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Parallel Numerical Linear Algebra for Future Extreme-Scale Systems

Deliverables

Final prototype software for different versions of the BLAS

Final implementation and evaluation of different versions of the BLAS based on the final specification from WP7.

Software for hybrid methods

Software for partitioning saddle point problems and overdetermined systems; improved block Cimmino methods incorporating new solvers from other deliverables. Includes extensive testing, documentation and benchmarking.

Software integration

Integration of the NLAFET library in the respective application environments.

Novel SVD algorithms

Prototypes for the standard SVD algorithm, the symmetric eigenvalue problem and the QDWH-based SVD algorithm.

Software for highly unsymmetric factorizations

Implementation of proposed methods from D3.4 on top of common task framework. Includes extensive testing, documentation and benchmarking.

Prototypes for tiled one-sided factorizations with algorithm-based fault tolerance

Prototype software for the Cholesky, LU, and QR factorizations with algorithm-based fault tolerance described in D6.6.

Prototype software for eigenvalue problem solvers

Prototypes for reduction to non-symmetric condensed forms (Hessenberg and Hessenberg-triangular), for the symmetric eigenvalue problem and the non-symmetric eigenvalue problems.

Prototype software, phase 2

Prototypes for Krylov-based iterative methods, and multilevel preconditioners.

Bidiagonal factorization

Prototypes for tw-sided bidiagonal factorization.

Prototype software, phase 1

Prototype software for sparse matrix-matrix multiplication and sparse low-rank matrix- approximation.

Prototype software for different versions of the BLAS

Implementation and evaluation of different versions of the BLAS based on the draft specification from WP7.

Software for symmetrically structured factorizations

Implementation and adaption of methods from D3.2 on top of the common task framework. Extension from symmetric to unsymmetric (but symmetrically structured) case. Includes extensive testing, documentation and benchmarking.

Prototypes for runtime systems exhibiting novel types of scheduling

Prototype for runtime systems capable of scheduling at a varying level of granularity/abstraction (i.e., basic kernels, BLAS, LAPACK, etc, and runtime systems that can execute the tasks along the critical path with an adaptive level of parallelism.

Algorithm design for symmetrically structured factorizations

Report on algorithm design and approaches to address issues around use of DAGs for sparse factorizations. Includes reporting on prototype code testing possible solutions.

Scalability and tunability of factorization algorithms

Report on scalability and tunability of the software implementing novel factorisation algorithms.

One-sided matrix factorizations

Report on tile algorithms and new experimental algorithms for matrix factorizations (LU, Cholesky, symmetric indefinite and QR). Includes reporting on and documentation of prototype code developed.

Analysis and algorithm design

Report on novel Krylov methods and multilevel preconditioners, focusing on numerical efficiency and theoretical properties.

Eigenvalue solvers for nonsymmetric problems

Evaluation of new eigenvalue solvers for the non-symmetric eigenvalue problem, including Krylov methods.

Draft specification for Hybrid BLAS

Requirements and draft specification for a common set of high performance linear algebra kernels on hybrid systems: Hybrid BLAS.

Evaluation of software prototypes

Report evaluating the prototype software developed in task 6.1 with regard to overall system performance on a selection of linear algebra algorithms.

Integration

Report on the integration of preconditioners and iterative methods from D4.3 into the NLAFET library. Evaluation of the parallel efficiency of the new preconditioned iterative solvers.

Algorithm design for highly unsymmetric factorizations

Report on experimental algorithms for parallel Markowitz ordering, analyzing various approaches and highlighting issues arising. Includes reporting on prototype code testing possible algorithms and solutions.

Evaluation of auto-tuning techniques

Report on the effect of applying the novel scheduling and auto-tuning prototypes to various linear algebra problems.

Novel methods for static and dynamic scheduling

Evaluation of existing and novel methods for static and dynamic scheduling in various types of HPC systems. Includes documentation of algorithms and prototypes developed.

Requirements analysis

Report describing the outcome of the requirements analysis for all applications.

First dissemination report

Dissemination report for the first reporting period, M1-M18.

Dissemination and community outreach plan

Plan for how to disseminate new results and interact with the broader community.

An off-line auto-tuning framework based on heuristic search

Review of techniques for pruning the search space in the context of auto-tuning, resulting in prototypes for an offline auto-tuning framework based on heuristic search. Includes reporting on optimal circumstances for switching scheduling approaches at run-time.

Second dissemination report

Final dissemination report, covering M19-M36.

Algorithm design for hybrid methods

Report on partitioning techniques and performance bottlenecks for hybrid methods including block Cimmino. Analysis of methods for saddle-point and overdetermined systems.

Eigenvalue problem solvers

Report on computation of eigenvectors and reordering of eigenvalues in Schur and generalized Schur forms. Includes evaluation of the scalability and tunability of the prototype software developed.

Theoretical bounds for communication in sparse operations

Report on theoretical lower bounds for key sparse matrix operations such as matrix-matrix multiplication and factorization.

Performance evaluation

Evaluation of the communication complexity and parallel performance of the prototypes from D4.3.

Final Hybrid BLAS specification

Revised versions of the BLAS specification based on collaborations with academic institutions and hardware vendors.

Algorithm-based fault tolerance techniques

Report on algorithm-based fault tolerance applied to the tiled Cholesky, LU, and/or QR factorizations.

Validation and evaluation

Evaluation of the NLAFET library in the context of the applications, leading to validation of the library and recommendations for future improvements.

Collaborative infrastructure

Project website (will contain both public parts, and parts that are restricted to the consortium), public and private source code repositories, bug tracking system, online forum, coding style, and developer guidelines.

Beta release of the NLAFET library

Beta release of parts of the NLAFET library and User's Guide.

Release of the NLAFET library

First complete release of the NLAFET library and associated User's Guide.

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Publications

A synchronization-free algorithm for parallel sparse triangular solves.

Author(s): W. Liu , A. Li , J. Hogg, I. Duff, B Vinter
Published in: Proceedings of Euro-Par 2016, Springer Lecture Notes, Issue 9833, 2016, Page(s) 617-630
DOI: 10.1007/978-3-319-43659-3_45

Creating a Standardised Set of Batched BLAS Routines

Author(s): Jack Dongarra and Sven Hammarling and Nicholas J. Higham and Samuel D. Relton and Pedro Valero-Lara and Mawussi Zounon
Published in: Proceedings of the Fourth Workshop on Sustainable Software for Science: Practice and Experiences (WSSSPE4, 2016), 2016

A 3D Parallel Algorithm for QR Decomposition

Author(s): Grey Ballard, James Demmel, Laura Grigori, Mathias Jacquelin, Nicholas Knight
Published in: Proceedings of the 30th on Symposium on Parallelism in Algorithms and Architectures - SPAA '18, 2018, Page(s) 55-65
DOI: 10.1145/3210377.3210415

An Auto-Tuning Framework for a NUMA-Aware Hessenberg Reduction Algorithm

Author(s): Mahmoud Eljammaly, Lars Karlsson, Bo Kågström
Published in: Companion of the 2018 ACM/SPEC International Conference on Performance Engineering - ICPE '18, 2018, Page(s) 5-8
DOI: 10.1145/3185768.3186304

Harnessing GPU’s Tensor Cores for Fast FP16 Arithmetic to Speedup Mixed-Precision Iterative Refinement Solvers

Author(s): Azzam Haidar, Stanimire Tomov, Jack Dongarra, Nick Higham
Published in: Proceedings of the International Conference for High Performance Computing, Networking, Storage, and Analysis, 2019

Distributed One-Stage Hessenberg Triangular Reduction with Wavefront Scheduling

Author(s): B. Adlerborn, L. Karlsson, and B. Kågström
Published in: NLAFET Working Papers, Issue 1, 2016

PDHGEQZ User Guide

Author(s): B. Adlerborn, B. Kågström, D. Kressner
Published in: NLAFET Working Notes, Issue 2, 2016

A Proposed API for Batched Basic Linear Algebra Subprograms

Author(s): J. Dongarra, I. Duff, J Hogg, M. Gates, A. Haidar, S. Hammarling, N. J. Higham, P.V.-Lara, S.D. Relton, S. Tomov, M. Zounon
Published in: 2016, ISSN 1749-9097

Experiments with sparse Cholesky using runtime systems

Author(s): I. Duff, J. Hogg, and F. Lopez
Published in: NLAFET Working Notes, Issue 7, 2016

Workshop on Batched, Reproducible, and Reduced Precision BLAS

Author(s): S. Hammarling
Published in: NLAFET Working Notes, Issue 4, 2016

Low rank approximation of a sparse matrix based on LU factorization with column and row tournament pivoting

Author(s): L. Grigori, S. Cayrols, and J. Demmel
Published in: NLAFET Working Notes, Issue 3, 2016

A Comparison of Potential Interfaces for Batched BLAS Computations

Author(s): S.D. Relton, P. Valero-Lara, and M. Zounon
Published in: NLAFET Working Notes, Issue 5, 2016

A new sparse solver using a posteriori threshold pivoting

Author(s): Jonathan Hogg
Published in: NLAFET Working Notes, Issue 6, 2016

Linear-time CUR approximation of BEM matrices

Author(s): Ayala, Alan; Claeys, Xavier; Grigori, Laura
Published in: https://hal.inria.fr/hal-01893036, Issue 1, 2018

A multilevel algebraic error estimator and the corresponding iterative solver with $p$-robust behavior

Author(s): Miraçi, Ani; Papež, Jan; Vohralík, Martin
Published in: https://hal.archives-ouvertes.fr/hal-02070981, Issue 1, 2019

Scalable Linear Solvers based on Enlarged Krylov subspaces with Dynamic Reduction of Search Directions

Author(s): Laura Grigori, Olivier Tissot
Published in: 2018

Reducing the communication and computational costs of enlarged Krylov subspaces conjugate gradient

Author(s): Laura Grigori, Olivier Tissot
Published in: 2017

A Multilevel Schwarz Preconditioner Based on a Hierarchy of Robust Coarse Spaces

Author(s): Hussam Al Daas, Laura Grigori, Pierre Jolivet, Pierre-Henri Tournier
Published in: 2019

Experiments with sparse Cholesky using a parametrized task graph implementation.

Author(s): Iain Duff and Florent Lopez
Published in: NLAFET Working Notes, Issue 14, 2017

PLASMA 17 Functionality Report: Parallel BLAS and Norms, Linear Systems and Least Squares, Mixed Precision and Matrix Inversion

Author(s): Maksims Abalenkovs, Jack Dongarra, Mark Gates, Azzam Haidar, Jakub Kurzak, Piotr Luszczek, Mawussi Zounon, Samuel Relton, Jakub Sistek, David Stevens, Ichitaro Yamazaki, Asim YarKhan
Published in: NLAFET Working Notes, Issue 15, 2017

Solving linear equations with messenger-field and conjugate gradients techniques – an application to CMB data analysis

Author(s): Jan Papez, Laura Grigori, Radoslav Stompor
Published in: NLAFET Working Notes, Issue 19, 2018

Sparse direct solution on parallel computers

Author(s): Iain Duff, Florent Lopez and Stojce Nakov
Published in: NLAFET Working Notes, Issue 17, 2017

Parallelization of the solve phase in a task-based Cholesky solver using a sequential task flow model

Author(s): Sébastien Cayrols, Iain Duff and Florent Lopez
Published in: NLAFET Working Notes, Issue 20, 2018

PLASMA 17 Performance Report. Linear Systems and Least Squares; Haswell, Knights Landing, POWER8.

Author(s): Maksims Abalenkovs, Negin Bagherpour, Jack Dongarra, Mark Gates, Azzam Haidar, Jakub Kurzak, Piotr Luszczek, Samuel Relton, Jakub Sistek, David Stevens, PanruoWu, Ichitaro Yamazaki, Asim YarKhan, Mawussi Zounon
Published in: NLAFET Working Notes, Issue 16, 2017

An Auto-Tuning Framework for a NUMA-Aware Hessenberg Reduction Algorithm

Author(s): Mahmoud Eljammaly, Lars Karlsson and Bo Kågström
Published in: NLAFET Working Notes, Issue 18, 2017

A new sparse symmetric indefinite solver using A Posteriori Threshold Pivoting

Author(s): Iain Duff, Jonathan Hogg and Florent Lopez
Published in: NLAFET Working Notes, Issue 21, 2018

Design and implementation of a parallel Markowitz threshold algorithm

Author(s): Timothy Davis, Iain Duff, Stojce Nakov
Published in: NLAFET Working Notes, Issue 22, 2019

Reducing the communication and computational costs of Enlarged Krylov subspaces Conjugate Gradient

Author(s): L. Grigori and O. Tissot
Published in: NLAFET Working Note, Issue 13, 2017

Evaluation of the Tunability of a New NUMA-Aware Hessenberg Reduction Algorithm

Author(s): M. Eljammaly, L. Karlsson, and B. Kågström
Published in: NLAFET Working Notes, Issue 8, 2017

Second Workshop on Batched, Reproducible, and Reduced Precision BLAS.

Author(s): Sven Hammarling
Published in: NLAFET Working Notes, Issue 12, 2017

Towards Highly Parallel and Compute-Bound Computation of Eigenvectors of Matrices in Schur Form

Author(s): B. Adlerborn, C.C. Kjelgaard Mikkelsen, L. Karlsson, and B. Kågström
Published in: NLAFET Working Notes, Issue 10, 2017

Robust solution of triangular linear systems

Author(s): C.C. Kjelgaard Mikkelsen and L. Karlsson
Published in: NLAFET Working Notes, Issue 9, 2017

Task-Based Parallel Algorithms for Reordering of Matrices in Real Schur Form

Author(s): M. Myllykoski, C.C. Kjelgaard Mikkelsen, L. Karlsson, and B. Kågström.
Published in: NLAFET Working Notes, Issue 11, 2017

Solving linear equations with messenger-field and conjugate gradient techniques: An application to CMB data analysis

Author(s): J. Papež, L. Grigori, R. Stompor
Published in: Astronomy & Astrophysics, Issue 620, 2018, Page(s) A59, ISSN 0004-6361
DOI: 10.1051/0004-6361/201832987

The Design and Performance of Batched BLAS on Modern High-Performance Computing Systems

Author(s): Jack Dongarra, Sven Hammarling, Nicholas J. Higham, Samuel D. Relton, Pedro Valero-Lara, Mawussi Zounon
Published in: Procedia Computer Science, Issue 108, 2017, Page(s) 495-504, ISSN 1877-0509
DOI: 10.1016/j.procs.2017.05.138

A Householder-Based Algorithm for Hessenberg-Triangular Reduction

Author(s): Zvonimir Bujanović, Lars Karlsson, Daniel Kressner
Published in: SIAM Journal on Matrix Analysis and Applications, Issue 39/3, 2018, Page(s) 1270-1294, ISSN 0895-4798
DOI: 10.1137/17m1153637

Distributed One-Stage Hessenberg-Triangular Reduction with Wavefront Scheduling

Author(s): Björn Adlerborn, Lars Karlsson, Bo Kågström
Published in: SIAM Journal on Scientific Computing, Issue 40/2, 2018, Page(s) C157-C180, ISSN 1064-8275
DOI: 10.1137/16m1103890

ALORA: Affine Low-Rank Approximations

Author(s): Alan Ayala, Xavier Claeys, Laura Grigori
Published in: Journal of Scientific Computing, Issue 79/2, 2019, Page(s) 1135-1160, ISSN 0885-7474
DOI: 10.1007/s10915-018-0885-5

PLASMA

Author(s): Jack Dongarra, Negin Bagherpour, Sven Hammarling, Jakub Šístek, David Stevens, Mawussi Zounon, Samuel D. Relton, Mark Gates, Azzam Haidar, Jakub Kurzak, Piotr Luszczek, Panruo Wu, Ichitaro Yamazaki, Asim Yarkhan, Maksims Abalenkovs
Published in: ACM Transactions on Mathematical Software, Issue 45/2, 2019, Page(s) 1-35, ISSN 0098-3500
DOI: 10.1145/3264491

Experiments with sparse Cholesky using a sequential task-flow implementation

Author(s): Iain Duff, Jonathan Hogg, Florent Lopez
Published in: Numerical Algebra, Control & Optimization, Issue 8/2, 2018, Page(s) 237-260, ISSN 2155-3297
DOI: 10.3934/naco.2018014

A Guide for Achieving High Performance with Very Small Matrices on GPU: A Case Study of Batched LU and Cholesky Factorizations

Author(s): Azzam Haidar, Ahmad Abdelfattah, Mawussi Zounon, Stanimire Tomov, Jack Dongarra
Published in: IEEE Transactions on Parallel and Distributed Systems, Issue 29/5, 2018, Page(s) 973-984, ISSN 1045-9219
DOI: 10.1109/tpds.2017.2783929

Low Rank Approximation of a Sparse Matrix Based on LU Factorization with Column and Row Tournament Pivoting

Author(s): Laura Grigori, Sebastien Cayrols, James W. Demmel
Published in: SIAM Journal on Scientific Computing, Issue 40/2, 2018, Page(s) C181-C209, ISSN 1064-8275
DOI: 10.1137/16m1074527

Parallel robust solution of triangular linear systems

Author(s): Carl Christian Kjelgaard Mikkelsen, Angelika Beatrix Schwarz, Lars Karlsson
Published in: Concurrency and Computation: Practice and Experience, 2017, Page(s) e5064, ISSN 1532-0626
DOI: 10.1002/cpe.5064

Fast synchronization-free algorithms for parallel sparse triangular solves with multiple right-hand sides

Author(s): Weifeng Liu, Ang Li, Jonathan D. Hogg, Iain S. Duff, Brian Vinter
Published in: Concurrency and Computation: Practice and Experience, Issue 29/21, 2017, Page(s) e4244, ISSN 1532-0626
DOI: 10.1002/cpe.4244

On solving separable block tridiagonal linear systems using a GPU implementation of radix-4 PSCR method

Author(s): M. Myllykoski, T. Rossi, J. Toivanen
Published in: Journal of Parallel and Distributed Computing, Issue 115, 2018, Page(s) 56-66, ISSN 0743-7315
DOI: 10.1016/j.jpdc.2018.01.004

Scalable eigenvector computation for the non-symmetric eigenvalue problem

Author(s): Angelika Schwarz, Lars Karlsson
Published in: Parallel Computing, Issue 85, 2019, Page(s) 131-140, ISSN 0167-8191
DOI: 10.1016/j.parco.2019.04.001

Symmetric Indefinite Linear Solver Using OpenMP Task on Multicore Architectures

Author(s): Ichitaro Yamazaki, Jakub Kurzak, Panruo Wu, Mawussi Zounon, Jack Dongarra
Published in: IEEE Transactions on Parallel and Distributed Systems, Issue 29/8, 2018, Page(s) 1879-1892, ISSN 1045-9219
DOI: 10.1109/tpds.2018.2808964

Optimized Batched Linear Algebra for Modern Architectures

Author(s): Jack Dongarra, Sven Hammarling, Nicholas J. Higham, Samuel D. Relton, Mawussi Zounon
Published in: Euro-Par 2017: Parallel Processing, Issue 10417, 2017, Page(s) 511-522
DOI: 10.1007/978-3-319-64203-1_37

Sparse Direct Solution on Parallel Computers

Author(s): Iain Duff, Florent Lopez, Stojce Nakov
Published in: Numerical Analysis and Optimization, Issue 235, 2018, Page(s) 67-98
DOI: 10.1007/978-3-319-90026-1_4

On the Tunability of a New Hessenberg Reduction Algorithm Using Parallel Cache Assignment

Author(s): Mahmoud Eljammaly, Lars Karlsson, Bo Kågström
Published in: Parallel Processing and Applied Mathematics, Issue 10777, 2018, Page(s) 579-589
DOI: 10.1007/978-3-319-78024-5_50

Experiments with Sparse Cholesky Using a Parametrized Task Graph Implementation

Author(s): Iain Duff, Florent Lopez
Published in: Parallel Processing and Applied Mathematics, Issue 10777, 2018, Page(s) 197-206
DOI: 10.1007/978-3-319-78024-5_18

The Design of Fast and Energy-Efficient Linear Solvers: On the Potential of Half-Precision Arithmetic and Iterative Refinement Techniques

Author(s): Azzam Haidar, Ahmad Abdelfattah, Mawussi Zounon, Panruo Wu, Srikara Pranesh, Stanimire Tomov, Jack Dongarra
Published in: Computational Science – ICCS 2018, Issue 10860, 2018, Page(s) 586-600
DOI: 10.1007/978-3-319-93698-7_45

Blocked Algorithms for Robust Solution of Triangular Linear Systems

Author(s): Carl Christian Kjelgaard Mikkelsen, Lars Karlsson
Published in: Parallel Processing and Applied Mathematics, Issue 10777, 2018, Page(s) 68-78
DOI: 10.1007/978-3-319-78024-5_7

A Task-Based Algorithm for Reordering the Eigenvalues of a Matrix in Real Schur Form

Author(s): Mirko Myllykoski
Published in: Parallel Processing and Applied Mathematics, Issue 10777, 2018, Page(s) 207-216
DOI: 10.1007/978-3-319-78024-5_19