Minimizing Communication for Eigenproblems and the Singular Value Decomposition

Algorithms have two costs: arithmetic and communication. The latter represents the cost of moving data, either between levels of a memory hierarchy, or between processors over a network. Communication often dominates arithmetic and represents a rapidly increasing proportion of the total cost, so we seek algorithms that minimize communication. In \cite{BDHS10} lower bounds were presented on the amount of communication required for essentially all $O(n^3)$-like algorithms for linear algebra, including eigenvalue problems and the SVD. Conventional algorithms, including those currently implemented in (Sca)LAPACK, perform asymptotically more communication than these lower bounds require. In this paper we present parallel and sequential eigenvalue algorithms (for pencils, nonsymmetric matrices, and symmetric matrices) and SVD algorithms that do attain these lower bounds, and analyze their convergence and communication costs.Comment: 43 pages, 11 figure

Fast computation of spectral projectors of banded matrices

We consider the approximate computation of spectral projectors for symmetric banded matrices. While this problem has received considerable attention, especially in the context of linear scaling electronic structure methods, the presence of small relative spectral gaps challenges existing methods based on approximate sparsity. In this work, we show how a data-sparse approximation based on hierarchical matrices can be used to overcome this problem. We prove a priori bounds on the approximation error and propose a fast algo- rithm based on the QDWH algorithm, along the works by Nakatsukasa et al. Numerical experiments demonstrate that the performance of our algorithm is robust with respect to the spectral gap. A preliminary Matlab implementation becomes faster than eig already for matrix sizes of a few thousand.Comment: 27 pages, 10 figure

Structured Eigenvalue Problems

Most eigenvalue problems arising in practice are known to be structured. Structure is often introduced by discretization and linearization techniques but may also be a consequence of properties induced by the original problem. Preserving this structure can help preserve physically relevant symmetries in the eigenvalues of the matrix and may improve the accuracy and efficiency of an eigenvalue computation. The purpose of this brief survey is to highlight these facts for some common matrix structures. This includes a treatment of rather general concepts such as structured condition numbers and backward errors as well as an overview of algorithms and applications for several matrix classes including symmetric, skew-symmetric, persymmetric, block cyclic, Hamiltonian, symplectic and orthogonal matrices

Solving Large Dense Symmetric Eigenproblem on Hybrid Architectures

Dense symmetric eigenproblem is one of the most significant problems in the numerical linear algebra that arises in numerous research fields such as bioinformatics, computational chemistry, and meteorology. In the past years, the problems arising in these fields become bigger than ever resulting in growing demands in both computational power as well as the storage capacities. In such problems, the eigenproblem becomes the main computational bottleneck for which solution is required an extremely high computational power. Modern computing architectures that can meet these growing demands are those that combine the power of the traditional multi-core processors and the general-purpose GPUs and are called hybrid systems. These systems exhibit very high performance when the data fits into the GPU memory ; however, if the volume of the data exceeds the total GPU memory, i.e. the data is out-of-core from the GPU perspective, the performance rapidly decreases. This dissertation is focused on the development of the algorithms that solve dense symmetric eigenproblems on the hybrid GPU-based architectures. In particular, it aims at developing the eigensolvers that exhibit very high performance even if a problem is out- of-core for the GPU. The developed out-of-core eigensolvers are evaluated and compared on real problems that arise in the simulation of molecular motions. In such problems the data, usually too large to fit into the GPU memory, are stored in the main memory and copied to the GPU memory in pieces. That approach results in the performance drop due to a slow interconnection and a high memory latency. To overcome this problem an approach that applies blocking strategy and re- designs the existing eigensolvers, in order to decrease the volume of data transferred and the number of memory transfers, is presented. This approach designs and implements a set of the block- oriented, communication-avoiding BLAS routines that overlap the data transfers with the number of computations performed. Next, these routines are applied to speed-up the following eigensolvers: the solver based on the multi-stage reduction to a tridiagonal form, the Krylov subspace-based method, and the spectral divide-and-conquer method. Although the out-of-core BLAS routines significantly improve the performance of these three eigensolvers, a careful re-design is required in order to tackle the solution of the large eigenproblems on the hybrid CPU-GPU systems. In the out-of-core multi-stage reduction approach, the factor that mostly influences the performance is the band size of the obtained band matrix. On the other hand, the Krylov subspace- based method, although it is based on the memory- bound BLAS-2 operations, is the fastest method if only a small subset of the eigenpairs is required. Finally, the spectral divide-and- conquer algorithm, which exhibits significantly higher arithmetic cost than the other two eigensolvers, achieves extremely high performance since it can be performed completely in terms of the compute-bound BLAS-3 operations. Furthermore, its high arithmetic cost is further reduced by exploiting the special structure of a matrix. Finally, the results presented in the dissertation show that the three out-of-core eigen- solvers, for a set of the specific macromolecular problems, significantly overcome the multi-core variants and attain high flops rate even if data do not fit into the GPU memory. This proves that it is possible to solve large eigenproblems on modest computing systems equipped with a single GPU

ChASE: Chebyshev Accelerated Subspace iteration Eigensolver for sequences of Hermitian eigenvalue problems

Solving dense Hermitian eigenproblems arranged in a sequence with direct solvers fails to take advantage of those spectral properties which are pertinent to the entire sequence, and not just to the single problem. When such features take the form of correlations between the eigenvectors of consecutive problems, as is the case in many real-world applications, the potential benefit of exploiting them can be substantial. We present ChASE, a modern algorithm and library based on subspace iteration with polynomial acceleration. Novel to ChASE is the computation of the spectral estimates that enter in the filter and an optimization of the polynomial degree which further reduces the necessary FLOPs. ChASE is written in C++ using the modern software engineering concepts which favor a simple integration in application codes and a straightforward portability over heterogeneous platforms. When solving sequences of Hermitian eigenproblems for a portion of their extremal spectrum, ChASE greatly benefits from the sequence's spectral properties and outperforms direct solvers in many scenarios. The library ships with two distinct parallelization schemes, supports execution over distributed GPUs, and it is easily extensible to other parallel computing architectures.Comment: 33 pages. Submitted to ACM TOM

Performance analysis and comparison of parallel eigensolvers on Blue Gene architectures

The solution of eigenproblems with dense, symmetric system matrices is a core task in many felds of computational science and engineering. As the problem complexity and thus the size of the matrices involved increases, the application of distributed memory supercomputer architectures and parallel algorithms becomes inevitable. Nearly all modern algorithms for eigensolving implement a tridiagonal reduction of the eigenproblem system matrix and a subsequentsolution of the tridigonalized eigenproblem. Additionally, back transformation of the eigenvectors is required if these are of interest. In the context of this thesis, implementations of two basically different approaches to the parallelsolution of eigenproblems were benchmarked, reviewed and compared with particular regard to their performance on the Blue Gene/P and Blue Gene/Q supercomputers JUGENE and JUQUEEN at the Forschungszentrum Jülich: ELPA, which implements an optimized version of the divide and conquer algorithm and Elemental which utilizes the PMRRR implementation of the MR3 algorithm. ELPA features two fundamentally different kinds of tridiagonalization, the standard one-stage and a two-stage approach. The comparision of thetwo-stage to the direct reduction was a primary concern in the performance analysis