Large Scale Eigenvalue Problems: Workshop Proceedings

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His core expertise is in numerical linear algebra, optimization, large-scale data analysis, and high performance computing. Over the last several years, he has successfully used techniques developed in these areas to solve problems in electronic structure calculations, nuclear structure calculations, cavity design for accelerator models, single-particle analysis for cryo-electron microscopy, single molecular diffractive imaging, phase retrieval, ptychography, etc.

Polynomial filtering can provide a highly effective means of computing all eigenvalues of a real symmetric or complex Hermitian matrix that are located in a given interval, anywhere in the spectrum. This paper describes a technique for tackling this problem by combining a Thick-Restart version of the Lanczos algorithm with deflation 'locking' and a new type of polynomial filters obtained from a least-squares technique. The resulting algorithm can be utilized in a 'spectrum-slicing' approach whereby a very large number of eigenvalues and associated eigenvectors of the matrix are computed by extracting eigenpairs located in different sub-intervals independently from one another.

B, 37, R is generalized to obtain a straightforward, surprisingly accurate, and scalable numerical representation for calculating the electronic wave functions of molecules. The Coulomb matrix elements are replaced with matrix elements obtained from the kinetic energy operator. A resolution-of-the-identity approximation renders the primitive one- and two-electron matrix elements diagonal; in other words, the Coulomb operator is local with respect to the grid indices.

The representation not only is numerically expedient, but also produces energies and properties superior to those calculated variationally. The method is particularly useful for computing a subset of eigenvalues, and their eigen- or Schur vectors, closest to a given shift. The proposed method is based on block iterations and can take advantage of a preconditioner if it is available.

It does not need to perform exact shift-and-invert transformation.

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Standard and generalized eigenproblems are handled in a unified framework. Our numerical experiments demonstrate that GPLHR is generally more robust and efficient than existing methods, especially if the available memory is limited. We are interested in the case in which the dimension of the invariant subspace is large e. These problems arise from, for example, density functional theory DFT based electronic structure calculations for complex materials.

The key feature of our algorithm is that it performs fewer Rayleigh—Ritz calculations compared to existing algorithms such as the locally optimal block preconditioned conjugate gradient or the Davidson algorithm. It is a block algorithm, and hence can take advantage of efficient BLAS3 operations and be implemented with multiple levels of concurrency.

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We discuss a number of practical issues that must be addressed in order to implement the algorithm efficiently on a high performance computer. With the help of our recently developed massively parallel DGDFT Discontinuous Galerkin Density Functional Theory methodology, we perform large-scale Kohn—Sham density functional theory calculations on phosphorene nanoribbons with armchair edges ACPNRs containing a few thousands to ten thousand atoms.

The use of DGDFT allows us to systematically achieve a conventional plane wave basis set type of accuracy, but with a much smaller number about 15 of adaptive local basis ALB functions per atom for this system. The relatively small number of degrees of freedom required to represent the Kohn—Sham Hamiltonian, together with the use of the pole expansion the selected inversion PEXSI technique that circumvents the need to diagonalize the Hamiltonian, results in a highly efficient and scalable computational scheme for analyzing the electronic structures of ACPNRs as well as their dynamics.

The total wall clock time for calculating the electronic structures of large-scale ACPNRs containing —10 atoms is only 10—25 s per self-consistent field SCF iteration, with accuracy fully comparable to that obtained from conventional planewave DFT calculations. New algorithms for iterative diagonalization procedures that solve for a small set of eigen-states of a large matrix are described.

Linear Algebra and its Applications

We present two algorithms suitable for calculating excited states that are close to a specified energy shift interior eigenvalues. One solver is based on the Davidson algorithm, a diagonalization procedure commonly used in quantum-chemical calculations.

Eigenvalue Problems and Applications to Structural Dynamics

The details of the algorithms, their computational scaling, and memory requirements are described. Lin, A. Huhs, and C. Yang, J. We believe the presence of these states makes their gap decrease more rapidly. We describe preconditioned iterative methods for estimating the number of eigenvalues of a Hermitian matrix within a given interval.

Such estimation is useful in a number of applications. In particular, it can be used to develop an efficient spectrum-slicing strategy to compute many eigenpairs of a Hermitian matrix. Our method is based on the Lanczos- and Arnoldi-type of iterations. We show that with a properly defined preconditioner, only a few iterations may be needed to obtain a good estimate of the number of eigenvalues within a prescribed interval.

We also demonstrate that the number of iterations required by the proposed preconditioned schemes is independent of the size and condition number of the matrix. The efficiency of the methods is illustrated on several problems arising from density functional theory based electronic structure calculations. We present two efficient iterative algorithms for solving the linear response eigenvalue problem arising fromthe time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations.

In particular, the nonsymmetric eigenvalue problem can be transformed into a product eigenvalue problem that is self-adjoint with respect to a K-inner product. This product eigenvalue problem can be solved efficiently by a modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient LOBPCG algorithm that make use of the K-inner product.

Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing

The solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. However, the other component of the eigenvector can be easily recovered in a postprocessing procedure. Therefore, the algorithms we present here are more efficient than existing algorithms that try to approximate both components of the eigenvectors simultaneously. The efficiency of the new algorithms is demonstrated by numerical examples. Computational Research. Scalable Solvers Group.

Chao Yang. Short Bio Chao Yang received his Ph. His research interests include: developing numerical algorithms and fast implementation to accelerate scientific simulations developing efficient and robust algorithms and implementations for solving inverse problems His core expertise is in numerical linear algebra, optimization, large-scale data analysis, and high performance computing. Journal Articles Y. Li, Z. Wen, C. Yang, Y.

Van Beeumen, O.

Marques, E. Ng, C. Yang, Z. Bai, L. Ge, O. Kononenko, Z. Li, C. Ng, L. Xiao, "Computing resonant modes of accelerator cavities by solving nonlinear eigenvalue problems via rational approximation", Journal of Computational Physics, , : , doi: Deusch, M.

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Shao, C. Yang, M.

Journal Articles

Clement, J. Zhang, C. Lewis, C. Yang, Edward F.

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Theory Comput. Huang, J. Sun, C. Yang, "Recursive integral method with Cayley transformation", Numerical Linear Algebra with Applications, July 10, , 25 : , doi: Meiyue Shao, Felipe H. Ke, A. Brewster, S. Yu, D. Ushizima, C. Yang, N. Banerjee, L. Lin, P. Suryanarayana, C.

Large Scale Eigenvalue Problems: Workshop Proceedings Large Scale Eigenvalue Problems: Workshop Proceedings
Large Scale Eigenvalue Problems: Workshop Proceedings Large Scale Eigenvalue Problems: Workshop Proceedings
Large Scale Eigenvalue Problems: Workshop Proceedings Large Scale Eigenvalue Problems: Workshop Proceedings
Large Scale Eigenvalue Problems: Workshop Proceedings Large Scale Eigenvalue Problems: Workshop Proceedings
Large Scale Eigenvalue Problems: Workshop Proceedings Large Scale Eigenvalue Problems: Workshop Proceedings

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