A High Performance Block Eigensolver for Nuclear Configuration Interaction Calculations

Abstract

Abstract-As on-node parallelism increases and the performance gap between the processor and the memory system widens, achieving high performance in large-scale scientific applications requires an architecture-aware design of algorithms and solvers. We focus on the eigenvalue problem arising in nuclear Configuration Interaction (CI) calculations, where a few extreme eigenpairs of a sparse symmetric matrix are needed. We consider a block iterative eigensolver whose main computational kernels are the multiplication of a sparse matrix with multiple vectors (SpMM), and tall-skinny matrix operations. We present techniques to significantly improve the SpMM and the transpose operation SpMM T by using the compressed sparse blocks (CSB) format. We achieve 3-4× speedup on the requisite operations over good implementations with the commonly used compressed sparse row (CSR) format. We develop a performance model that allows us to correctly estimate the performance of our SpMM kernel implementations, and we identify cache bandwidth as a potential performance bottleneck beyond DRAM. We also analyze and optimize the performance of LOBPCG kernels (inner product and linear combinations on multiple vectors) and show up to 15× speedup over using high performance BLAS libraries for these operations. The resulting high performance LOBPCG solver achieves 1.4× to 1.8× speedup over the existing Lanczos solver on a series of CI computations on high-end multicore architectures (Intel Xeons). We also analyze the performance of our techniques on an Intel Xeon Phi Knights Corner (KNC) processor.

Toward scalable many-body calculations for nuclear open quantum systems using the Gamow Shell Model

“…Similarly to Fig. (3), total calculation time scales very well for 8 He, when storage and on-the-fly methods show a typical ratio of 90% and 75% with respect to perfect scaling, respectively, while it is not the case for that of 8 Be. While these performances seem to be low at first sight for 8 Be, they follow the uneven pattern of memory distribution among nodes, provided by the full storage calculation (see the left panel of Fig.(3)).…”

“…However, the situation is very different in practice. In fact, one obtains a fairly large value for the ratio of average and maximal MPI communication times to total times, of 20%-50% for 8 He and 30%-70% for 8 Be (see Fig.(5)). Moreover, obtained values for average and maximal MPI times are comparable for full, partial and on-the-fly methods (see Fig.(5)).…”

“…In order to test the performance of the new version of the GSM code, we consider the 8 He and 8 Be nuclei. Their model space consists of the 4 He core with valence nucleons, four valence neutrons for 8 He and two valence protons and two valence neutrons for 8 Be nuclei.…”

“…In order to test the performance of the new version of the GSM code, we consider the 8 He and 8 Be nuclei. Their model space consists of the 4 He core with valence nucleons, four valence neutrons for 8 He and two valence protons and two valence neutrons for 8 Be nuclei. The core is mimicked by a Woods-Saxon potential and the interaction used is that of Furuchi-Horiuchi-Tamagaki type, which has been recently used for the description of light nuclei in GSM [4].…”

Developing a New Storage Format and a Warp-Based SpMV Kernel for Configuration Interaction Sparse Matrices on the GPU

“…The Compressed Sparse Blocks (CSB) format [13,21,22] is used for storing sparse matrices. The CSB format partitions the n × n matrix into n 2 /z 2 equal-sized z × z square blocks using a block size parameter z.…”