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 ).
Three-dimensional topological insulators feature Dirac-like surface states which are topologically protected against the influence of weak quenched disorder. Here we investigate the effect of surface disorder beyond the weak-disorder limit using large-scale numerical simulations. We find two qualitatively distinct regimes: Moderate disorder destroys the Dirac cone and induces diffusive met allic behavior at the surface. Even more remarkably, for strong surface disorder a Dirac cone reappears, as new weakly disordered "surface" states emerge in the sample beneath the disordered surface layer, which can be understood in terms of an interface between a topological and an Anderson insulator. Together, this demonstrates the drastic effect of disorder on topological surface states, which cannot be captured within effective two-dimensional models for the surface states alone. 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.   An efficient power flow solution is highly important for power system analyses. Traditionally, LU decomposition-based direct method is dominant in solving linear systems in the power flow analysis. However, with the increasing scale and complexity, direct methods may suffer scalability issues, especially under parallel implementation. Therefore, iterative methods such as Conjugate Gradient (CG) are gaining more attention for their scalability and feasibility for parallelisation. To efficiently apply an iterative solver like CG, a preconditioner is usually required. A polynomial-based Chebyshev preconditioner is discussed in this work. An essential parameter in Chebyshev preconditioning is the maximum eigenvalue of the linear system. However, direct calculation of the eigenvalues is too time-consuming to be practical. Therefore, this work proposes a method to estimate the largest eigenvalue to save the time spent on eigenvalue calculation. Thus, Chebyshev preconditioning will be practical to use in power system analyses. This work first proves that the eigenvalues of power system applications range within (0, 2] after a normalisation step, then demonstrates the eigenvalue estimation accuracy, and finally presents the performance improvement. The test shows an average 98.92 runtime saving for the Chebyshev preconditioner, and a 40.99 runtime saving for precondition and iterative solving process. Abstract Conjugate gradient (CG) methods to solve sparse systems of linear equations play an important role in numerical methods for solving discretized partial differential equations. The large size and the condition of many technical or physical applications in this area result in the need for efficient parallelization and preconditioning techniques of the CG method, in particular on massively parallel machines. Here, the data distribution and the communication scheme for the sparse matrix operations of the preconditioned CG are based on the analysis of the indices of the non-zero elements. Polynomial preconditioning is shown to reduce global synchronizations considerably, and a fully local incomplete Cholesky preconditioner is presented. On a PARAGON XP S 10 with 138 processors, the developed parallel methods outperform diagonally scaled CG markedly with respect to both scaling behavior and execution time for many matrices from real finite element applications. Efficient and stable algorithms for the calculation of spectral quantities and correlation functions are some of the key tools in computational condensed matter physics. In this article we review basic properties and recent developments of Chebyshev expansion based algorithms and the Kernel Polynomial Method. Characterized by a resource consumption that scales linearly with the problem dimension these methods enjoyed growing popularity over the last decade and found broad application not only in physics. Representative examples from the fields of disordered systems, strongly correlated electrons, electron-phonon interaction, and quantum spin systems we discuss in detail. In addition, we illustrate how the Kernel Polynomial Method is successfully embedded into other numerical techniques, such as Cluster Perturbation Theory or Monte Carlo simulation. Summary Estimating the number of eigenvalues located in a given interval of a large sparse Hermitian matrix is an important problem in certain applications, and it is a prerequisite of eigensolvers based on a divide-and-conquer paradigm. Often, an exact count is not necessary, and methods based on stochastic estimates can be utilized to yield rough approximations. This paper examines a number of techniques tailored to this specific task. It reviews standard approaches and explores new ones based on polynomial and rational approximation filtering combined with a stochastic procedure. We also discuss how the latter method is particularly well-suited for the FEAST eigensolver. Copyright © 2016 John Wiley & Sons, Ltd. The power of density functional theory is often limited by the high computational demand in solving an eigenvalue problem at each self-consistent-field (SCF) iteration. The method presented in this paper replaces the explicit eigenvalue calculations by an approximation of the wanted invariant subspace, obtained with the help of well-selected Chebyshev polynomial filters. In this approach, only the initial SCF iteration requires solving an eigenvalue problem, in order to provide a good initial subspace. In the remaining SCF iterations, no iterative eigensolvers are involved. Instead, Chebyshev polynomials are used to refine the subspace. The subspace iteration at each step is easily five to ten times faster than solving a corresponding eigenproblem by the most efficient eigen-algorithms. Moreover, the subspace iteration reaches self-consistency within roughly the same number of steps as an eigensolver-based approach. This results in a significantly faster SCF iteration. Programmable graphics processing units (GPUs) currently offer the best ratio of floating point computational throughput to price for commodity processors, outdistancing same-generation CPUs by an order of magnitude, which has in turn led to their widespread adoption in a variety of computationally demanding fields. Adapting power system simulations to these processors is complicated by the unique hardware architecture of GPUs, which precludes the usage of direct linear system solvers based on Gaussian elimination. Krylov subspace methods are better suited to the GPU architecture, yet the ill-conditioned nature of power system matrices requires substantial preconditioning to ensure robustness of these methods. To reduce the time spent on preconditioning, we have developed a GPU-based preconditioner designed specifically to handle the large, sparse matrices typically encountered in power system simulations. The preconditioning technique used, based on Chebyshev polynomials, is described in detail, as are the design decisions made in adapting this algorithm to the GPU. Evaluation of the performance of the GPU-based preconditioner on a variety of sparse matrices, ranging in size from 30 x 30 to 3948 x 3948, shows significant computational savings relative to a CPU-based implementation of the same preconditioner and a typical incomplete LU (ILU) preconditioner. a b s t r a c t Traditionally, linear equations in power system applications are solved by direct methods based on LU decomposition. With the development of advanced power system controls, the industrial and research community is more interested in simulating larger, interconnected power grids. Iterative methods such as the conjugate gradient method have been applied to power system applications in the literature for its parallelism potential with larger systems. Preconditioner, used for preconditioning the linear system for a better convergence rate in iterative computations, is an indispensable part of iterative solving process. This work implemented a polynomial preconditioner Chebyshev preconditioner with graphic processing unit (GPU), and integrated a GPU-based conjugate gradient solver. Results show that GPU-based Chebyshev preconditioner can reach around 46× speedup for the largest test system, and conjugate gradient can gain more than 4× speedup. This demonstrates great potentials for GPU application in power system