Accelerating nuclear configuration interaction calculations through a preconditioned block iterative eigensolver

Shao, Meiyue; Aktulga, Hasan Metin; Yang, Chao; Ng, Esmond G.; Maris, Pieter; Vary, James P.

doi:10.1016/j.cpc.2017.09.004

Cited by 63 publications

(93 citation statements)

References 36 publications

(80 reference statements)

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“…8. We remark that without the preconditioner and initial guess techniques we adopted, LOBPCG's slow convergence rate leads to similar or worse solution times compared to Lanczos/FO, wiping out the gains from replacing SpMVs with SpMMs (for a detailed analyses, please see Shao et al [35]). In this regard, for nuclear CI calculations, inexpensive solves with smaller N max truncations and low cost preconditioners are crucial for the performance benefits observed with LOBPCG.…”

Section: Nm6mentioning

confidence: 99%

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A High Performance Block Eigensolver for Nuclear Configuration Interaction Calculations

Aktulga

Afibuzzaman

Williams

et al. 2017

IEEE Trans. Parallel Distrib. Syst.

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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.

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Section: Nm6mentioning

confidence: 99%

“…Full details and analyses of initial guesses and preconditioning techniques used are discussed by Shao et al [35]. We describe them briefly here:…”

Section: Putting It Altogethermentioning

confidence: 99%

A High Performance Block Eigensolver for Nuclear Configuration Interaction Calculations

Aktulga

Afibuzzaman

Williams

et al. 2017

IEEE Trans. Parallel Distrib. Syst.

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“…We demonstrated the improvement by several numerical examples performed on single processor and multi-core systems using OpenMP parallelization. Most of the techniques discussed here can help improve the stability and performance of a distributed memory parallel implementation of the LOBPCG algorithm also (see, e.g., [14] for a recent implementation on distributed memory systems). However, the performance of such an implementation often depends on the parallelization of the sparse matrix-vector multiplications (SpMV).…”

Section: Discussionmentioning

confidence: 99%

A Robust and Efficient Implementation of LOBPCG

Duersch¹,

Shao²,

Yang³

et al. 2018

SIAM J. Sci. Comput.

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Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is widely used to compute eigenvalues of large sparse symmetric matrices. The algorithm can suffer from numerical instability if it is not implemented with care. This is especially problematic when the number of eigenpairs to be computed is relatively large. In this paper we propose an improved basis selection strategy based on earlier work by Hetmaniuk and Lehoucq as well as a robust convergence criterion which is backward stable to enhance the robustness. We also suggest several algorithmic optimizations that improve performance of practical LOBPCG implementations. Numerical examples confirm that our approach consistently and significantly outperforms previous competing approaches in both stability and speed.from which the approximate solution to the eigenvalue problem is extracted can become linearly dependent. This problem becomes progressively worse when the number of eigenpairs to be computed becomes relatively large (e.g., hundreds or thousands). For example, in electronic structure calculations, the number of desired eigenpairs is proportional to the number of atoms in the system, which can grow to several thousands [13]. Hence remedies for improving numerical stability are of practical interest.A strategy proposed in the work of Hetmaniuk and Lehoucq [8] addresses this issue. Their strategy is based on performing additional orthogonalization to ensure that the preconditioned gradient is numerically B-orthogonal to both the current and the previous approximations to the desired eigenvectors. However, this strategy can become expensive when the number of eigenpairs to be computed is relatively large. More importantly, reliability can still be severely compromised due to numerical instability within the orthogonalization steps.This paper presents an efficient and reliable implementation of LOBPCG. We develop a number of techniques to significantly enhance the Hetmaniuk-Lehoucq (HL) orthogonalization strategy in both efficiency and reliability. We also adopt an alternative convergence criterion to ensure achievable error control in computed eigenpairs. For simplicity, we assume that both A and B are real matrices. But our techniques naturally carry over to complex Hermitian matrices.The rest of this paper is organized as follows. In Section 2, we describe the basic LOBPCG algorithm. In Section 3, we discuss numerical difficulties one may encounter in LOBPCG and the HL strategy for overcoming these difficulties. In Section 4, we present our techniques for improving the HL strategy. In Section 5, we present additional techniques for improving all other aspects of LOBPCG. Finally, in Section 6, we report numerical experimental results to illustrate the effectiveness of our techniques.

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“…Recent developments of the nucleon-nucleon (NN) and three-nucleon (3N) interactions, derived from chiral effective field theory, have yielded major progress [7][8][9][10][11][12][13][14]. These, together with the utilization of massively parallel computing resources (e.g., see [15][16][17][18]), have placed ab initio large-scale simulations at the frontier of nuclear structure and reaction explorations. Among other successful many-body theories, the ab initio no-core shell-model (NCSM) approach, which has considerably advanced our understanding and capability of achieving first-principles descriptions of low-lying states in light nuclear systems (e.g., see [19][20][21][22][23]), has over the last decade taken center stage in the development of microscopic tools for studying the structure of atomic nuclei.…”

Section: Introductionmentioning

confidence: 99%

“…In Sec. III we present results for elastic scattering of protons as well as neutrons from the "closed shell" nuclei 4 He and 16 O in the energy regime between 100 and 200 MeV. Then we apply the formulation to the "open shell" nuclei 12 C and 6 He.…”

Section: Introductionmentioning

confidence: 99%

Ab initio folding potentials for nucleon-nucleus scattering based on no-core shell-model one-body densities

et al. 2019

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Background: Calculating microscopic optical potentials for elastic nucleon-nucleus scattering has already led to large body of work in the past. For folding first-order calculations the nucleon-nucleon (NN) interaction and the one-body density of the nucleus were taken as input to rigorous calculations in a spectator expansion of the multiple scattering series.Purpose: Based on the Watson expansion of the multiple scattering series we employ a nonlocal translationally invariant nuclear density derived from a chiral next-to-next-to-leading order (NNLO) and the very same interaction for consistent full-folding calculation of the effective (optical) potential for nucleon-nucleus scattering for light nuclei.Methods: The first order effective (optical) folding potential is computed by integrating over the nonlocal, translationally invariant NCSM one-body density and the off-shell Wolfenstein amplitudes A and C. The resulting nonlocal potential serves as input for a momentum-space Lippmann-Schwinger equation, whose solutions are summed to obtain the nucleon-nucleus scattering observables. Results:We calculate scattering observables, such as total, reaction, and differential cross sections as well as the analyzing power and the spin-rotation parameter, for elastic scattering of protons and neutrons from 4 He, 6 He, 12 C, and 16 O, in the energy regime between 100 and 200 MeV projectile kinetic energy, and compare to available data.Conclusions: Our calculations show that the effective nucleon-nucleus potential obtained from the first-order term in the spectator expansion of the multiple scattering expansion describes experiments very well to about 60 degrees in the center-of-mass frame, which coincides roughly with the validity of the NNLO chiral interaction used to calculate both the NN amplitudes and the one-body nuclear density.

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Accelerating nuclear configuration interaction calculations through a preconditioned block iterative eigensolver

Cited by 63 publications

References 36 publications

A High Performance Block Eigensolver for Nuclear Configuration Interaction Calculations

A High Performance Block Eigensolver for Nuclear Configuration Interaction Calculations

A Robust and Efficient Implementation of LOBPCG

Ab initio folding potentials for nucleon-nucleus scattering based on no-core shell-model one-body densities

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