Increasing the Performance of the Jacobi--Davidson Method by Blocking

Röhrig-Zöllner, Melven; Thies, Jonas; Kreutzer, Moritz; Alvermann, Andreas; Pieper, Andreas; Basermann, Achim; Hager, Georg; Wellein, Gerhard; Fehske, Holger

doi:10.1137/140976017

Cited by 25 publications

(38 citation statements)

References 35 publications

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“…More recently, Liu et al [1] investigated strategies to improve the performance of SpMM 1 using SIMD (AVX/SSE) instructions for Stokesian dynamics simulation of biological macromolecules on modern multicore CPUs. Röhrig-Zöllner et al [19] discuss performance optimization techniques for the block Jacobi-Davidson method to compute a few eigenpairs of large-scale sparse matrices, and report reduced time-to-solution using block methods over single vector counterparts for quantum mechanics problems and PDEs. Finally, Anzt et al [20] describe an SpMM implementation based on the SELLC matrix format, and show that performance improvements in the SpMM kernel can translate into performance improvements in a block eigensolver running on GPUs.…”

Section: Introductionmentioning

confidence: 99%

A High Performance Block Eigensolver for Nuclear Configuration Interaction Calculations

Aktulga

Afibuzzaman

Williams

et al. 2017

IEEE Trans. Parallel Distrib. Syst.

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Abstract-As on-node parallelism increases and the performance gap between the processor and the memory system widens, achieving high performance in large-scale scientific applications requires an architecture-aware design of algorithms and solvers. We focus on the eigenvalue problem arising in nuclear Configuration Interaction (CI) calculations, where a few extreme eigenpairs of a sparse symmetric matrix are needed. We consider a block iterative eigensolver whose main computational kernels are the multiplication of a sparse matrix with multiple vectors (SpMM), and tall-skinny matrix operations. We present techniques to significantly improve the SpMM and the transpose operation SpMM T by using the compressed sparse blocks (CSB) format. We achieve 3-4× speedup on the requisite operations over good implementations with the commonly used compressed sparse row (CSR) format. We develop a performance model that allows us to correctly estimate the performance of our SpMM kernel implementations, and we identify cache bandwidth as a potential performance bottleneck beyond DRAM. We also analyze and optimize the performance of LOBPCG kernels (inner product and linear combinations on multiple vectors) and show up to 15× speedup over using high performance BLAS libraries for these operations. The resulting high performance LOBPCG solver achieves 1.4× to 1.8× speedup over the existing Lanczos solver on a series of CI computations on high-end multicore architectures (Intel Xeons). We also analyze the performance of our techniques on an Intel Xeon Phi Knights Corner (KNC) processor.

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Section: Introductionmentioning

confidence: 99%

A High Performance Block Eigensolver for Nuclear Configuration Interaction Calculations

Aktulga

Afibuzzaman

Williams

et al. 2017

IEEE Trans. Parallel Distrib. Syst.

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“…For instance, by combining multiple, consecutive sparse matrix-vector (SpMV) multiplications to a sparse matrix-multiple-vector (SpMMV) multiplication, the matrix entries are loaded only once and used for the multiple vectors, which reduces the overall memory traffic and consequently increases performance of this memory-bound operation. This has first been analytically shown in Gropp et al (1999) and is used in many applications; see, e.g., Röhrig-Zöllner et al (2015); Kreutzer et al (2018).…”

Section: Applicationmentioning

confidence: 95%

Performance Engineering for a Tall & Skinny Matrix Multiplication Kernels on GPUs

Ernst¹,

Hager²,

Thies

et al. 2020

Parallel Processing and Applied Mathematics

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General matrix-matrix multiplications (GEMM) in vendor-supplied BLAS libraries are best optimized for square matrices but often show bad performance for tall & skinny matrices, which are much taller than wide. NVIDIA's current CUBLAS implementation delivers only a fraction of the potential performance (as given by the roofline model) in this case. We describe the challenges and key properties of an implementation that can achieve perfect performance. We further evaluate different approaches of parallelization and thread distribution, and devise a flexible, configurable mapping scheme. A code generation approach enables a simultaneously flexible and specialized implementation with autotuning. This results in perfect performance for a large range of matrix sizes in the domain of interest, and at least 2/3 of maximum performance for the rest on an NVIDIA Volta GPGPU.

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“…In contrast to the standard Jacobi-Davidson method, which determines the sought eigenpairs one-by-one, the block Jacobi-Davison method in ESSEX [24] computes them by groups. Here we will consider only the real standard eigenvalue problem Av i = v i λ i .…”

Section: Using Higher Precision For Robust and Fast Orthogonalizationmentioning

confidence: 99%

Benefits from using mixed precision computations in the ELPA-AEO and ESSEX-II eigensolver projects

Alvermann

Basermann

Bungartz

et al. 2019

Japan J. Indust. Appl. Math.

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We first briefly report on the status and recent achievements of the ELPA-AEO (Eigenvalue Solvers for Petaflop Applications -Algorithmic Extensions and Optimizations) and ESSEX II (Equipping Sparse Solvers for Exascale) projects. In both collaboratory efforts, scientists from the application areas, mathematicians, and computer scientists work together to develop and make available efficient highly parallel methods for the solution of eigenvalue problems. Then we focus on a topic addressed in both projects, the use of mixed precision computations to enhance efficiency. We give a more detailed description of our approaches for benefiting from either lower or higher precision in three selected contexts and of the results thus obtained.Keywords ELPA-AEO · ESSEX · eigensolver · parallel · mixed precision IntroductionEigenvalue computations are at the core of simulations in various application areas, including quantum physics and electronic structure computations. Being able to best utilize the capabilities of current and emerging high-end computing systems is essential for further improving such simulations with respect to space/time resolution or by including additional effects in the models. Given these needs, the ELPA-AEO and ESSEX-II projects contribute to the development and implementation of efficient highly parallel methods for eigenvalue problems, in different contexts.Both projects are aimed at adding new features (concerning, e.g., performance and resilience) to previously developed methods and at providing additional functionality with new methods. Building on the results of the first ESSEX funding phase [14,34], ESSEX-II again focuses on iterative methods for very large eigenproblems arising, e.g., in quantum physics. ELPA-AEO's main application area is electronic structure computation, and for these moderately sized eigenproblems direct methods are often superior. Such methods are available in the widely used ELPA library [19], which had originated in an earlier project [2] and is being improved further and extended with ELPA-AEO.In Sections 2 and 3 we briefly report on the current state and on recent achievement in the two projects, with a focus on aspects that may be of particular interest to prospective users of the software or the underlying methods.Mixed precision in the ELPA-AEO and ESSEX-II projects 3In Section 4 we turn to computations involving different precisions. Looking at three examples from the two projects we describe how lower or higher precision is used to reduce the computing time. The ELPA-AEO projectIn the ELPA-AEO project, chemists, mathematicians and computer scientists from the Max Planck Computing and Data Facility in Garching, the Fritz Haber Institute of the Max Planck Society in Berlin, the Technical University of Munich, and the University of Wuppertal collaborate to provide highly scalable methods for solving moderately-sized (n 10 6 ) Hermitian eigenvalue problems. Such problems arise, e.g., in electronic structure computations, and during the earlier ELPA project, efficient...

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Increasing the Performance of the Jacobi--Davidson Method by Blocking

Cited by 25 publications

References 35 publications

A High Performance Block Eigensolver for Nuclear Configuration Interaction Calculations

A High Performance Block Eigensolver for Nuclear Configuration Interaction Calculations

Performance Engineering for a Tall & Skinny Matrix Multiplication Kernels on GPUs

Benefits from using mixed precision computations in the ELPA-AEO and ESSEX-II eigensolver projects

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