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Journal of Chemical Theory and Computation

Accelerating Coupled-Cluster Calculations with GPUs: An Implementation of the Density-Fitted CCSD(T) Approach for Heterogeneous Computing Architectures Using OpenMP Directives

Dipayan Datta, Mark S. Gordon

Published Oct 25, 2023• 19 citations
Publisher

Abstract

An algorithm is presented for the coupled-cluster singles, doubles, and perturbative triples correction [CCSD(T)] method based on the density fitting or the resolution-of-the-identity (RI) approximation for performing calculations on heterogeneous computing platforms composed of multicore CPUs and graphics processing units (GPUs). The directive-based approach to GPU offloading offered by the OpenMP application programming interface has been employed to adapt the most compute-intensive terms in the RI-CCSD amplitude equations with computational costs scaling as O ( N O 2 N V 4 ), O ( N O 3 N V 3 ), and O ( N O 4 N V 2 ) (where N O and N V denote the numbers of correlated occupied and virtual orbitals, respectively) and the perturbative triples correction to execute on GPU architectures. The pertinent tensor contractions are performed using an accelerated math library such as cuBLAS or hipBLAS. Optimal strategies are discussed for splitting large data arrays into tiles to fit them into the relatively small memory space of the GPUs, while also minimizing the low-bandwidth CPU–GPU data transfers. The performance of the hybrid CPU–GPU RI-CCSD(T) code is demonstrated on pre-exascale supercomputers composed of heterogeneous nodes equipped with NVIDIA Tesla V100 and A100 GPUs and on the world’s first exascale supercomputer named “Frontier”, the nodes of which consist of AMD MI250X GPUs. Speedups within the range 4–8× relative to the recently reported CPU-only algorithm are obtained for the GPU-offloaded terms in the RI-CCSD amplitude equations. Applications to polycyclic aromatic hydrocarbons containing 16–66 carbon atoms demonstrate that the acceleration of the hybrid CPU–GPU code for the perturbative triples correction relative to the CPU-only code increases with the molecule size, attaining a speedup of 5.7× for the largest circumovalene molecule (C 66 H 20 ). The GPU-offloaded code enables the computation of the perturbative triples correction for the C 60 molecule using the cc-pVDZ/aug-cc-pVTZ-RI basis sets in 7 min on Frontier when using 12,288 AMD GPUs with a parallel efficiency of 83.1%.

DOI
10.1021/acs.jctc.3c00876

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An algorithm is presented for the coupled-cluster singles, doubles, and perturbative triples correction [CCSD(T)] method based on the density fitting or the resolution-of-the-identity (RI) approximation for performing calculations on heterogeneous computing platforms composed of multicore CPUs and graphics processing units (GPUs). The directive-based approach to GPU offloading offered by the OpenMP application programming interface has been employed to adapt the most compute-intensive terms in the RI-CCSD amplitude equations with computational costs scaling as O ( N O 2 N V 4 ), O ( N O 3 N V 3 ), and O ( N O 4 N V 2 ) (where N O and N V denote the numbers of correlated occupied and virtual orbitals, respectively) and the perturbative triples correction to execute on GPU architectures. The pertinent tensor contractions are performed using an accelerated math library such as cuBLAS or hipBLAS. Optimal strategies are discussed for splitting large data arrays into tiles to fit them into the relatively small memory space of the GPUs, while also minimizing the low-bandwidth CPU–GPU data transfers. The performance of the hybrid CPU–GPU RI-CCSD(T) code is demonstrated on pre-exascale supercomputers composed of heterogeneous nodes equipped with NVIDIA Tesla V100 and A100 GPUs and on the world’s first exascale supercomputer named “Frontier”, the nodes of which consist of AMD MI250X GPUs. Speedups within the range 4–8× relative to the recently reported CPU-only algorithm are obtained for the GPU-offloaded terms in the RI-CCSD amplitude equations. Applications to polycyclic aromatic hydrocarbons containing 16–66 carbon atoms demonstrate that the acceleration of the hybrid CPU–GPU code for the perturbative triples correction relative to the CPU-only code increases with the molecule size, attaining a speedup of 5.7× for the largest circumovalene molecule (C 66 H 20 ). The GPU-offloaded code enables the computation of the perturbative triples correction for the C 60 molecule using the cc-pVDZ/aug-cc-pVTZ-RI basis sets in 7 min on Frontier when using 12,288 AMD GPUs with a parallel efficiency of 83.1%.
Efficient scalable implementations are presented for two single-reference coupled-cluster (CC) methods with non-iterative corrections for triple excitations, which accurately describe single bond dissociations in molecules in contrast to the standard CCSD(T) approach. These methods are (a) the completely renormalized coupled-cluster theory, CR-CC(2,3), and (b) the ΛCCSD(T) approach, the triples energy corrections in both of which are formulated using the left eigenvector of the CC similarity transformed Hamiltonian. The density fitting approximation is employed for factorizing the two-electron repulsion integrals (ERIs), which eliminates memory bottlenecks associated with the storage of the four-index ERIs by allowing an integral-direct algorithm. Massively parallel algorithms have been developed for the above methods within the GAMESS quantum chemistry program using a hybrid MPI and OpenMP based parallelization model. All compute-intensive steps in the CCSD amplitude and the left eigenvector equations, as well as the triples corrections in both methods, have been adapted to execute on graphical processing units (GPUs) using the OpenMP target directives. Performance of these codes is demonstrated on petascale supercomputers equipped with NVIDIA A100 GPUs. While the CR-CC(2,3) and ΛCCSD(T) computations take roughly twice as much time as the standard CCSD(T) method, the GPU-offloaded codes are shown to impart 3.5-5× accelerations relative to the CPU-only parallel codes, thereby significantly reducing the time-to-solution for the CR-CC(2,3) and ΛCCSD(T) methods.
A parallel algorithm is described for the coupled-cluster singles and doubles method augmented with a perturbative correction for triple excitations [CCSD(T)] using the resolution-of-the-identity (RI) approximation for two-electron repulsion integrals (ERIs). The algorithm bypasses the storage of four-center ERIs by adopting an integral-direct strategy. The CCSD amplitude equations are given in a compact quasi-linear form by factorizing them in terms of amplitude-dressed three-center intermediates. A hybrid MPI/OpenMP parallelization scheme is employed, which uses the OpenMP-based shared memory model for intranode parallelization and the MPI-based distributed memory model for internode parallelization. Parallel efficiency has been optimized for all terms in the CCSD amplitude equations. Two different algorithms have been implemented for the rate-limiting terms in the CCSD amplitude equations that entail and -scaling computational costs, where N O and N V denote the number of correlated occupied and virtual orbitals, respectively. One of the algorithms assembles the four-center ERIs requiring N V 4 and N O 2 N V 2 -scaling memory costs in a distributed manner on a number of MPI ranks, while the other algorithm completely bypasses the assembling of quartic memory-scaling ERIs and thus largely reduces the memory demand. It is demonstrated that the former memory-expensive algorithm is faster on a few hundred cores, while the latter memory-economic algorithm shows a better strong scaling in the limit of a few thousand cores. The program is shown to exhibit a near-linear scaling, in particular for the compute-intensive triples correction step, on up to 8000 cores. The performance of the program is demonstrated via calculations involving molecules with 24–51 atoms and up to 1624 atomic basis functions. As the first application, the complete basis set (CBS) limit for the interaction energy of the π-stacked uracil dimer from the S66 data set has been investigated. This work reports the first calculation of the interaction energy at the CCSD(T)/aug-cc-pVQZ level without local orbital approximation. The CBS limit for the CCSD correlation contribution to the interaction energy was found to be −8.01 kcal/mol, which agrees very well with the value −7.99 kcal/mol reported by Schmitz, Hättig, and Tew [ Phys. Chem. Chem. Phys. 2014, 16, 22167−22178]. The CBS limit for the total interaction energy was estimated to be −9.64 kcal/mol.
We adapt an algorithm for singles and doubles coupled cluster (CCSD) that uses density fitting or Cholesky decomposition (CD) in the construction and contraction of all electron repulsion integrals (ERIs) for use on heterogeneous compute nodes consisting of a multicore central processing unit (CPU) and at least one graphics processing unit (GPU). The use of approximate three-index ERIs ameliorates two of the major difficulties in designing scientific algorithms for GPUs: (1) the extremely limited global memory on the devices and (2) the overhead associated with data motion across the bus. For the benzene trimer described by an aug-cc-pVDZ basis set, the use of a single NVIDIA Tesla C2070 (Fermi) GPU accelerates a CD-CCSD computation by a factor of 2.1, relative to the multicore CPU-only algorithm that uses six highly efficient Intel Core i7-3930K CPU cores. The use of two Fermi GPUs provides an acceleration of 2.89, which is comparable to that observed when using a single NVIDIA Kepler K20c GPU (2.73).
Abstract A high‐performance implementation of the coupled‐cluster singles, doubles, and perturbative triples [CCSD(T)] is developed in the Massively Parallel Quantum Chemistry program. Novel features include: (1) reduced memory requirements via a density‐fitting (DF) CCSD implementation utilizing distributed lazy evaluation for tensors with more than two unoccupied indices and (2) the ability to utilize efficiently many‐core nodes (Intel Xeon Phi) and heterogeneous nodes with multiple NVIDIA GPUs on each node. All data that are greater than quadratic in the system size are distributed among processes. Excellent strong scaling is observed on distributed‐memory computers equipped with conventional CPUs, Intel Xeon Phi processors, and heterogeneous nodes with multiple NVIDIA GPUs Canonical CCSD(T) energies can be evaluated for systems containing 200 electrons and 1000 basis functions in a few days using a small size commodity cluster, with even larger computations possible on leadership‐class computing resources.
Heterogeneous nodes composed of a multicore CPU and at least one graphics processing unit (GPU) are increasingly common in high-performance scientific computing, and significant programming effort is currently being undertaken to port existing scientific algorithms to these unique architectures. We present implementations for two many-body quantum chemistry methods on heterogeneous nodes: the coupled-cluster with single and double excitations (CCSD) and time-dependent configuration interaction with single and double excitations (TD-CISD) methods. Both methods can be implemented on a computer as a series of dense matrix-matrix multiplications, operations that GPUs are particularly adept at performing. The GPU-accelerated CCSD algorithm is as much as 4.3 times faster than the corresponding CPU algorithm and 9.7 times faster than the algorithm in the Molpro package. The TD-CISD algorithm is accelerated by as much as a factor of 3.9 by GPUs. Enhanced performance is achieved by overlapping CPU and GPU computations.
The iterative solution of the coupled-cluster with single and double excitations (CCSD) equations is a very time-consuming component of the ``gold standard'' in quantum chemistry, the CCSD(T) method. In an effort to accelerate accurate quantum mechanical calculations, we explore two implementation strategies for the iterative solution of the CC equations on graphics procesing units (GPUs). We consider a communication-avoiding algorithm for the spin-free coupled cluster doubles (CCD) equations followed by a low-storage algorithm for the spin-free CCSD equations. In the communication-avoiding algorithm, the entire iterative procedure for the CCD method is performed on the GPU, resulting in accelerations of a factor of 4-5 relative to the pure CPU algorithm. The low-storage CCSD algorithm requires that a minimum of 4o2v2+2ov elements be stored on the device, where o and v represent the number of orbitals occupied and unoccupied in the reference configuration, respectively. The algorithm masks the transfer time for copying large amounts of data to the GPU by overlapping GPU and CPU computations. The per-iteration costs of this hybrid GPU/CPU algorithm are up to 4.06 times less than those of the pure CPU algorithm and up to 10.63 times less than those of the CCSD implementation found in the Molpro electronic structure package. These results provide insight into how to organize communication and computation as to maximize utilization of a GPU and multicore CPU at the same time.