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Journal of Parallel and Distributed Computing

A massively parallel tensor contraction framework for coupled-cluster computations

Edgar Solomonik, Devin A. Matthews, Jeff R. Hammond, John F. Stanton, James Weldon Demmel

Published Jun 24, 2014• 192 citations
Publisher
DOI
10.1016/j.jpdc.2014.06.002

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We have developed a symbolic manipulation program and program generator (tensor contraction engine or TCE) that abstracts and automates the time-consuming, error-prone processes of deriving the working equations of a well-defined model of second-quantized many-electron theories and synthesizing efficient parallel computer programs on the basis of these equations. Provided an ansatz of a many-electron theory model, TCE performs valid contractions of creation and annihilation operators according to Wick's theorem, consolidates identical terms, and reduces the expressions into the form of multiple tensor contractions acted upon by permutation operators. It subsequently determines the binary contraction order for each multiple tensor contraction with the minimal operation and memory cost, factorizes common binary contractions (defines intermediate tensors), and identifies reusable intermediates. The resulting ordered list of binary tensor contractions, additions, and index permutations is translated into an optimized program that is combined with the NWCHEM and UTCHEM computational chemistry software packages. The programs synthesized by TCE take advantage of spin symmetry (within the spin−orbital formalisms), real Abelian point-group symmetry, and index permutation symmetry at every stage of the calculations to minimize the number of arithmetic operations and storage requirement, adjust the peak local memory usage by index-range tiling, and support parallel I/O interfaces and dynamic load balancing for parallel executions. We demonstrate the utility of TCE through automatic derivation and implementation of parallel programs for a range of predictive computational methods configuration-interaction theory (CISD, CISDT, CISDTQ), generalized many-body perturbation theory [MBPT(2), MBPT(3), MBPT(4)], and coupled-cluster theory (LCCD, CCD, LCCSD, CCSD, QCISD, CCSDT, and CCSDTQ), some for the first time and discuss the performance of the implemented programs.
Using GPU-based HPC platforms efficiently for coupled cluster computations is a challenge due to heterogeneous hardware structures. The constant need to adapt software to these structures and the required man-hours makes the systematization of high-performance code development desirable, even more so for higher-order coupled cluster. This is generally achieved by introducing a high-level representation of the problem, which is then translated into low-level instructions for the hardware using a compiler/translator component. Designing such software comes with another challenge: Allowing efficient implementation by capturing key symmetries of tensors while retaining abstraction from the hardware. We review ways to address these two challenges while presenting the design decisions that led us to the development of a general-order coupled cluster code generator. The systematically produced code shows excellent weak scaling behavior, running on up to 1200 GPUs using the distributed memory tensor library ExaTENSOR. We present an open-source modular tensor framework ″tenpi″ for coupled cluster code development with diagrammatic derivation, visualization module, symbolic algebra, intermediate optimization, and support for multiple tensor backends. Tenpi brings higher-order CC functionality to the massively parallel ExaCorr module of the DIRAC code for relativistic molecular calculations.