Chebyshev filter diagonalization is well established in quantum chemistry and quantum physics to compute bulks of eigenvalues of large sparse matrices. Choosing a block vector implementation, we investigate optimization opportunities on the new class of high-performance compute devices featuring both high-bandwidth and low-bandwidth memory. We focus on the transparent access to the full address space supported by both architectures under consideration: Intel Xeon Phi “Knights Landing” and Nvidia “Pascal” “Volta.” After a thorough performance analysis of the single-device implementations using the roofline model we propose two optimizations: (1) Subspace blocking is applied for improved performance and data access efficiency. We also show that it allows transparently handling problems much larger than the high-bandwidth memory without significant performance penalties. (2) Pipelining of communication and computation phases of successive subspaces is implemented to hide communication costs without extra memory traffic. As an application scenario we perform filter diagonalization studies for topological quantum matter. Performance numbers on up to 2048 nodes of the Oakforest-PACS and Piz Daint supercomputers are presented, achieving beyond 500 Tflop s for computing (10^2 ) inner eigenvalues of sparse matrices of dimension (4 10^9 ).
The computation of bulks of eigenvalues of large sparse matrices is very data intensive, both in terms of bandwidth demands (i.e., low computational intensity) and data set sizes. Subspace projection using polynomial filters based on the Chebyshev iteration is an efficient approach for the computation of extremal and interior eigenvalues in quantum physics and quantum chemistry. Application areas include inner eigenvalue problems in the context of modeling graphene or topological insulator materials @cite_24 @cite_8 or the solution of an eigenvalue problem in density functional theory @cite_25 . Beyond eigenvalue computations, Chebyshev polynomials can be used as acceleration techniques for linear solvers (see, e.g., @cite_3 @cite_15 ) in various application areas (e.g., power flow modeling @cite_20 @cite_11 @cite_2 ). Moreover, the closely related kernel polynomial method (KPM) (see @cite_16 for a review on KPM and its relation to Chebyshev polynomials) also relies on evaluating those polynomials to calculate spectral properties of sparse matrices, such as the density of states @cite_23 @cite_10 .
@cite_8 @cite_3 @cite_24 @cite_23 @cite_2 @cite_15 @cite_16 @cite_10 @cite_25 @cite_20 @cite_11
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We study Chebyshev filter diagonalization as a tool for the computation of many interior eigenvalues of very large sparse symmetric matrices. In this technique the subspace projection onto the target space of wanted eigenvectors is approximated with filter polynomials obtained from Chebyshev expansions of window functions. After the discussion of the conceptual foundations of Chebyshev filter diagonalization we analyze the impact of the choice of the damping kernel, search space size, and filter polynomial degree on the computational accuracy and effort, before we describe the necessary steps towards a parallel high-performance implementation. Because Chebyshev filter diagonalization avoids the need for matrix inversion it can deal with matrices and problem sizes that are presently not accessible with rational function methods based on direct or iterative linear solvers. To demonstrate the potential of Chebyshev filter diagonalization for large-scale problems of this kind we include as an example the computation of the 102 innermost eigenpairs of a topological insulator matrix with dimension 109 derived from quantum physics applications. Abstract : The present paper deals with the problem of computing a few of the eigenvalues with largest (or smallest) real parts, of a large sparse nonsymmetric matrix. We present a general acceleration technique based on Chebyshev polynomials and discuss its practical application to Arnoldi's method and the subspace iteration method. The resulting algorithms are compared with the classical ones in a few experiments which exhibit a sharp superiority of the Arnoldi-Chebyshev approach. The Kernel Polynomial Method (KPM) is a well-established scheme in quantum physics and quantum chemistry to determine the Eigen value density and spectral properties of large sparse matrices. In this work we demonstrate the high optimization potential and feasibility of peta-scale heterogeneous CPU-GPU implementations of the KPM. At the node level we show that it is possible to decouple the sparse matrix problem posed by KPM from main memory bandwidth both on CPU and GPU. To alleviate the effects of scattered data access we combine loosely coupled outer iterations with tightly coupled block sparse matrix multiple vector operations, which enables pure data streaming. All optimizations are guided by a performance analysis and modelling process that indicates how the computational bottlenecks change with each optimization step. Finally we use the optimized node-level KPM with a hybrid-parallel framework to perform large-scale heterogeneous electronic structure calculations for novel topological materials on a pet scale-class Cray XC30 system. Chebyshev polynomial approximations are an efficient and numerically stable way to calculate properties of the very large Hamiltonians important in computational condensed matter physics. The present paper derives an optimal kernel polynomial which enforces positivity of density of states and spectral estimates, achieves the best energy resolution, and preserves normalization. This kernel polynomial method (KPM) is demonstrated for electronic structure and dynamic magnetic susceptibility calculations. For tight binding Hamiltonians of Si, we show how to achieve high precision and rapid convergence of the cohesive energy and vacancy formation energy by careful attention to the order of approximation. For disordered XXZ-magnets, we show that the KPM provides a simpler and more reliable procedure for calculating spectral functions than Lanczos recursion methods. Polynomial approximations to Fermi projection operators are also proposed.