Research projects "ESSEX" and "ESSEX II" - "Equipping Sparse Solvers for Exascale"

Researchers

Martin Galgon
Sarah Huber
Lukas Krämer
Bruno Lang

Duration and funding

January 2013 to December 2015 (ESSEX),
since January 2016 (ESSEX II).

From Jan 2013 to Dec 2018 this research was funded by Deutsche Forschungsgemeinschaft within the German Priority Programme 1648 "Software for Exascale Computing".

Description

The aim of the ESSEX project is to develop programming concepts and numerical methods for the solution of large scale sparse eigenvalue problems. The choice of the considered algorithms (left picture below) is driven by applications from physics, and the exascale challenges of extreme parallelism, energy efficiency, and resiliency are addressed by coherent software design between the Building Blocks, Algorithms, and Applications layers (right picture). The first ESSEX project was a joint research effort of groups at the Universities of Erlangen-Nuremberg, Greifswald and Wuppertal, and DLR Köln. The ESSEX II team also includes groups at the Universities of Tsukuba and Tokyo in Japan.

The work of the Wuppertal group originally was based on the FEAST algorithm for computing a bulk of eigenvalues in the interior of the spectrum. This algorithm has been introduced in [E. Polizzi: Density-matrix-based algorithm for solving eigenvalue problems. Phys. Rev. B 79:115112 (2009)]. In each iteration of the algorithm, a contour integral is used to approximately project the current "search space" onto the desired eigenspace, and a Rayleigh-Ritz type method is used to extract eigenvalue and eigenvector approximations from the resulting subspace.

Our research so far included work on

the robustness of the "core" FEAST algorithm, in particular
- reliable counts for the eigenvalues in the search interval,
- criteria for detecting convergence and mechanisms for locking,
- control of the searchspace dimension, and
- the quality of the projector associated with the method,
a framework providing efficient variants for applying the projector, in particular
- efficient solvers for the linear systems arising in the contour-based approach (jointly with the DLR group),
- polynomial-based approaches for realizing the projector,
- Sakurai-Sugiura-type (moment-based) projectors,
orthogonality issues when eigenpairs are computed by subsets to make use of an additional level of parallelism.
adaptivity in choosing the type and parameters of the projectors in order to achieve minimum overall effort,
the use of mixed precision for saving energy and/or time and for rubustness reasons, and
resilience issues (techniques for detecting and handling errors that become more frequent on extreme-scale systems).

We have provided a scalable parallel implementation based on the highly optimized building blocks that are developed within the ESSEX project, and applied it to research problems from quantum physics.

The research on projection-based algorithms is continued. Currently our investigations focus on

the placement of the poles in the rational projectors and
the software infrastructure.

Project-related publications

[1]	Sarah Huber, Yasunori Futamura, Martin Galgon, Akira Imakura, Bruno Lang, and Tetsuya Sakurai. Flexible subspace iteration with moments for an effective contour integration-based eigensolver. Numer. Linear Algebra Appl., 29:e2447:1--15, 2022. [ DOI \| Abstract ]
[2]	Christie L. Alappat, Andreas Alvermann, Achim Basermann, Holger Fehske, Yasunori Futamura, Martin Galgon, Georg Hager, Sarah Huber, Akira Imakura, Masatoshi Kawai, Moritz Kreutzer, Bruno Lang, Kengo Nakajima, Melven Röhrig-Zöllner, Tetsuya Saturai, Faisal Shahzad, Jonas Thies, and Gerhard Wellein. ESSEX: Equipping sparse solvers for exascale. In Hans-Joachim Bungartz, Severin Reiz, Benjamin Uekermann, Philipp Neumann, and Wolfgang E. Nagel, editors, Software for Exascale Computing -- SPPEXA 2016--2019, volume 136 of LNCSE, pages 143--187. Springer Nature Switzerland AG, Cham, 2020. [ DOI \| Abstract ]
[3]	Andreas Alvermann, Achim Basermann, Hans-Joachim Bungartz, Christian Carbogno, Dominik Ernst, Holger Fehske, Yasunori Futamura, Martin Galgon, Georg Hager, Sarah Huber, Thomas Huckle, Akihiro Ida, Akira Imakura, Masatoshi Kawai, Simone Köcher, Moritz Kreutzer, Pavel Kus, Bruno Lang, Hermann Lederer, Valeriy Manin, Andreas Marek, Kengo Nakajima, Lydia Nemec, Karsten Reuter, Michael Rippl, Melven Röhrig-Zöllner, Tetsuya Sakurai, Matthias Scheffler, Christoph Scheurer, Faisal Shahzad, Danilo Simoes Brambila, Jonas Thies, and Gerhard Wellein. Benefits from using mixed precision computations in the ELPA-AEO and ESSEX-II eigensolver projects. Japan J. Indust. Appl. Math., 36(2):699--717, 2019. [ DOI \| Abstract ]
[4]	Martin Galgon, Sarah Huber, and Bruno Lang. Mixed precision in subspace iteration-based eigensolvers. Proc. Appl. Math. Mech., 18:e201800334, December 2018. [ DOI \| Abstract ]
[5]	Lukas Krämer and Bruno Lang. Convergence of integration-based methods for the solution of standard and generalized Hermitian eigenvalue problems. Electron. Trans. Numer. Anal., 48:183--201, 2018. [ DOI \| Abstract ]
[6]	Martin Galgon, Lukas Krämer, and Bruno Lang. Improving projection-based eigensolvers via adaptive techniques. Numer. Linear Algebra Appl., 25(1):e2124, January 2018. [ DOI \| Abstract ]
[7]	Martin Galgon, Lukas Krämer, Bruno Lang, Andreas Alvermann, Holger Fehske, Andreas Pieper, Georg Hager, Moritz Kreutzer, Faisal Shahzad, Gerhard Wellein, Achim Basermann, Melven Röhrig-Zöllner, and Jonas Thies. Improved coefficients for polynomial filtering in ESSEX. In Tetsuya Sakurai, Shao-Liang Zhang, Toshiyuki Imamura, Yusaku Yamamoto, Yoshinobu Kuramashi, and Takeo Hoshi, editors, Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing. Proc. EPASA 2015, Tsukuba, Japan, September 2015, volume 117 of LNCSE. Springer International Publishing, Cham, 2017. [ DOI \| Abstract ]
[8]	Moritz Kreutzer, Jonas Thies, Melven Röhrig-Zöllner, Andreas Pieper, Faisal Shahzad, Martin Galgon, Achim Basermann, Holger Fehske, Georg Hager, and Gerhard Wellein. GHOST: Building blocks for high performance sparse linear algebra on heterogeneous systems. Int. J. Parallel Prog., October 2016. [ http \| Abstract ]
[9]	Andreas Pieper, Moritz Kreutzer, Andreas Alvermann, Martin Galgon, Holger Fehske, Georg Hager, Bruno Lang, and Gerhard Wellein. High-performance implementation of Chebyshev filter diagonalization for interior eigenvalue computations. J. Comput. Phys., 325:226--243, 2016. [ DOI \| Abstract ]
[10]	Moritz Kreutzer, Jonas Thies, Andreas Pieper, Andreas Alvermann, Martin Galgon, Melven Röhrig-Zöllner, Faisal Shahzad, Achim Basermann, Alan R. Bishop, Holger Fehske, Georg Hager, Bruno Lang, and Gerhard Wellein. Performance engineering and energy efficiency of building blocks for large, sparse eigenvalue computations on heterogeneous supercomputers. In Hans-Joachim Bungartz, Philipp Neumann, and Wolfgang E. Nagel, editors, Software for Exascale Computing -- SPPEXA 2013--2015, volume 113 of LNCSE, pages 317--338. Springer, Switzerland, 2016. [ DOI \| Abstract ]
[11]	Jonas Thies, Martin Galgon, Faisal Shahzad, Andreas Alvermann, Moritz Kreutzer, Andreas Pieper, Melven Röhrig-Zöllner, Achim Basermann, Holger Fehske, Georg Hager, Bruno Lang, and Gerhard Wellein. Towards an exascale enabled sparse solver repository. In Hans-Joachim Bungartz, Philipp Neumann, and Wolfgang E. Nagel, editors, Software for Exascale Computing -- SPPEXA 2013--2015, volume 113 of LNCSE, pages 295--316. Springer, Switzerland, 2016. [ DOI \| Abstract ]
[12]	Martin Galgon, Lukas Krämer, Jonas Thies, Achim Basermann, and Bruno Lang. On the parallel iterative solution of linear systems arising in the FEAST algorithm for computing inner eigenvalues. Parallel Comput., 49:153--163, 2015. [ DOI \| Abstract ]
[13]	Martin Galgon, Lukas Krämer, Bruno Lang, Andreas Alvermann, Holger Fehske, and Andreas Pieper. Improving robustness of the FEAST algorithm and solving eigenvalue problems from graphene nanoribbons. Proc. Appl. Math. Mech., 14(1):821--822, December 2014. [ DOI \| Abstract ]
[14]	Andreas Alvermann, Achim Basermann, Holger Fehske, Martin Galgon, Georg Hager, Moritz Kreutzer, Lukas Krämer, Bruno Lang, Andreas Pieper, Melven Röhrig-Zöllner, Faisal Shahzad, Jonas Thies, and Gerhard Wellein. ESSEX: Equipping sparse solvers for exascale. In Luís Lopes et al., editors, Euro-Par 2014: Parallel Processing Workshops, volume 8806 of LNCS, pages 577--588. Springer, 2014. [ DOI \| Abstract ]

Bergische Universität Wuppertal Fakultät für Mathematik und Naturwissenschaften Angewandte Informatik - Algorithmik and IMACM

Research projects "ESSEX" and "ESSEX II" - "Equipping Sparse Solvers for Exascale"

Researchers

Duration and funding

Description

Project-related publications

See also

Bergische Universität Wuppertal
Fakultät für Mathematik und Naturwissenschaften
Angewandte Informatik - Algorithmik and IMACM