Extending Constraint Preconditioners for Saddle Point Problems

The problem of finding good preconditioners for the numerical solution of a certain important class of indefinite linear systems is considered. These systems are of a block 2 by 2 saddle point structure. In "Constraint preconditioning for indefinite linear systems" SIAM J. Matrix Anal. Appl., 21 (2000), Keller, Gould and Wathen introduced the idea of using constraint preconditioners that have a specific 2 by 2 block structure for the case of the (2,2) matrix block being zero. We shall extend this idea by allowing the (2,2) block to be non-zero. Results concerning the spectrum and form of the eigenvectors are presented, as are numerical results to validate our conclusions

Incomplete factorization constraint preconditioners for saddle-point matrices

We consider the application of the conjugate gradient method to the solution of large symmetric, indefinite linear systems. Special emphasis is put on the use of constraint preconditioners and a new factorization that can reduce the number of flops required by the preconditioning step. Results concerning the eigenvalues of the preconditioned matrix and its minimum polynomial are given. Numerical experiments validate these conclusions

Optimal solvers for PDE-Constrained Optimization

Optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering, in particular in problems of design. The solution of such PDE-constrained optimization problems is usually a major computational task. Here we consider simple problems of this type: distributed control problems in which the 2- and 3-dimensional Poisson problem is the PDE. The large dimensional linear systems which result from discretization and which need to be solved are of saddle-point type. We introduce two optimal preconditioners for these systems which lead to convergence of symmetric Krylov subspace iterative methods in a number of iterations which does not increase with the dimension of the discrete problem. These preconditioners are block structured and involve standard multigrid cycles. The optimality of the preconditioned iterative solver is proved theoretically and verified computationally in several test cases. The theoretical proof indicates that these approaches may have much broader applicability for other partial differential equations

On implicit-factorization constraint preconditioners

Recently Dollar and Wathen [14] proposed a class of incomplete factorizations for saddle-point problems, based upon earlier work by Schilders [40]. In this paper, we generalize this class of preconditioners, and examine the spectral implications of our approach. Numerical tests indicate the efficacy of our preconditioners

On solving trust-region and other regularised subproblems in optimization

The solution of trust-region and regularisation subproblems which arise in unconstrained optimization is considered. Building on the pioneering work of Gay, Mor´e and Sorensen, methods which obtain the solution of a sequence of parametrized linear systems by factorization are used. Enhancements using high-order polynomial approximation and inverse iteration ensure that the resulting method is both globally and asymptotically at least superlinearly convergent in all cases, including in the notorious hard case. Numerical experiments validate the effectiveness of our approach. The resulting software is available as packages TRS and RQS as part of the GALAHAD optimization library, and is especially designed for large-scale problems

Using constraint preconditioners with regularized saddle-point problems

The problem of finding good preconditioners for the numerical solution of a certain important class of indefinite linear systems is considered. These systems are of a 2 by 2 block (KKT) structure in which the (2,2) block (denoted by -C) is assumed to be nonzero. In Constraint preconditioning for indefinite linear systems , SIAM J. Matrix Anal. Appl., 21 (2000), Keller, Gould and Wathen introduced the idea of using constraint preconditioners that have a specific 2 by 2 block structure for the case of C being zero. We shall give results concerning the spectrum and form of the eigenvectors when a preconditioner of the form considered by Keller, Gould and Wathen is used but the system we wish to solve may have C \neq 0 . In particular, the results presented here indicate clustering of eigenvalues and, hence, faster convergence of Krylov subspace iterative methods when the entries of C are small; such situations arise naturally in interior point methods for optimization and we present results for such problems which validate our conclusions.\ud \ud The first author's work was supported by the OUCL Doctorial Training Accoun

A Bramble-Pasciak-like method with applications in optimization

Saddle-point systems arise in many applications areas, in fact in any situation where an extremum principle arises with constraints. The Stokes problem describing slow viscous flow of an incompressible fluid is a classic example coming from partial differential equations and in the area of Optimization such problems are ubiquitous.\ud In this manuscript we show how new approaches for the solution of saddle-point systems arising in Optimization can be derived from the Bramble-Pasciak Conjugate Gradient approach widely used in PDEs and more recent generalizations thereof. In particular we derive a class of new solution methods based on the use of Preconditioned Conjugate Gradients in non-standard inner products and demonstrate how these can be understood through more standard machinery. We show connections to Constraint Preconditioning and give the results of numerical computations on a number of standard Optimization test examples

The Persistent Southern Disadvantage in Us Early Life Mortality, 1965‒2014

Background: Recent studies of US adult mortality demonstrate a growing disadvantage among southern states. Few studies have examined long-term trends and geographic patterns in US early life (ages 1 to 24) mortality, ages at which key risk factors and causes of death are quite different than among adults. Objective: This article examines trends and variations in early life mortality rates across US states and census divisions. We assess whether those variations have changed over a 50-year time period and which causes of death contribute to contemporary geographic disparities. Methods: We calculate all-cause and cause-specific death rates using death certificate data from the Multiple Cause of Death files, combining public-use files from 1965‒2004 and restricted data with state geographic identifiers from 2005‒2014. State population (denominator) data come from US decennial censuses or intercensal estimates. Results: Results demonstrate a persistent mortality disadvantage for young people (ages 1 to 24) living in southern states over the last 50 years, particularly those located in the East South Central and West South Central divisions. Motor vehicle accidents and homicide by firearm account for most of the contemporary southern disadvantage in US early life mortality. Contribution: Our results illustrate that US children and youth living in the southern United States have long suffered from higher levels of mortality than children and youth living in other parts of the country. Our findings also suggest the contemporary southern disadvantage in US early life mortality could potentially be reduced with state-level policies designed to prevent deaths involving motor vehicles and firearms

The Unexpected Role of Evolving Longitudinal Electric Fields in Generating Energetic Electrons in Relativistically Transparent Plasmas

Superponderomotive-energy electrons are observed experimentally from the interaction of an intense laser pulse with a relativistically transparent target. For a relativistically transparent target, kinetic modeling shows that the generation of energetic electrons is dominated by energy transfer within the main, classically overdense, plasma volume. The laser pulse produces a narrowing, funnel-like channel inside the plasma volume that generates a field structure responsible for the electron heating. The field structure combines a slowly evolving azimuthal magnetic field, generated by a strong laser-driven longitudinal electron current, and, unexpectedly, a strong propagating longitudinal electric field, generated by reflections off the walls of the funnel-like channel. The magnetic field assists electron heating by the transverse electric field of the laser pulse through deflections, whereas the longitudinal electric field directly accelerates the electrons in the forward direction. The longitudinal electric field produced by reflections is 30 times stronger than that in the incoming laser beam and the resulting direct laser acceleration contributes roughly one third of the energy transferred by the transverse electric field of the laser pulse to electrons of the super-ponderomotive tail

Learning to Segments Objects Candidates

Recent object detection systems rely on two critical steps: (1) a set of object proposals is predicted as efficiently as possible, and (2) this set of candidate proposals is then passed to an object classifier. Such approaches have been shown they can be fast, while achieving the state of the art in detection performance. In this paper, we propose a new way to generate object proposals, introducing an approach based on a discriminative convolutional network. Our model is trained jointly with two objectives: given an image patch, the first part of the system outputs a class-agnostic segmentation mask, while the second part of the system outputs the likelihood of the patch being centered on a full object. At test time, the model is efficiently applied on the whole test image and generates a set of segmentation masks, each of them being assigned with a corresponding object likelihood score. We show that our model yields significant improvements over state-of-the-art object proposal algorithms. In particular, compared to previous approaches, our model obtains substantially higher object recall using fewer proposals. We also show that our model is able to generalize to unseen categories it has not seen during training. Unlike all previous approaches for generating object masks, we do not rely on edges, superpixels, or any other form of low-level segmentation