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## Challenges and Advances in Solving Eigenproblems for Electronic-Structure Theory

#### CECAM-HQ-EPFL, Lausanne, Switzerland

#### Organisers

Numerically solving (generalized) eigenvalue problems is one the fundamental tasks in computational science with broad applications in quantum mechanics [1]** **and machine learning** **(e.g., principal component analysis) [2]. In particular, eigenvalue solvers play a fundamental role in electronic-structure theory at the density-functional theory (DFT) level, in which this step constitutes the dominant computational bottleneck [3]. Besides critically limiting the time-to-solution (code efficiency), the cubic scaling of traditional eigenvalue solvers with problem size limits which physical systems can be addressed from first-principles codes within reasonable computational effort at all. To tackle this challenge, different strategies (iterative solvers, multi-step solvers, etc.) are suitable for different DFT codes, e.g. for those using a plane wave expansion or for those using local orbitals. To date, many electronic-structure software packages, e.g., CP2K, FHI-aims, GPAW, NWChem, Quantum ESPRESSO, and Siesta, utilize external, highly optimized eigensolver libraries for the solution of eigenproblems, e.g., EigenExa** **and ELPA [4]. These open-source libraries were developed over the last decade with the specific goal to accelerate and enable novel first-principles calculations by leveraging synergetic collaborations between mathematicians, High-Performance-Computing (HPC) experts, and electronic-structure code developers. With that, these libraries often serve as lighthouse examples of how software engineering and science should work hand-in-hand to enable more efficient numerical simulations.

Currently, significant new challenges have appeared on the horizon with the upcoming deployment of (pre-)Exascale HPC systems featuring heterogeneous architectures with hundreds of cores per nodes as well as multi-purpose accelerators such as GPUs and Field Programmable Arrays (FPGAs). Harnessing the computational power available on these resources requires significant efforts to port existing algorithms to novel architectures and to design new algorithms tailored to these architectures [5-7]. In this context, novel algorithms [8, 9] with lower scaling exponent (but larger prefactor) seem to be particularly promising for large-scale calculations. Tackling these challenges necessitates an even tighter collaboration between mathematicians, HPC experts, first-principles code developers and even hardware and software vendors. For instance, achieving optimal “energy-to-solution” requires not only novel algorithms, but also (i) a strong coupling between the eigensolver and the electronic-structure code (which sets the actual requirements, e.g., the desired numerical precision [7,10]) and (ii) a strong coupling between the eigensolver and vendor-specific libraries in order to monitor the numerical efficiency (of both CPUs and accelerators) as well the actual energy consumption. In recent years, first steps to tackle these challenges have already been undertaken, e.g., by adding support for GPU accelerators [11] or by allowing for calculations at different levels of precision** **[8]. Similarly, the development of generalized libraries, e.g ELSI [4] and Eigenkernel [12], which give access to different eigensolvers in a unified fashion, has facilitated and accelerated the integration in first-principles codes.

Bringing eigensolvers and electronic-structure codes to exascale architectures requires further, considerable community efforts in the coming years. To facilitate and coordinate these efforts, this workshop aims at bringing together the key players from different fields, as detailed in the Sec. Structure.

## References

**Germany**

Christian Carbogno (Fritz-Haber-Institut der Max-Planck-Gesellschaft) - Organiser

Andreas Marek (Max Planck Computing and Data Facility) - Organiser

**United States**

Volker Blum (Duke University) - Organiser