A projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a Hermitian matrix

Large-scale ab initio calculations based on three levels of parallelization

Given a convex and differentiable objective Q(M) for a real symmetric matrix M in the positive definite (PD) cone-used to compute Mahalanobis distances-we propose a fast general metric learning framework that is entirely projection-free. We first assume that M resides in a space S of generalized graph Laplacian matrices corresponding to balanced signed graphs. M ∈ S that is also PD is called a graph metric matrix. Unlike low-rank metric matrices common in the literature, S includes the important diagonal-only matrices as a special case. The key theorem to circumvent full eigen-decomposition and enable fast metric matrix optimization is Gershgorin disc perfect alignment (GDPA): given M ∈ S and diagonal matrix S, where Sii = 1/vi and v is the first eigenvector of M, we prove that Gershgorin disc left-ends of similarity transform B = SMS-1 are perfectly aligned at the smallest eigenvalue λmin. Using this theorem, we replace the PD cone constraint in the metric learning problem with tightest possible linear constraints per iteration, so that the alternating optimization of the diagonal / off-diagonal terms in M can be solved efficiently as linear programs via the Frank-Wolfe method. We update v using Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) with warm start as entries in M are optimized successively. Experiments show that our graph metric optimization is significantly faster than cone-projection schemes, and produces competitive binary classification performance.

An h-adaptive finite element solver for the calculations of the electronic structures

<abstract> By utilizing an inner-outer iteration strategy, a shift-splitting (SS) iteration method to solve a class of large sparse linear matrix equation $ AXB = C $ is proposed in this work. Two convergence theorems for differential forms are studied in depth. Moreover, the quasi-optimal parameters which minimize the upper bound for the spectral radius of SS iteration matrix are given. Two numerical examples illustrate the high-efficiency of SS iteration method, especially when coefficient matrices are ill-conditioned. </abstract>

Bundle block adjustment of large-scale remote sensing data with Block-based Sparse Matrix Compression combined with Preconditioned Conjugate Gradient

Many graph-based algorithms in high performance computing (HPC) use approximate solutions due to having algorithms that are computationally expensive or serial in nature. Neural acceleration, i.e., the process of speeding up approximation computation elements via artificial neural networks, is relatively new and has not focused on HPC graph-based algorithms. In this paper, we propose a starting point for applying models for neural acceleration to graph-based HPC algorithms utilizing an understanding of the connectivity computational pattern, recursive neural networks, and graph neural networks. We demonstrate these techniques on the problem related to the utility functions of sparse matrix ordering and fill-in (i.e., zero elements becoming nonzero during factorization) calculations. The problem of sparse matrix ordering is commonly used for issues related to load balancing, improving memory reuse, or reducing computational and memory costs in direct sparse linear solver methods. These utility functions are ideal for demonstration as they comprise a number of different graph-based subproblems, and thus demonstrate the usefulness of our method over a wide range. We show that we can accurately approximate the best ordering and the nonzero count of the sparse factorization matrix while speeding up the calculation by as much as 30.3× over the traditional serial method.

In this paper we consider various stability properties of real invariant lagrangian subspaces for real matrices which are either symmetric or skew-symmetric in a real quadratic form which may be symmetric or skew-symmetric itself. In particular, apart from ordinary stability we shall consider strong stability, which seems to be more desirable from a numerical point of view. For the classes of matrices we consider here stable subspaces are not always strongly stable, in contrast with the previous work. We shall completely characterize strongly stable invariant lagrangian subspaces, and in many cases also the stable ones. Invariant lagrangian subspaces with other stability properties, such as Lipschitz stability, are characterized as well.KeywordsInvariant SubspaceReal EigenvalueReal MatriceStrong StabilityStable ElementThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Hamiltonian matrices with respect to a nondegenerate skewsymmetric or skewhermitian indefinite inner product in finite dimensional real, complex, or quaternion vector spaces are studied. Subspaces that are simultaneously invariant for the matrices and neutral in the indefinite inner product are of special interest. The dimension of maximal (by inclusion) such subspaces is identified in terms of the canonical forms and sign characteristics. Criteria for uniqueness of maximal invariant neutral subspaces are given. The important special case of invariant Lagrangian subspaces is treated separately. Comparisons are made between real, complex, and quaternion contexts; for example, for complex Hamiltonian matrices with respect to a nondegenerate skewhermitian inner product in a finite dimensional complex vector space, the (complex) dimension of (complex) maximal invariant neutral subspaces is compared to the (quaternion) dimension of (quaternion) maximal invariant neutral subspaces, and necessary and sufficient conditions are given for the two dimensions to coincide (this is not always the case).

Recursion relations for the extended Krylov subspace method

The extended Krylov subspace method and orthogonal Laurent polynomials

This paper discusses the design and development of a code to calculate the eigenvalues of a large sparse real unsymmetric matrix that are the rightmost, leftmost, or are of the largest modulus. A subspace iteration algorithm is used to compute a sequence of sets of vectors that converge to an orthonormal basis for the invariant subspace corresponding to the required eigenvalues. This algorithm is combined with Chebychev acceleration if the rightmost or leftmost eigenvalues are sought, or if the eigenvalues of largest modulus are known to be the rightmost or leftmost eigenvalues. An option exists for computing the corresponding eigenvectors. The code does not need the matrix explicitly since it only requires the user to multiply sets of vectors by the matrix. Sophisticated and novel iteration controls, stopping criteria, and restart facilities are provided. The code is shown to be efficient and competitive on a range of test problems.

Triangular factorization of general N × N Hermitian matrices requires O(N 3) operations. We present a procedure that factors certain structured matrices in O(N 2) operations. The family of structured matrices considered in this paper includes Toeplitz and Hankel matrices, as well as a variety of ‘Toeplitz-like’ and ‘Hankel-like’ matrices (e.g. inverses, sums and products of Toeplitz and Hankel matrices).

We present an efficient implementation of the analytical nuclear gradient of linear-response time-dependent density functional theory (LR-TDDFT) with the frozen core approximation (FCA). This implementation is realized based on the Hutter’s formalism and the plane wave pseudopotential method. Numerical results demonstrate that the LR-TDDFT/FCA method using a small subset of Kohn–Sham occupied orbitals are accurate enough to reproduce the LR-TDDFT results. Here, the FCA remarkably reduces the computational cost in solving the LR-TDDFT eigenvalue equation. Another challenge in the calculations of analytical nuclear gradients for LR-TDDFT is the solution of the Z-vector equation, for which the Davidson algorithm is a popular choice. While, for large systems the standard Davidson algorithm exhibits a low convergence rate. In order to overcome this problem, we generalize the two-level Davidson algorithm to solve linear equation problems. A more stable performance is achieved with this new algorithm. Our method should encourage further studies of excited-state properties with LR-TDDFT in the plane wave basis.

The representation of an integral operator in wavelet bases is a large numerical sparse matrix. Thus, a suitable approach is needed that converts the matrix from full storage mode into sparse storage mode, which not only saves the storage, but also saves the execution time. In this paper, the storage of this kind of matrix is discussed in the symmetric case. First, we present a standard storage approach. Then, a modified one is proposed, which is based on a special decomposition of the symmetric matrix. When iterative solvers are used to solve the corresponding linear system, the main cost is the multiplication of the symmetric sparse matrix and a given vector. The modified sparse storage mode is optimized for this purpose and reduces nearly half of the number of multiplications.

The energy consumption of supercomputers is one of the critical problems for the upcoming Exascale supercomputing era. The awareness of power an energy consumption is required on both software and hardware side. This poster deals with the energy consumption evaluation of the Total-Finite Element Tearing and Interconnect (TFETI) based solvers [2] of linear systems implemented in PERMON toolbox [1], which is an established method for solving real-world engineering problems, and with the energy consumption evaluation of the BLAS routines. The experiments performed in the poster deal with CPU frequency. This work is performed in the scope of the READEX project (Runtime Exploitation of Application Dynamism for Energy-efficient eXascale computing) [6]. The measurements were performed on the Intel Xeon E5-2680 (Intel Haswell micro-architecture) based Taurus system installed at TU Dresden. The system contains over 1400 nodes that have an FPGA-based power instrumentation called HDEEM (High Definition Energy Efficiency Monitoring), that allows for fine-grained and more accurate power and energy measurements. The measurements can be accessed through the HDEEM library, allowing developers to take energy measurements before and after the region of interest. We have evaluated the effect of the CPU frequency on the energy consumption of the TFETI solver for a linear elasticity 3D cube synthetic benchmark. On the dualized problem MPFX=MPd, we have evaluated the effect of frequency tuning on the energy consumption of the essential processing kernels of the TFETI method. There are two main phases in TFETI — preprocessing and solve. In preprocessing it is necessary to regularize the stiffness matrix K and factorize it and to assemble the G and GGT matrices and the second one to factorize. Both operations belong to the most time and also energy consuming operations. The solve employs the Preconditioned Conjugate Gradient (PCG) algorithm, which consists of sparse matrix-vector multiplications (by F, P, M L , M D matrices) and vector dot products and AXPY functions. In each iteration, we need to apply the direct solver twice, i.e., for forward and backward solves for the pseudoinverse K+ action and for the coarse problem solution, the (GGT)−1 action. The multiplication by the dense Schur complement matrix adds an additional operator with different computational characteristics, potentially increasing the exploitable dynamism. The poster provides results for two types of frequency tuning: (1) static tuning and (2) dynamic tuning. For static tuning experiments, the frequency is set before execution and kept constant during the runtime. For dynamic tuning, the frequency is changed during the program execution to adapt the system to the actual needs of the application. The poster shows that static tuning brings up 11.84% energy savings when compared to default CPU settings (the highest clock rate). The dynamic tuning improves this further by up to 2.68%. In total, the approach presented in this paper shows the potential to save up to 14.52% of energy for TFETI based solvers, see Table1. Another energy consumption evaluations were done with selected Sparse and Dense BLAS Level 1, 2 and 3 routines. For benchmarking we have used a set of matrices from University Florida collection [4]. We have employed AXPY, Sparse Matrix-Vector, Sparse MatrixMatrix, Dense Matrix-Vector, Dense Matrix-Matrix and Sparse Matrix-Dense Matrix multiplication routines from Intel Math Kernel Library (MKL) [3]. The measured characteristics illustrate the different energy consumption of BLAS routines, as some operations are memory-bounded and others are compute-bounded. Based on our recommendations one can explore dynamic frequency switching to achieve significant energy savings up to 23%, for more details see Table 2.