HPC libraries for solving dense symmetric eigenvalue problems

Research topic and goals

Many applications for example in Density Functional Theory (DFT) used in physics, chemistry, and materials science have to compute eigenvalues and eigenvectors of dense symmetric matrices. There are several libraries available for that task in high performance computing: ScaLAPACK, EigenExa, ELPA, and Elemental. On the other hand there exist different supercomputers like K-computer, BlueGene/Q, and Intel clusters with XeonPhi accelerators or with GPU accelerators. The eigensolver routines from the different libraries behave differently on the different computer architectures. In the first step, we evaluate the several aspects of the routines on the existing production and prototype supercomputers available to give the users recommendations which routine to use for their specific task. At the moment, evaluation is planned with respect to performance, accuracy and reproducibility in hybrid parallel execution. In the next step, we adapt the routines to the architectures.

We expect to provide the JLESC community with an overview of pros and cons of existing software, and a perspective view of better adaptation to present and planned computers and accelerators.

For the goal of the project, we want to publish results of the evaluations so that users can choose the best software for their computer. In the next step tuning of the existing software should be done and the new results are made available to the community.

Results for 2015/2016

This project was submitted in August 2015, and started from January 2016, officially. The porting process has already started for couple of eigensolver libraries and on couple of systems. We selected three libraries, ELPA, Elemental and EigenExa, and available resources, the K computer, JUQUEEN, and JURECA.

The present status is summarized in the following table. We confirmed the libraries are ported and perform on the systems; Hybrid means MPI/OpenMP hybrid parallel mode, Pure MPI means only MPI parallel mode.

HW Platform ELPA (2015.11) Elemental (0.85) EigenExa (2.4a)
K Hybrid 1 NA 2 Hybrid
JUQUEEN Hybrid Pure MPI 3 Hybrid
JURECA Hybrid Hybrid Hybrid

Results for 2016/2017

  1. We have done benchmarks using EigenExa, ELPA, and Elemental on available platforms, K, JUQUEEN and JURECA.
  2. Some of the combinations are not available due to compilation problems or runtime software. We will also continue the compilation if we get a new version or a new compiler environment.
  3. On the latter half of 2016/2017, we analyzed the correlation between the number of expected eigenspectrum and computing time cost. It affects to select best dense eigensolver for application users.
    • EigenExa very good if the full eigenspectrum is wanted
    • Libraries have to be tuned for the architecture
    • Best performance with the library tuned for the machine
    • On JUQUEEN and JURECA hybrid parallelization with moderate OpenMP part preferred
    • On K computer EigenExa best with as little MPI parallelization as necessary
    • On K computer ELPA2 best with pure MPI parallelization
    • If only 5 percent of eigenspectrum needed even on K computer ELPA2 pure MPI becomes fastest
  4. GPU or accelerator-based eigenvalue solver like MAGMA or ELPA must be examined in 2017/2018. Currently, we have not yet confirmed to build the ELPA GPU extension on JURECA. Also, we would enhance the EigenExa library with an acceleration of GPUs.

Software update and descriptions:

  • EigenExa : 2.4a1_rewind (officially 2.4 has been released) supports calculation of a part of eigenspectrum like ELPA
  • ELPA : 2016.005.003 includes new QR decomposition based block householder transformation.
  • Elemental: 0.87 allows to compute arbitrary part of the eigenspectrum

Visits and meetings

In the 4th JLESC meeting at Bonn, we had a pre-meeting of this project with regard to a research update and planning in 2016 of each member.

Frequent e-mail exchanges between Toshiyuki Imamura and Inge Gutheil. In the 5th JLESC meeting at Lyon, both met and discussed about this project. Also, 6th JLESC meeting was hosted by AICS and Inge Gutheil visited AICS.

Impact and publications

We have no publication yet.

    Future plans


      1. On the K computer, ELPA2 performs in a hybrid parallel fashion, but we confirmed that ELPA1 performs in the pure MPI mode.

      2. Elemental needs C++10, but C++10 functionality is not supported on the current compiler on the K computer.

      3. Running in the pure MPI mode is confirmed on JUQUEEN and JURECA at the moment.