Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

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Iterative Methods for Sparse Linear Systems

http://home.eps.hw.ac.uk/~ab226/tmp/implicit/lele_92.pdf

Compact finite difference schemes with spectral-like resolution

Recently, a number of researchers have investigated a class of graph partitioning algorithms that reduce the size of the graph by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partition for the original graph [Bui and Jones, Proc. of the 6th SIAM Conference on Parallel Processing for Scientific Computing, 1993, 445--452; Hendrickson and Leland, A Multilevel Algorithm for Partitioning Graphs, Tech. report SAND 93-1301, Sandia National Laboratories, Albuquerque, NM, 1993]. From the early work it was clear that multilevel techniques held great promise; however, it was not known if they can be made to consistently produce high quality partitions for graphs arising in a wide range of application domains. We investigate the effectiveness of many different choices for all three phases: coarsening, partition of the coarsest graph, and refinement. In particular, we present a new coarsening heuristic (called heavy-edge heuristic) for which the size of the partition of the coarse graph is within a small factor of the size of the final partition obtained after multilevel refinement. We also present a much faster variation of the Kernighan--Lin (KL) algorithm for refining during uncoarsening. We test our scheme on a large number of graphs arising in various domains including finite element methods, linear programming, VLSI, and transportation. Our experiments show that our scheme produces partitions that are consistently better than those produced by spectral partitioning schemes in substantially smaller time. Also, when our scheme is used to compute fill-reducing orderings for sparse matrices, it produces orderings that have substantially smaller fill than the widely used multiple minimum degree algorithm.

/pdf/a-fast-and-high-quality-multilevel-scheme-for-partitioning-2i240yved4.pdf

A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs

An iterative method is given for solving Ax ~ffi b and minU Ax b 112, where the matrix A is large and sparse. The method is based on the bidiagonalization procedure of Golub and Kahan. It is analytically equivalent to the standard method of conjugate gradients, but possesses more favorable numerical properties. Reliable stopping criteria are derived, along with estimates of standard errors for x and the condition number of A. These are used in the FORTRAN implementation of the method, subroutine LSQR. Numerical tests are described comparing I~QR with several other conjugate-gradient algorithms, indicating that I~QR is the most reliable algorithm when A is ill-conditioned.

/pdf/lsqr-an-algorithm-for-sparse-linear-equations-and-sparse-2ef90mvttx.pdf

LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares

This paper is a survey of the theory and methods of photogrammetric bundle adjustment, aimed at potential implementors in the computer vision community. Bundle adjustment is the problem of refining a visual reconstruction to produce jointly optimal structure and viewing parameter estimates. Topics covered include: the choice of cost function and robustness; numerical optimization including sparse Newton methods, linearly convergent approximations, updating and recursive methods; gauge (datum) invariance; and quality control. The theory is developed for general robust cost functions rather than restricting attention to traditional nonlinear least squares.

/pdf/bundle-adjustment-a-modern-synthesis-449tfs1h4g.pdf

Bundle Adjustment - A Modern Synthesis

Computer solution of large sparse positive definite systems

Let M be a mesh consisting of $n^2 $ squares called elements, formed by subdividing the unit square $(0,1) \times (0,1)$ into $n^2 $ small squares of side ${1 / h}$, and having a node at each of the $(n + 1)^2 $ grid points. With M we associate the $N \times N$ symmetric positive definite system $Ax = b$, where $N = (n + 1)^2 $, each $x_i $ is associated with a node of M, and $A_{ij} \ne 0$ if and only if $x_i $ and $x_j $ are associated with nodes of the same element. If we solve the equations via the standard symmetric factorization of A, then $O(n^4 )$ arithmetic operations are required if the usual row by row (banded) numbering scheme is used, and the storage required is $O(n^3 )$. In this paper we present an unusual numbering of the mesh (unknowns) and show that if we avoid operating on zeros, the $LDL^T $ factorization of A can be computed using the same standard algorithm in $O(n^3 )$ arithmetic operations. Furthermore, the storage required is only $O(n^2 \log _2 n)$. Finally, we prove that all ord...

Nested Dissection of a Regular Finite Element Mesh

Computer Solution of Large Sparse Positive Definite

Solution of sparse linear least squares problems using givens rotations

When reality is modelled by computation, matrices are often the connection between the continuous physical world and the finite algorithmic one. Usually, the more detailed the model, the bigger the matrix; however, efficiency demands that every possible advantage be exploited. The articles in this volume are based on recent research on sparse matrix computations. They examine graph theory as it connects to linear algebra, parallel computing, data structures, geometry and both numerical and discrete algorithms. The articles are grouped into three general categories: graph models of symmetric matrices and factorizations; graph models of algorithms on nonsymmetric matrices; and parallel sparse matrix algorithms.

Alan George

Papers

Computer solution of large sparse positive definite systems

Nested Dissection of a Regular Finite Element Mesh

Computer Solution of Large Sparse Positive Definite

Solution of sparse linear least squares problems using givens rotations

Graph theory and sparse matrix computation