Stable Sparse Orthogonal Factorization of Ill-Conditioned Banded Matrices for Parallel Computing

Sequential and parallel algorithms based on the LU factorization or the QR factorization have been intensely studied and widely used in the problems of computation with large-scale ill-conditioned banded matrices. Great concerns on existing methods include ill-conditioning, sparsity of factor matrices, computational complexity, and scalability. In this dissertation, we study a sparse orthogonal factorization of a banded matrix motivated by parallel computing. Specifically, we develop a process to factorize a banded matrix as a product of a sparse orthogonal matrix and a sparse matrix which can be transformed to an upper triangular matrix by column permutations. We prove that the proposed process requires low complexity, and it is numerically stable, maintaining similar stability results as the modified Gram-Schmidt process. On this basis, we develop a parallel algorithm for the factorization in a distributed computing environment. Through an analysis of its performance, we show that the communication costs reach the theoretical least upper bounds, while its parallel complexity or speedup approaches the optimal bound. For an ill-conditioned banded system, we construct a sequential solver that breaks it down into small-scale underdetermined systems, which are solved by the proposed factorization with high accuracy. We also implement a parallel solver with strategies to treat the memory issue appearing in extra large-scale linear systems of size over one billion. Numerical experiments confirm the theoretical results derived in this thesis, and demonstrate the superior accuracy and scalability of the proposed solvers for ill-conditioned linear systems, comparing to the most commonly used direct solvers

ELSI: A Unified Software Interface for Kohn-Sham Electronic Structure Solvers

Solving the electronic structure from a generalized or standard eigenproblem is often the bottleneck in large scale calculations based on Kohn-Sham density-functional theory. This problem must be addressed by essentially all current electronic structure codes, based on similar matrix expressions, and by high-performance computation. We here present a unified software interface, ELSI, to access different strategies that address the Kohn-Sham eigenvalue problem. Currently supported algorithms include the dense generalized eigensolver library ELPA, the orbital minimization method implemented in libOMM, and the pole expansion and selected inversion (PEXSI) approach with lower computational complexity for semilocal density functionals. The ELSI interface aims to simplify the implementation and optimal use of the different strategies, by offering (a) a unified software framework designed for the electronic structure solvers in Kohn-Sham density-functional theory; (b) reasonable default parameters for a chosen solver; (c) automatic conversion between input and internal working matrix formats, and in the future (d) recommendation of the optimal solver depending on the specific problem. Comparative benchmarks are shown for system sizes up to 11,520 atoms (172,800 basis functions) on distributed memory supercomputing architectures.Comment: 55 pages, 14 figures, 2 table

A bibliography on parallel and vector numerical algorithms

This is a bibliography of numerical methods. It also includes a number of other references on machine architecture, programming language, and other topics of interest to scientific computing. Certain conference proceedings and anthologies which have been published in book form are listed also

Solution of partial differential equations on vector and parallel computers

The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed

The scheduling of sparse matrix-vector multiplication on a massively parallel dap computer

An efficient data structure is presented which supports general unstructured sparse matrix-vector multiplications on a Distributed Array of Processors (DAP). This approach seeks to reduce the inter-processor data movements and organises the operations in batches of massively parallel steps by a heuristic scheduling procedure performed on the host computer. The resulting data structure is of particular relevance to iterative schemes for solving linear systems. Performance results for matrices taken from well known Linear Programming (LP) test problems are presented and analysed

Lanczos eigensolution method for high-performance computers

The theory, computational analysis, and applications are presented of a Lanczos algorithm on high performance computers. The computationally intensive steps of the algorithm are identified as: the matrix factorization, the forward/backward equation solution, and the matrix vector multiples. These computational steps are optimized to exploit the vector and parallel capabilities of high performance computers. The savings in computational time from applying optimization techniques such as: variable band and sparse data storage and access, loop unrolling, use of local memory, and compiler directives are presented. Two large scale structural analysis applications are described: the buckling of a composite blade stiffened panel with a cutout, and the vibration analysis of a high speed civil transport. The sequential computational time for the panel problem executed on a CONVEX computer of 181.6 seconds was decreased to 14.1 seconds with the optimized vector algorithm. The best computational time of 23 seconds for the transport problem with 17,000 degs of freedom was on the the Cray-YMP using an average of 3.63 processors

Load-balanced parallel banded-system solvers

AbstractSolving banded systems is important in the applications of science and engineering. This paper presents a load-balancing strategy for solving banded systems in parallel when the number of processors used is small. An optimization-based load-balancing analysis is given to determine how many loads should be assigned to each processor in order to minimize the time requirement. Some experimentations are carried out on the nCUBE 2E multiprocessor to demonstrate the speedup advantage of the proposed load-balancing strategy. The speedup improvement ratio ranges from 47% to 66% (from 12% to 24%) when using 4 (8) processors

Distributed Finite Element Analysis Using a Transputer Network

The principal objective of this research effort was to demonstrate the extraordinarily cost effective acceleration of finite element structural analysis problems using a transputer-based parallel processing network. This objective was accomplished in the form of a commercially viable parallel processing workstation. The workstation is a desktop size, low-maintenance computing unit capable of supercomputer performance yet costs two orders of magnitude less. To achieve the principal research objective, a transputer based structural analysis workstation termed XPFEM was implemented with linear static structural analysis capabilities resembling commercially available NASTRAN. Finite element model files, generated using the on-line preprocessing module or external preprocessing packages, are downloaded to a network of 32 transputers for accelerated solution. The system currently executes at about one third Cray X-MP24 speed but additional acceleration appears likely. For the NASA selected demonstration problem of a Space Shuttle main engine turbine blade model with about 1500 nodes and 4500 independent degrees of freedom, the Cray X-MP24 required 23.9 seconds to obtain a solution while the transputer network, operated from an IBM PC-AT compatible host computer, required 71.7 seconds. Consequently, the $80,000 transputer network demonstrated a cost-performance ratio about 60 times better than the$15,000,000 Cray X-MP24 system