Newton-Picard Gauss-Seidel

Newton-Picard methods are iterative methods that work well for computing roots of nonlinear equations within a continuation framework. This project presents one of these methods and includes the results of a computation involving the Brusselator problem performed by an implementation of the method. This work was done in collaboration with Andrew Salinger at Sandia National Laboratories

Inexact Newton Methods Applied to Under-Determined Systems

Consider an under-determined system of nonlinear equations F(x)=0, F:R^m→R^n, where F is continuously differentiable and m \u3e n. This system appears in a variety of applications, including parameter-dependent systems, dynamical systems with periodic solutions, and nonlinear eigenvalue problems. Robust, efficient numerical methods are often required for the solution of this system. Newton\u27s method is an iterative scheme for solving the nonlinear system of equations F(x)=0, F:R^n→R^n. Simple to implement and theoretically sound, it is not, however, often practical in its pure form. Inexact Newton methods and globalized inexact Newton methods are computationally efficient variations of Newton\u27s method commonly used on large-scale problems. Frequently, these variations are more robust than Newton\u27s method. Trust region methods, thought of here as globalized exact Newton methods, are not as computationally efficient in the large-scale case, yet notably more robust than Newton\u27s method in practice. The normal flow method is a generalization of Newton\u27s method for solving the system F:R^m→R^n, m \u3e n. Easy to implement, this method has a simple and useful local convergence theory; however, in its pure form, it is not well suited for solving large-scale problems. This dissertation presents new methods that improve the efficiency and robustness of the normal flow method in the large-scale case. These are developed in direct analogy with inexact-Newton, globalized inexact-Newton, and trust-region methods, with particular consideration of the associated convergence theory. Included are selected problems of interest simulated in MATLAB

A Numerical Study of Globalizations of Newton-GMRES Methods

Newton\u27s method is at the core of many algorithms used for solving nonlinear equations. A globalized Newton method is an implementation of Newton\u27s method augmented with ``globalization procedures\u27 intended to enhance the likelihood of convergence to a solution from an arbitrary initial guess. A Newton-GMRES method is an implementation of Newton\u27s method in which the iterative linear algebra method GMRES is used to solve approximately the linear system that characterizes the Newton step. A globalized Newton-GMRES method combines both globalization procedures and the GMRES scheme to develop robust and efficient algorithms for solving nonlinear equations. The aim of this project is to describe the development of some globalized Newton-GMRES methods and to compare their performances on a few benchmark fluid flow problems

Functional implications of the intertarsal joint shape in a terrestrial ( Coturnix coturnix ) versus a semi-aquatic bird ( Callonetta leucophrys )

International audienceAs birds have a diversity of locomotor behaviors, their skeleton is subjected to a variety of mechanical constraints (gravitational, aerodynamic and sometimes hydrodynamic forces). Yet, only minor modifications in post-cranial skeleton shape are observed across the diversity of avian species in comparison with other vertebrates. The goal of this study was to explore potential morphological adjustments that allow locomotion in different habitats in Anatidae. Specifically, we compared a strictly terrestrial bird, the common quail Coturnix coturnix, and a semi-aquatic bird, the ringed teal Callonetta leucophrys, to explore whether their anatomy reflects the constraints of locomotion in different habitats (water vs. land). We compared the tibiotarsus and the tarsometatarsus shape between the two species using a geometric morphometric approach. Our data illustrate distinct differences between species with a more medially oriented intertarsal joint in the ringed teal than in the common quail, which may be linked to the kinematics of walking and paddling. This study lays the foundations to understand the functional requirements for moving in both terrestrial and aquatic environments in Anatidae, and suggests morphological characteristics of the bird hindlimb skeleton that may help to predict the motions it is capable of

Optimal General Matchings

Given a graph $G=(V,E)$ and for each vertex $v \in V$ a subset $B(v)$ of the set $\{0,1,\ldots, d_G(v)\}$, where $d_G(v)$ denotes the degree of vertex $v$ in the graph $G$, a $B$-factor of $G$ is any set $F \subseteq E$ such that $d_F(v) \in B(v)$ for each vertex $v$, where $d_F(v)$ denotes the number of edges of $F$ incident to $v$. The general factor problem asks the existence of a $B$-factor in a given graph. A set $B(v)$ is said to have a {\em gap of length} $p$ if there exists a natural number $k \in B(v)$ such that $k+1, \ldots, k+p \notin B(v)$ and $k+p+1 \in B(v)$. Without any restrictions the general factor problem is NP-complete. However, if no set $B(v)$ contains a gap of length greater than $1$, then the problem can be solved in polynomial time and Cornuejols \cite{Cor} presented an algorithm for finding a $B$-factor, if it exists. In this paper we consider a weighted version of the general factor problem, in which each edge has a nonnegative weight and we are interested in finding a $B$-factor of maximum (or minimum) weight. In particular, this version comprises the minimum/maximum cardinality variant of the general factor problem, where we want to find a $B$-factor having a minimum/maximum number of edges. We present an algorithm for the maximum/minimum weight $B$-factor for the case when no set $B(v)$ contains a gap of length greater than $1$. This also yields the first polynomial time algorithm for the maximum/minimum cardinality $B$-factor for this case

Construction and Performance of a Micro-Pattern Stereo Detector with Two Gas Electron Multipliers

The construction of a micro-pattern gas detector of dimensions 40x10 cm**2 is described. Two gas electron multiplier foils (GEM) provide the internal amplification stages. A two-layer readout structure was used, manufactured in the same technology as the GEM foils. The strips of each layer cross at an effective crossing angle of 6.7 degrees and have a 406 um pitch. The performance of the detector has been evaluated in a muon beam at CERN using a silicon telescope as reference system. The position resolutions of two orthogonal coordinates are measured to be 50 um and 1 mm, respectively. The muon detection efficiency for two-dimensional space points reaches 96%.Comment: 21 pages, 17 figure

Effect of the disorder in graphene grain boundaries: A wave packet dynamics study

Chemical vapor deposition (CVD) on Cu foil is one of the most promising methods to produce graphene samples despite of introducing numerous grain boundaries into the perfect graphene lattice. A rich variety of GB structures can be realized experimentally by controlling the parameters in the CVD method. Grain boundaries contain non-hexagonal carbon rings (4, 5, 7, 8 membered rings) and vacancies in various ratios and arrangements. Using wave packet dynamic (WPD) simulations and tight-binding electronic structure calculations, we have studied the effect of the structure of GBs on the transport properties. Three model GBs with increasing disorder were created in the computer: a periodic 5-7 GB, a "serpentine" GB, and a disordered GB containing 4, 8 membered rings and vacancies. It was found that for small energies (E = EF ± 1 eV) the transmission decreases with increasing disorder. Four membered rings and vacancies are identified as the principal scattering centers. Revealing the connection between the properties of GBs and the CVD growth method may open new opportunities in the graphene based nanoelectronics. © 2013 Elsevier B.V. All rights reserved

Theoretical analysis of the role of chromatin interactions in long-range action of enhancers and insulators

Long-distance regulatory interactions between enhancers and their target genes are commonplace in higher eukaryotes. Interposed boundaries or insulators are able to block these long distance regulatory interactions. The mechanistic basis for insulator activity and how it relates to enhancer action-at-a-distance remains unclear. Here we explore the idea that topological loops could simultaneously account for regulatory interactions of distal enhancers and the insulating activity of boundary elements. We show that while loop formation is not in itself sufficient to explain action at a distance, incorporating transient non-specific and moderate attractive interactions between the chromatin fibers strongly enhances long-distance regulatory interactions and is sufficient to generate a euchromatin-like state. Under these same conditions, the subdivision of the loop into two topologically independent loops by insulators inhibits inter-domain interactions. The underlying cause of this effect is a suppression of crossings in the contact map at intermediate distances. Thus our model simultaneously accounts for regulatory interactions at a distance and the insulator activity of boundary elements. This unified model of the regulatory roles of chromatin loops makes several testable predictions that could be confronted with \emph{in vitro} experiments, as well as genomic chromatin conformation capture and fluorescent microscopic approaches.Comment: 10 pages, originally submitted to an (undisclosed) journal in May 201

Cones, pringles, and grain boundary landscapes in graphene topology

A polycrystalline graphene consists of perfect domains tilted at angle {\alpha} to each other and separated by the grain boundaries (GB). These nearly one-dimensional regions consist in turn of elementary topological defects, 5-pentagons and 7-heptagons, often paired up into 5-7 dislocations. Energy G({\alpha}) of GB computed for all range 0<={\alpha}<=Pi/3, shows a slightly asymmetric behavior, reaching ~5 eV/nm in the middle, where the 5's and 7's qualitatively reorganize in transition from nearly armchair to zigzag interfaces. Analysis shows that 2-dimensional nature permits the off-plane relaxation, unavailable in 3-dimensional materials, qualitatively reducing the energy of defects on one hand while forming stable 3D-landsapes on the other. Interestingly, while the GB display small off-plane elevation, the random distributions of 5's and 7's create roughness which scales inversely with defect concentration, h ~ n^(-1/2)Comment: 9 pages, 4 figure