Abstract not found

QR factorization is most often used as a “black box ” algorithm, but is in fact an elegant computation on a factor graph. By computing a rooted clique tree on this graph, the computation can be parallelized across subtrees, which forms the basis of so-called multifrontal QR methods. By judiciously choosing the order in which variables are eliminated in the clique tree computation, we show that one straightforwardly obtains a method for performing inference in distributed sensor networks. One obvious application is distributed localization and mapping with a team of robots. We phrase the problem as inference on a large-scale Gaussian Markov Random Field induced by the measurement factor graph, and show how multifrontal QR on this graph solves for the global map and all the robot poses in a distributed fashion. The method is illustrated using both small and large-scale simulations, and validated in practice through actual robot experiments.

Abstract. We propose and analyze a new tool to help solve sparse linear least-squares problems minx �Ax − b�2. Our method is based on asparseQR factorization of a low-rank perturbation Ä of A. Moreprecisely, we show that the R factor of Ä is an effective preconditioner for the least-squares problem minx �Ax−b�2, when solved using LSQR. We propose applications for the new technique. When A is rank deficient we can add rows to ensure that the preconditioner is well-conditioned without column pivoting. When A is sparse except for a few dense rows we can drop these dense rows from A to obtain Ä. Another application is solving an updated or downdated problem. If R is a good preconditioner for the original problem A, it is a good preconditioner for the updated/downdated problem Ä. We can also solve what-if scenarios, where we want to find the solution if a column of the original matrix is changed/removed. We present a spectral theory that analyzes the generalized spectrum of the pencil (A ∗ A, R ∗ R) and analyze the applications. 1.

SuiteSparseQR is a sparse QR factorization package based on the multifrontal method. Within each frontal matrix, LAPACK and the multithreaded BLAS enable the method to obtain high performance on multicore architectures. Parallelism across different frontal matrices is handled with Intel’s Threading Building Blocks library. The symbolic analysis and ordering phase preeliminates singletons by permuting the input matrix into the form [R11 R12; 0 A22] where R11 is upper triangular with diagonal entries above a given tolerance. Next, the fill-reducing ordering, column elimination tree, and frontal matrix structures are found without requiring the formation of the pattern of A T A. Rank-detection is performed within each frontal matrix using Heath’s method, which does not require column pivoting. The resulting sparse QR factorization obtains a substantial fraction of the theoretical peak performance of a multicore computer.

This paper studies a primal-dual interior/exterior-point path-following approach for linearprogramming that is motivated on using an iterative solver rather than a direct solver for the search direction. We begin with the usual perturbed primal-dual optimality equations Fu(x, y, z) = 0. Under nondegeneracy assumptions, this nonlinear system is well-posed,i.e. it has a nonsingular Jacobian at optimality and is not necessarily ill-conditioned as the iterates approach optimality. We use a simple preprocessing step to eliminate boththe primal and dual feasibility equations. This results in a single bilinear equation that maintains the well-posedness property. We then apply both a direct solution techniqueas well as a preconditioned conjugate gradient method (PCG), within an inexact Newton framework, directly on the linearized equations. This is done without forming the usualnormal equations, NEQ, or augmented system. Sparsity is maintained. The work of aniteration for the PCG approach consists almost entirely in the (approximate) solution of this well-posed linearized system. Therefore, improvements depend on efficient preconditioning.

In this report a new version of the multifrontal sparse QR factorization routine sqr, originally by Matstoms, for general sparse matrices is described and evaluated. In the previous version the orthogonal factor Q is discarded due to storage considerations. The new version provides Q and uses the multifrontal structure to store this orthogonal factor in a compact way. A new data class with overloaded operators is implemented in Matlab to provide an easy usage of the compact orthogonal factors. This implicit way of storing the orthogonal factor also results in faster computation and application of Q and Q T . Examples are given, where the new version is up to four times faster when computing only R and up to 1000 times faster when computing both Q and R, than the built-in function qr in Matlab. The sqr package is available at URL: http://www.mai.liu.se/~milun/sls/. Key words: QR factorization, sparse problems, multifrontal method, orthogonal factorization. 1 Introduction. Let A 2 IR...