The ChASE library for large Hermitian eigenvalue problems
Research topics and goals
In this project we propose to focus on the modern implementation of one of the oldest subspace iteration methods: the Chebyshev Accelerated Subspace Eigensolver (ChASE), recently developed at JSC. ChASE is tailored to compute a fraction not exceeding 20% of the extremal spectrum of dense Hermitian eigenproblems.
Scaling the ChASE for large Hermitian eigenvalue problems
State of the art eigensolvers used in Computational Science and Engineering present evidence of widely varying performance and serious limitations when it comes to scalability over thousands of cores. Scalability is often an issue intrinsic to the algorithms granularity which determines the amount of parallelism available. Ultimately, though, it is the rigid structure of the libraries imposing limitations on the optimal use of the hierarchical structure of the caches that creates serious restrictions to the level of scalability these routines can achieve.
An appealing alternative to traditional algorithms for dense and sparse problems is a class of modern iterative eigensolvers based on subspace iteration with spectral projection. Thanks to their simple algorithmic structure and extended granularity, this class of algorithms are experiencing a revival as an alternative to traditional dense solvers. Moreover, at the subroutine level the internal tasks can be designed so as to use almost exclusively low level kernels from specialized libraries, such as BLAS, guaranteeing performance portability.
The objective of this topic is to port the ChASE library to the largest computing architectures currently available, generalize its interface so as to allow multiple parallelization schemes, and extend its applicability by supporting the solution of large Hermitian eigenvalue problems.
Optimizing ChASE for BetheSalpeter computations on multiGPUs
Accurate predictions for the interaction of light with matter are crucial, for instance, to design novel optoelectronic or photovoltaic materials and to understand their fundamental properties. To this end, atomistic firstprinciples methods are capable of computing absorption spectra or the complex frequencydependent dielectric function. More specifically, by solving the BetheSalpeter equation for the optical polarization function, computational materials scientists can take excitonic effects into account when simulating optical properties of real materials. However, for modern complex systems such as lanthanumaluminum oxide or indium oxide, this approach leads to large dense matrices with sizes reaching up to 400,000.
To compute experimentally relevant exciton binding energies, only a fraction of the spectrum, corresponding to the lowest eigenpairs of these matrices, are of interest (Fuchs et al. 2008). Due to its large computational cost, direct diagonalization using outofthebox libraries is not a workable option. In these instances iterative schemes provide a feasible alternative route, allowing to efficiently compute the lowest portion of the eigenspectrum by exploiting the massively parallel architecture of modern supercomputers.
In this project we propose to test and customize ChASE, so as to facilitate the computation of the desired lowest eigenpairs of large dense eigenproblems on hybrid architectures. The positive outcome of the project will allow to draw conclusions with implications for electronhole separation in solar cell absorbers and the overall optical properties of a material in the vicinity of the absorption onset. If this part of the project is successful, we envision exploring a foldedspectrum method to also access the highenergy range of the spectrum, which may become important to detect, for instance, Fano resonances.
Contributions
 Port and validate the robustness of the reference version of the ChASE library on Blue Waters;
 Extend the reference implementation of ChASE to distributed memory platforms by porting the new hybrid parallelization of the Chebyshev filter;
 Extend the hybrid parallelization to the RayleighRitz projection, QR factorization and residual computation;
 Testing the scalability of ChASE when solving extremely large Hermitian eigenvalue problems extracted from specific applications.
Results for 2015/2016
This project was discussed for the first time during the JLESC meeting in December of 2015 in Bonn. Since then, both PIs set up the necessary testing infrastructure which involved creating an account for A. Schleife on JURECA, a 23 GB Globus file transfer from Blue Waters to JURECA, and compiling the existing conjugategradient eigensolver (KSCG) based code on JURECA. The transferred files were used to extract a number of matrices of increasing size to be diagonalized outside the Jena BSE code by using the existing ChASE eigensolver for comparison with the KSCG solver. We completed these steps and made sure to check the correctness of the solutions for the relative matrices. Already at this stage the ChASE solver showed to outperform KSCG outofthebox despite operating in suboptimal conditions. These results were presented at the JLESC workshop in Lyon, France (2016).
Results for 2016/2017
In a second phase, the ChASE library underwent a restructuring of the parallelization scheme in order to accommodate for a new paradigm based on MPI+CUDA. The initial effort concentrated on redesigning only the Chebyshev filter, which is the most computationally intensive routine of the ChASE solver.The rewriting effort was successfully tested on the JURECA nodes hosting 2 x K40 NVIDIA GPU cards using the same set of matrices created in the first phase of the project. The tests, run over up to 64 computing nodes showed the potential for ChASE to scale over hundreds of computing nodes and making use of multiGPU cards per each node. The results were presented at the JLESC workshop in Kobe, Japan (2016).
In the following phase, working accounts were opened on Blue Waters for the JSC team members. The JSC and UIUC teams met for a week in order to start integrating the ChASE solver into the Jena BSE code. First the ChASE solver was templated to work both in single (SP) and double precision (DP). Numerical tests showed that the Jena BSE does not require eigenpairs with a high precision so that ChASE can just operate in SP. The second step of this integration was successful carried out on one computing node showing that, already at this stage, Jena BSE + ChASE is about five times faster than Jena BSE using the KSCG solver. In the next step we will fully integrate the MPI+CUDA version of ChASE and run tests on Blue Waters. Eventually we plan to demonstrate the potential of the new solver by tackling a physical system of unprecedented size which will require thousands of computing nodes.
Results for 2017/2018
In the third phase of this project, the ChASE library was completely rewritten in order to separate the actual ChASE algorithm from the numerical kernels. As a C++ program such a separation is accomplished via a pure abstract class which defines the interface for the C++ numerical kernels. The ChASE algorithm implementation then uses this interface and all parallel operations and handling of data is performed by the subclass of the pure abstract ChASE class. Such rewrite allowed the library to accommodate several different parallelization scheme from MPI+CUDA to pure MPI using the linear algebra kernels and data distribution from the Elemental library. The new ChASE library has been tested using several different eigenvalue problems, as well as on sequences of eigenvalue problems where it outperforms even the most advanced direct eigensolvers. These results and the library description will appear in a manuscript submission in early March 2018.
The new library is currently being tested over a new set of very large eigenproblems generated by the Jena BSE code. Such problems are described by matrices whose rank ranges from 300k up to 1000k. In this phase of the project we plan to solve these eigenproblems using as many computing nodes and GPU cards on Blue Waters as feasible for the size of the eigenproblem under scrutiny. The results of these tests will be presented in a publication, we plan to submit just before the JLESC meeting in Barcelona on April 1719.
Visits and meetings
 March 3 – 11 2017: Jan Winkelmann and Edoardo Di Napoli visited Andre Schleife in Urbana, Champaign. During the visit several important details were tackled, paving the way to a promising integration of the ChASE library into Jena BSE code.
Compute resource needs
 Test account on JURECA for Matrix generation in the first phase of the project
 Regular account on Blue Waters in the successive phases of the project.
Impact and publications
 One journal publication presenting the new ChASE library to be submitted in March 2018
 One conference publication presenting results related to the use of ChASE to solve very large eigenvalue problems generated by the BSE Jena code to be submitted in April 2018
Future plans
The next step is to fully integrate a hybrid parallelization of the ChASE library into the Jena BSE code and test scalability and efficiency of the ChASE library on Blue Waters when used to solve eigenproblems of increasing size and complexity. This plan include a full customization of ChASE parameters to adapt seamlessly to the Jena BSE requirements. One of the the major obstacles in this sense comes from the absence of a parallel I/O for the Jena code. This is an essential task that will have to be tackled before full integration can be achieved. We propose to achieve such result by modifying the Jena BSE code in such a way as to accommodate the HDF5 standard. Allocating resources to such task and possibly an enlargement of the collaboration effort to the developers of the HDF5 library are envisaged.
References

Fuchs, F., C. Rödl, A. Schleife, and F. Bechstedt. 2008. “Efficient \MathcalO(N^2) Approach to Solve the BetheSalpeter Equation For Excitonic Bound States.” Phys. Rev. B 78: 085103. doi:10.1103/PhysRevB.78.085103.
@article{FuchsEtAl2008, author = {Fuchs, F. and R\"odl, C. and Schleife, A. and Bechstedt, F.}, doi = {10.1103/PhysRevB.78.085103}, journal = {Phys. Rev. B}, numpages = {13}, pages = {085103}, title = {Efficient $\mathcal{O}({N}^{2})$ approach to solve the BetheSalpeter equation for excitonic bound states}, url = {http://link.aps.org/doi/10.1103/PhysRevB.78.085103}, volume = {78}, year = {2008} }