CP2K: An electronic structure and molecular dynamics software package - Quickstep: Efficient and accurate electronic structure calculations - Kühne, Thomas; Iannuzzi, Marcella; Ben, Mauro Del; Rybkin, Vladimir V.; Seewald, Patrick; Stein, Frederick; Laino, Teodoro; Khaliullin, Rustam Z.; Schütt, Ole; Schiffmann, Florian; Golze, Dorothea; Wilhelm, Jan; et al. (The Journal of Chemical Physics)
CP2K is an open source electronic structure and molecular dynamics software package to perform atomistic simulations of solid-state, liquid, molecular, and biological systems. It is especially aimed at massively parallel and linear-scaling electronic structure methods and state-of-theart ab initio molecular dynamics simulations. Excellent performance for electronic structure calculations is achieved using novel algorithms implemented for modern high-performance computing systems. This review revisits the main capabilities of CP2K to perform efficient and accurate electronic structure simulations. The emphasis is put on density functional theory and multiple post–Hartree–Fock methods using the Gaussian and plane wave approach and its augmented all-electron extension.

A Submatrix-Based Method for Approximate Matrix Function Evaluation in the Quantum Chemistry Code CP2K - Lass, Michael; Schade, Robert; Kühne, Thomas; Plessl, Christian (Proc. International Conference for High Performance Computing, Networking, Storage and Analysis (SC))
Electronic structure calculations based on density-functional theory (DFT) represent a significant part of today's HPC workloads and pose high demands on high-performance computing resources. To perform these quantum-mechanical DFT calculations on complex large-scale systems, so-called linear scaling methods instead of conventional cubic scaling methods are required. In this work, we take up the idea of the submatrix method and apply it to the DFT computations in the software package CP2K. For that purpose, we transform the underlying numeric operations on distributed, large, sparse matrices into computations on local, much smaller and nearly dense matrices. This allows us to exploit the full floating-point performance of modern CPUs and to make use of dedicated accelerator hardware, where performance has been limited by memory bandwidth before. We demonstrate both functionality and performance of our implementation and show how it can be accelerated with GPUs and FPGAs.

Accurate Sampling with Noisy Forces from Approximate Computing - Rengaraj, Varadarajan; Lass, Michael; Plessl, Christian; Kühne, Thomas (Computation)
In scientific computing, the acceleration of atomistic computer simulations by means of custom hardware is finding ever-growing application. A major limitation, however, is that the high efficiency in terms of performance and low power consumption entails the massive usage of low precision computing units. Here, based on the approximate computing paradigm, we present an algorithmic method to compensate for numerical inaccuracies due to low accuracy arithmetic operations rigorously, yet still obtaining exact expectation values using a properly modified Langevin-type equation.

OpenCL Implementation of Cannon's Matrix Multiplication Algorithm on Intel Stratix 10 FPGAs - Gorlani, Paolo; Kenter, Tobias; Plessl, Christian (Proceedings of the International Conference on Field-Programmable Technology (FPT))
Stratix 10 FPGA cards have a good potential for the acceleration of HPC workloads since the Stratix 10 product line introduces devices with a large number of DSP and memory blocks. The high level synthesis of OpenCL codes can play a fundamental role for FPGAs in HPC, because it allows to implement different designs with lower development effort compared to hand optimized HDL. However, Stratix 10 cards are still hard to fully exploit using the Intel FPGA SDK for OpenCL. The implementation of designs with thousands of concurrent arithmetic operations often suffers from place and route problems that limit the maximum frequency or entirely prevent a successful synthesis. In order to overcome these issues for the implementation of the matrix multiplication, we formulate Cannon's matrix multiplication algorithm with regard to its efficient synthesis within the FPGA logic. We obtain a two-level block algorithm, where the lower level sub-matrices are multiplied using our Cannon's algorithm implementation. Following this design approach with multiple compute units, we are able to get maximum frequencies close to and above 300 MHz with high utilization of DSP and memory blocks. This allows for performance results above 1 TeraFLOPS.

A General Algorithm to Calculate the Inverse Principal p-th Root of Symmetric Positive Definite Matrices - Richters, Dorothee; Lass, Michael; Walther, Andrea; Plessl, Christian; Kühne, Thomas (Communications in Computational Physics)
We address the general mathematical problem of computing the inverse p-th root of a given matrix in an efficient way. A new method to construct iteration functions that allow calculating arbitrary p-th roots and their inverses of symmetric positive definite matrices is presented. We show that the order of convergence is at least quadratic and that adaptively adjusting a parameter q always leads to an even faster convergence. In this way, a better performance than with previously known iteration schemes is achieved. The efficiency of the iterative functions is demonstrated for various matrices with different densities, condition numbers and spectral radii.

Using Approximate Computing for the Calculation of Inverse Matrix p-th Roots - Lass, Michael; Kühne, Thomas; Plessl, Christian (Embedded Systems Letters)
Approximate computing has shown to provide new ways to improve performance and power consumption of error-resilient applications. While many of these applications can be found in image processing, data classification or machine learning, we demonstrate its suitability to a problem from scientific computing. Utilizing the self-correcting behavior of iterative algorithms, we show that approximate computing can be applied to the calculation of inverse matrix p-th roots which are required in many applications in scientific computing. Results show great opportunities to reduce the computational effort and bandwidth required for the execution of the discussed algorithm, especially when targeting special accelerator hardware.

A Massively Parallel Algorithm for the Approximate Calculation of Inverse p-th Roots of Large Sparse Matrices - Lass, Michael; Mohr, Stephan; Wiebeler, Hendrik; Kühne, Thomas; Plessl, Christian (Proc. Platform for Advanced Scientific Computing (PASC) Conference)
We present the submatrix method, a highly parallelizable method for the approximate calculation of inverse p-th roots of large sparse symmetric matrices which are required in different scientific applications. Following the idea of Approximate Computing, we allow imprecision in the final result in order to utilize the sparsity of the input matrix and to allow massively parallel execution. For an n x n matrix, the proposed algorithm allows to distribute the calculations over n nodes with only little communication overhead. The result matrix exhibits the same sparsity pattern as the input matrix, allowing for efficient reuse of allocated data structures. We evaluate the algorithm with respect to the error that it introduces into calculated results, as well as its performance and scalability. We demonstrate that the error is relatively limited for well-conditioned matrices and that results are still valuable for error-resilient applications like preconditioning even for ill-conditioned matrices. We discuss the execution time and scaling of the algorithm on a theoretical level and present a distributed implementation of the algorithm using MPI and OpenMP. We demonstrate the scalability of this implementation by running it on a high-performance compute cluster comprised of 1024 CPU cores, showing a speedup of 665x compared to single-threaded execution.
Keywords: approximate computing; linear algebra; matrix inversion; matrix p-th roots; numeric algorithm; parallel computing

Flexible FPGA design for FDTD using OpenCL - Kenter, Tobias; Förstner, Jens; Plessl, Christian (Proc. Int. Conf. on Field Programmable Logic and Applications (FPL))
Compared to classical HDL designs, generating FPGA with high-level synthesis from an OpenCL specification promises easier exploration of different design alternatives and, through ready-to-use infrastructure and common abstractions for host and memory interfaces, easier portability between different FPGA families. In this work, we evaluate the extent of this promise. To this end, we present a parameterized FDTD implementation for photonic microcavity simulations. Our design can trade-off different forms of parallelism and works for two independent OpenCL-based FPGA design flows. Hence, we can target FPGAs from different vendors and different FPGA families. We describe how we used pre-processor macros to achieve this flexibility and to work around different shortcomings of the current tools. Choosing the right design configurations, we are able to present two extremely competitive solutions for very different FPGA targets, reaching up to 172 GFLOPS sustained performance. With the portability and flexibility demonstrated, code developers not only avoid vendor lock-in, but can even make best use of real trade-offs between different architectures.
Keywords: tet_topic_hpc

Microdisk Cavity FDTD Simulation on FPGA using OpenCL - Kenter, Tobias; Plessl, Christian (Proc. Workshop on Heterogeneous High-performance Reconfigurable Computing (H2RC))