numerical algorithms are being redesigned to take advantage of the architectural features of emerging multicore processors. These new classes of algorithms express fine-grained asynchronous parallelism and hence reduce the cost of synchronization. We performed the first extensive performance study of a recently proposed parallel programming model, called Concurrent Collections (CnC). In CnC, the programmer expresses her computation in terms of application-specific operations, partially-ordered by semantic scheduling constraints. The CnC model is well-suited to expressing asynchronous-parallel algorithms, so we evaluate CnC using two dense linear algebra algorithms in this style for execution on state-of-the-art mul- ticore systems. Our implementations in CnC was able to match and in some cases even exceed competing vendor-tuned and domain specific library codes. We combine these two distinct research efforts by expressing FMM in CnC, our approach tries to marry performance with productivity that will be critical on future systems. Looking forward, we would like to extend this to distributed memory machines, specifically implement FMM in the new distributed CnC, distCnC to express fine-grained paral- lelism which would require significant effort in alternative models.Ph.D

Ginkgo: A Modern Linear Operator Algebra Framework for High Performance Computing

© ACM, YYYY. This is the author's version of the work "Anzt, H., Cojean, T., Flegar, G., Göbel, F., Grützmacher, T., Nayak, P., ... & Quintana-Ortí, E. S. (2022). Ginkgo: A modern linear operator algebra framework for high performance computing. ACM Transactions on Mathematical Software (TOMS), 48(1), 1-33". It is posted here by permission of ACM for your personal use. Not for redistribution. The definitive version was published in ACM Transactions on Mathematical Software, {VOL48, ISS 1, (MAR 2022)} http://doi.acm.org/10.1145/3480935"[EN] In this article, we present GINKGO, a modern C++ math library for scientific high performance computing. While classical linear algebra libraries act on matrix and vector objects, Gnswo's design principle abstracts all functionality as linear operators," motivating the notation of a "linear operator algebra library" GINKGO'S current focus is oriented toward providing sparse linear algebra functionality for high performance graphics processing unit (GPU) architectures, but given the library design, this focus can be easily extended to accommodate other algorithms and hardware architectures. We introduce this sophisticated software architecture that separates core algorithms from architecture-specific backends and provide details on extensibility and sustainability measures. We also demonstrate GINKGO'S usability by providing examples on how to use its functionality inside the MFEM and deal.ii finite element ecosystems. Finally, we offer a practical demonstration of GINKGO'S high performance on state-of-the-art GPU architectures.This work was supported by the "Impuls und Vernetzungsfond of the Helmholtz Association" under grant VH-NG-1241. G. Flegar and E. S. Quintana-Orti were supported by project TIN2017-82972-R of the MINECO and FEDER and the H2020 EU FETHPC Project 732631 "OPRECOMP". This researchwas also supported by the Exascale Computing Project (17-SC-20-SC), a collaborative effort of the U.S. Department of Energy Office of Science and the National Nuclear Security Administration. The experiments on the NVIDIA A100 GPU were performed on the HAICORE@KIT partition, funded by the "Impuls und Vernetzungsfond" of the Helmholtz Association. The experiments on the AMD MI100 GPU were performed on Tulip, an early-access platform hosted by HPE.Anzt, H.; Cojean, T.; Flegar, G.; Göbel, F.; Grützmacher, T.; Nayak, P.; Ribizel, T.... (2022). Ginkgo: A Modern Linear Operator Algebra Framework for High Performance Computing. ACM Transactions on Mathematical Software. 48(1):1-33. https://doi.org/10.1145/348093513348