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Register a new userUsing SIMD and SIMT vectorization to evaluate sparse chemical kinetic Jacobian matrices and thermochemical source terms
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Accurately predicting key combustion phenomena in reactive-flow simulations, e.g., lean blow-out, extinction/ignition limits and pollutant formation, necessitates the use of detailed chemical kinetics. The large size and high levels of numerical stiffness typically present in chemical kinetic models relevant to transportation/power-generation applications make the efficient evaluation/factorization of the chemical kinetic Jacobian and thermochemical source-terms critical to the performance of reactive-flow codes. Here we investigate the performance of vectorized evaluation of constant-pressure/volume thermochemical source-term and sparse/dense chemical kinetic Jacobians using single-instruction, multiple-data (SIMD) and single-instruction, multiple thread (SIMT) paradigms. These are implemented in pyJac, an open-source, reproducible code generation platform. A new formulation of the chemical kinetic governing equations was derived and verified, resulting in Jacobian sparsities of 28.6-92.0% for the tested models. Speedups of 3.40-4.08x were found for shallow-vectorized OpenCL source-rate evaluation compared with a parallel OpenMP code on an avx2 central processing unit (CPU), increasing to 6.63-9.44x and 3.03-4.23x for sparse and dense chemical kinetic Jacobian evaluation, respectively. Furthermore, the effect of data-ordering was investigated and a storage pattern specifically formulated for vectorized evaluation was proposed; as well, the effect of the constant pressure/volume assumptions and varying vector widths were studied on source-term evaluation performance. Speedups reached up to 17.60x and 45.13x for dense and sparse evaluation on the GPU, and up to 55.11x and 245.63x on the CPU over a first-order finite-difference Jacobian approach. Further, dense Jacobian evaluation was up to 19.56x and 2.84x times faster than a previous version of pyJac on a CPU and GPU, respectively.
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Characterizing macromolecular kinetics from molecular dynamics (MD) simulations requires a distance metric that can distinguish slowly-interconverting states. Here we build upon diffusion map theory and define a kinetic distance for irreducible Markov processes that quantifies how slowly molecular conformations interconvert. The kinetic distance can be computed given a model that approximates the eigenvalues and eigenvectors (reaction coordinates) of the MD Markov operator. Here we employ the time-lagged independent component analysis (TICA). The TICA components can be scaled to provide a kinetic map in which the Euclidean distance corresponds to the kinetic distance. As a result, the question of how many TICA dimensions should be kept in a dimensionality reduction approach becomes obsolete, and one parameter less needs to be specified in the kinetic model construction. We demonstrate the approach using TICA and Markov state model (MSM) analyses for illustrative models, protein conformation dynamics in bovine pancreatic trypsin inhibitor and protein-inhibitor association in trypsin and benzamidine.
This work describes the SIMD vectorization of the force calculation of the Lennard-Jones potential with Intel AVX2 and AVX-512 instruction sets. Since the force-calculation kernel of the molecular dynamics method involves indirect access to memory, the data layout is one of the most important factors in vectorization. We find that the Array of Structures (AoS) with padding exhibits better performance than Structure of Arrays (SoA) with appropriate vectorization and optimizations. In particular, AoS with 512-bit width exhibits the best performance among the architectures. While the difference in performance between AoS and SoA is significant for the vectorization with AVX2, that with AVX-512 is minor. The effect of other optimization techniques, such as software pipelining together with vectorization, is also discussed. We present results for benchmarks on three CPU architectures: Intel Haswell (HSW), Knights Landing (KNL), and Skylake (SKL). The performance gains by vectorization are about 42% on HSW compared with the code optimized without vectorization. On KNL, the hand-vectorized codes exhibit 34% better performance than the codes vectorized automatically by the Intel compiler. On SKL, the code vectorized with AVX2 exhibits slightly better performance than that with vectorized AVX-512.
Properties that are necessarily formulated within pure (symmetric) expectation values are difficult to calculate for projector quantum Monte Carlo approaches, but are critical in order to compute many of the important observable properties of electronic systems. Here, we investigate an approach for the sampling of unbiased reduced density matrices within the Full Configuration Interaction Quantum Monte Carlo dynamic, which requires only small computational overheads. This is achieved via an independent replica population of walkers in the dynamic, sampled alongside the original population. The resulting reduced density matrices are free from systematic error (beyond those present via constraints on the dynamic itself), and can be used to compute a variety of expectation values and properties, with rapid convergence to an exact limit. A quasi-variational energy estimate derived from these density matrices is proposed as an accurate alternative to the projected estimator for multiconfigurational wavefunctions, while its variational property could potentially lend itself to accurate extrapolation approaches in larger systems.
Sparse matrix-vector multiplication (spMVM) is the most time-consuming kernel in many numerical algorithms and has been studied extensively on all modern processor and accelerator architectures. However, the optimal sparse matrix data storage format is highly hardware-specific, which could become an obstacle when using heterogeneous systems. Also, it is as yet unclear how the wide single instruction multiple data (SIMD) units in current multi- and many-core processors should be used most efficiently if there is no structure in the sparsity pattern of the matrix. We suggest SELL-C-sigma, a variant of Sliced ELLPACK, as a SIMD-friendly data format which combines long-standing ideas from General Purpose Graphics Processing Units (GPGPUs) and vector computer programming. We discuss the advantages of SELL-C-sigma compared to established formats like Compressed Row Storage (CRS) and ELLPACK and show its suitability on a variety of hardware platforms (Intel Sandy Bridge, Intel Xeon Phi and Nvidia Tesla K20) for a wide range of test matrices from different application areas. Using appropriate performance models we develop deep insight into the data transfer properties of the SELL-C-sigma spMVM kernel. SELL-C-sigma comes with two tuning parameters whose performance impact across the range of test matrices is studied and for which reasonable choices are proposed. This leads to a hardware-independent (catch-all) sparse matrix format, which achieves very high efficiency for all test matrices across all hardware platforms.
Processing-using-DRAM has been proposed for a limited set of basic operations (i.e., logic operations, addition). However, in order to enable the full adoption of processing-using-DRAM, it is necessary to provide support for more complex operations. In this paper, we propose SIMDRAM, a flexible general-purpose processing-using-DRAM framework that enables massively-parallel computation of a wide range of operations by using each DRAM column as an independent SIMD lane to perform bit-serial operations. SIMDRAM consists of three key steps to enable a desired operation in DRAM: (1) building an efficient majority-based representation of the desired operation, (2) mapping the operation input and output operands to DRAM rows and to the required DRAM commands that produce the desired operation, and (3) executing the operation. These three steps ensure efficient computation of any arbitrary and complex operation in DRAM. The first two steps give users the flexibility to efficiently implement and compute any desired operation in DRAM. The third step controls the execution flow of the in-DRAM computation, transparently from the user. We comprehensively evaluate SIMDRAMs reliability, area overhead, operation throughput, and energy efficiency using a wide range of operations and seven diverse real-world kernels to demonstrate its generality. Our results show that SIMDRAM provides up to 5.1x higher operation throughput and 2.5x higher energy efficiency than a state-of-the-art in-DRAM computing mechanism, and up to 2.5x speedup for real-world kernels while incurring less than 1% DRAM chip area overhead. Compared to a CPU and a high-end GPU, SIMDRAM is 257x and 31x more energy-efficient, while providing 93x and 6x higher operation throughput, respectively.
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