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

In this paper, we analyze the main features and discuss the tuning of the algorithms for the direct solution of sparse linear systems on distributed memory computers developed in the context of a long term European research project. The algorithms use a multifrontal approach and are especially designed to cover a large class of problems. The problems can be symmetric positive definite, general symmetric, or unsymmetric matrices, both possibly rank deficient, and they can be provided by the user in several formats. The algorithms achieve high performance by exploiting parallelism coming from the sparsity in the problem and that available for dense matrices. The algorithms use a dynamic distributed task scheduling technique to accommodate numerical pivoting and to allow the migration of computational tasks to lightly loaded processors. Large computational tasks are divided into subtasks to enhance parallelism. Asynchronous communication is used throughout the solution process to efficiently overlap communication with computation.
We illustrate our design choices by experimental results obtained on an SGI Origin 2000 and an IBM SP2 for test matrices provided by industrial partners in the PARASOL project.

/pdf/a-fully-asynchronous-multifrontal-solver-using-distributed-1xlwbmzqj5.pdf

A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling

The subject of sparse matrices has its roots in such diverse fields as management science, power systems analysis, surveying, circuit theory, and structural analysis. Mathematical models in all these areas give rise to very large systems of linear equations which could not be solved were it not for the fact that the matrices contain relatively few non-zero entries. Only comparatively recently, in the last fifteen years or so, has it become apparent that the equations can be solved even when the pattern is irregular, and it is primarily the solution of such problems that is considered in this book. The subject is intensely practical and this book is written with practicalities ever in mind. Whenever two methods are applicable, their rival merits are considered, and conclusions are based on practical experience where theoretical comparison is not possible. Non-numeric computing techniques have been included, as well as frequent illustrations, in an attempt to bridge the usually wide gap between the printed page and the working computer code. Despite this practical bias, it is recognized that many aspects of the subject are of interest in their own right, and the book aims to be suitable also for a student course, probably at MSc level. Exercises have been included to illustrate and strengthen understanding of the material, as well as to extend it. Efficient use of sparsity is a key to solving large problems in many fields. This book will supply both insight and answers for those attempting to solve these problems.

Direct methods for sparse matrices

We investigate several ways to improve the performance of sparse LU factorization with partial pivoting, as used to solve unsymmetric linear systems. We introduce the notion of unsymmetric supernodes to perform most of the numerical computation in dense matrix kernels. We introduce unsymmetric supernode-panel updates and two-dimensional data partitioning to better exploit the memory hierarchy. We use Gilbert and Peierls's depth-first search with Eisenstat and Liu's symmetric structural reductions to speed up symbolic factorization. 
We have developed a sparse LU code using all these ideas. We present experiments demonstrating that it is significantly faster than earlier partial pivoting codes. We also compare its performance with UMFPACK, which uses a multifrontal approach; our code is very competitive in time and storage requirements, especially for large problems.

/pdf/a-supernodal-approach-to-sparse-partial-pivoting-30rkhd31hw.pdf

A Supernodal Approach to Sparse Partial Pivoting

We present the main algorithmic features in the software package SuperLU_DIST, a distributed-memory sparse direct solver for large sets of linear equations. We give in detail our parallelization strategies, with a focus on scalability issues, and demonstrate the software's parallel performance and scalability on current machines. The solver is based on sparse Gaussian elimination, with an innovative static pivoting strategy proposed earlier by the authors. The main advantage of static pivoting over classical partial pivoting is that it permits a priori determination of data structures and communication patterns, which lets us exploit techniques used in parallel sparse Cholesky algorithms to better parallelize both LU decomposition and triangular solution on large-scale distributed machines.

/pdf/superlu-dist-a-scalable-distributed-memory-sparse-direct-34h5mah2x2.pdf

SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems

This article deals with the problem of factoring a large sparse positive definite matrix on a multiprocessor system. The processors are assumed to have substantial local memory but no globally shared memory. They communicate among themselves and with a host processor through message passing. Our primary interest is in designing an algorithm which exploits parallelism, rather than in exploiting features of the underlying topology of the hardware. However, part of our study is aimed at determining, for certain sparse matrix problems, whether hardware based on the binary hypercube topology adequately supports the communication requirements for such problems. Numerical results from experiments conducted on a hypercube multiprocessor are included.

Sparse Cholesky factorization on a local-memory multiprocessor

Let $Ax = b$ be a large sparse nonsingular system of linear equations to be solved using Gaussian elimination with partial pivoting. The factorization obtained can be expressed in the form $A = P_1 M_1 P_2 M_2 \cdots P_{n - 1} M_{n - 1} U$, where $P_k $ is an elementary permutation matrix reflecting the row interchange that occurs at step k during the factorization, $M_k $ is a unit lower triangular matrix whose kth column contains the multipliers, and U is an upper triangular matrix.Consider the kth step of the elimination. Suppose we replace the structure of row k of the partially reduced matrix by the union of the structures of those rows which are candidates for the pivot row and then perform symbolically Gaussian elimination without partial pivoting. Assume that this is done at each step k, and let $\bar L$ and $\bar U$ denote the resulting lower and upper triangular matrices, respectively. Then the structures of $\bar L$ and $\bar U$, respectively, contain the structures of $\Sigma _{k = 1}^{n - 1} M_k $ and U. This paper describes an algorithm which determines the structures of $\bar L$ and $\bar U$, and sets up an efficient data structure for them. Since the algorithm depends only on the structure of A, the data structure can be created in advance of the actual numerical computation, which can then be performed very efficiently using the fixed storage scheme. Although the data structure is more generous than it needs to be for any specific sequence $P_1 ,P_2 , \cdots ,P_{n - 1} $, experiments indicate that the approach is competitive with conventional methods. Another important point is that the storage scheme is large enough to accommodate the $QR$ factorization of A, so it is also useful in the context of computing a sparse orthogonal decomposition of A. The algorithm is shown to execute in time bounded by $|\bar L| + |\bar U|$, where $|M|$ denotes the number of nonzeros in the matrix M.

Symbolic factorization for sparse Gaussian elimination with partial pivoting

A row-oriented implementation of Gaussian elimination with partial pivoting on a local-memory multiprocessor is described. In the absence of pivoting, the initial data loading of the node processors leads to a balanced computation. However, if interchanges occur, the computational loads on the processors may become unbalanced, leading to inefficiency. A simple load-balancing scheme is described which is inexpensive and which maintains computational balance in the presence of pivoting. Using some reasonable assumptions about the probability of pivoting occurring, an analysis of the communication costs of the algorithm is developed, along with an analysis of the computation performed in each node processor. This model is then used to derive the expected speedup of the algorithm. Finally, experiments using an Intel hypercube are presented in order to demonstrate the extent to which the analytical model predicts the performance.

Gaussian elimination with partial pivoting and load balancing on a multiprocessor

Algorithms and software for solving sparse symmetric positive definite systems on serial computers have reached a high state of development. In this paper, we present algorithms for performing sparse Cholesky factorization and sparse triangular solutions on a shared-memory multiprocessor computer, along with some numerical experiments demonstrating their performance on a Sequent Balance 8000 system.

Solution of sparse positive definite systems on a shared-memory multiprocessor

For a general m by n sparse matrix A, a new scheme is proposed for the structural representation of the factors of its sparse orthogonal decomposition by Householder transformations. The storage scheme is row-oriented and is based on the structure of the upper triangular factor obtained in the decomposition. The storage of the orthogonal matrix factor is particularly efficient in that the overhead required is only $m + n$ items, independent of the actual number of nonzeros in the factor. The same scheme is applicable to sparse orthogonal factorization by Givens rotations, and also to the recent implementation of sparse Gaussian elimination with partial pivoting developed by George and Ng (this Journal, 1987, to appear). Experimental results are provided to compare the sparse Gaussian elimination using the new storage scheme with that proposed by George and Ng.

A data structure for sparse QR and LU factorizations

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