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Mathematics

Portland State University

2011

Articles 1 - 13 of 13

Full-Text Articles in Physical Sciences and Mathematics

Reliable A-Posteriori Error Estimators For Hp-Adaptive Finite Element Approximations Of Eigenvalue/Leigenvector Problems, Stefano Giani, Luka Grubisic, Jeffrey S. Ovall Dec 2011

Reliable A-Posteriori Error Estimators For Hp-Adaptive Finite Element Approximations Of Eigenvalue/Leigenvector Problems, Stefano Giani, Luka Grubisic, Jeffrey S. Ovall

Mathematics and Statistics Faculty Publications and Presentations

We present reliable a-posteriori error estimates for hp-adaptive finite element approxima- tions of eigenvalue/eigenvector problems. Starting from our earlier work on h adaptive finite element approximations we show a way to obtain reliable and efficient a-posteriori estimates in the hp-setting. At the core of our analysis is the reduction of the problem on the analysis of the associated boundary value problem. We start from the analysis of Wohlmuth and Melenk and combine this with our a-posteriori estimation framework to obtain eigenvalue/eigenvector approximation bounds.

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A Review On Convolutions Of Gamma Random Variables, Baha-Eldin Khaledi, Subhash C. Kochar Dec 2011

A Review On Convolutions Of Gamma Random Variables, Baha-Eldin Khaledi, Subhash C. Kochar

Mathematics and Statistics Faculty Publications and Presentations

Due to its wide range of applications, the topic of the distribution theory of convolutions of Gamma random variables has attracted the attention of many researchers. In this paper we review some of the latest developments on this problem.

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Biomimetic Broadband Antireflection Gratings On Solar-Grade Multicrystalline Silicon Wafers, Blayne M. Phillips, Peng Jiang, Bin Jiang Nov 2011

Biomimetic Broadband Antireflection Gratings On Solar-Grade Multicrystalline Silicon Wafers, Blayne M. Phillips, Peng Jiang, Bin Jiang

Mathematics and Statistics Faculty Publications and Presentations

The authors report a simple and scalable bottom-up technique for fabricating broadband antireflection gratings on solar-grade multicrystalline silicon (mc-Si) wafers. A Langmuir-Blodgett process is developed to assemble close-packed silica microspheres on rough mc-Si substrates. Subwavelength moth-eye pillars can then be patterned on mc-Si by using the silica microspheres as structural template. Hemispherical reflectance measurements show that the resulting mc-Si gratings exhibit near zero reflection for a wide range of wavelengths. Both experimental results and theoretical prediction using a rigorous coupled-wave analysis model show that close-packed moth-eye arrays exhibit better antireflection performance than non-close-packed arrays due to a smoother refractive index …

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Determination Of The Electric Field Intensity And Space Charge Density Versus Height Prior To Triggered Lightning, Christopher J. Biagi, Martin A. Uman, Jay Gopalakrishnan, J. D. Hill, Vladimir A. Rakov, T. Ngin, Douglas M. Jordan Aug 2011

Determination Of The Electric Field Intensity And Space Charge Density Versus Height Prior To Triggered Lightning, Christopher J. Biagi, Martin A. Uman, Jay Gopalakrishnan, J. D. Hill, Vladimir A. Rakov, T. Ngin, Douglas M. Jordan

Mathematics and Statistics Faculty Publications and Presentations

We infer the vertical profiles of space charge density and electric field intensity above ground by comparing modeling and measurements of the ground-level electric field changes caused by elevating grounded lightning-triggering wires. The ground-level electric fields at distances of 60 m and 350 m were measured during six wire launches that resulted in triggered lightning. The wires were launched when ground-level electric fields ranged from 3.2 to 7.6 kV m−1 and the triggering heights ranged from 123 to 304 m. From wire launch time to lightning initiation time, the ground-level electric field reduction at 60 m ranged from 2.2 …

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Integer Optimization And Computational Algebraic Topology, Bala Krishnamoorthy Apr 2011

Integer Optimization And Computational Algebraic Topology, Bala Krishnamoorthy

Systems Science Friday Noon Seminar Series

We present recently discovered connections between integer optimization, or integer programming (IP), and homology. Under reasonable assumptions, these results lead to efficient solutions of several otherwise hard-to-solve problems from computational topology and geometric analysis. The main result equates the total unimodularity of the boundary matrix of a simplicial complex to an algebraic topological condition on the complex (absence of relative torsion), which is often satisfied in real-life applications . When the boundary matrix is totally unimodular, the problem of finding the shortest chain homologous under Z (ring of integers) to a given chain, which is inherently an integer program, can …

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Modal Logic And Its Applications, Explained Using Puzzles And Examples, Marek Perkowski Mar 2011

Modal Logic And Its Applications, Explained Using Puzzles And Examples, Marek Perkowski

Systems Science Friday Noon Seminar Series

The talk introduces Modal Logic as an extension of classical propositional and First Order Logics. We discuss motivations of Lewis to create modal logic system, axioms and rules of proof. Several examples illustrate deriving theorems from axioms. "Muddy Children" puzzle is used to explain the principles of dealing with uncertainty problems where a temporal lack of response is used as additional information. Other examples include "Narrow Bridge" problem/game which relates to the problem of necessary evil in the world, robot planning and law and robot morality problems, especially related to military robots and use of force by police. Kripke semantics …

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A Second Elasticity Element Using The Matrix Bubble, Jay Gopalakrishnan, Johnny Guzmán Jan 2011

A Second Elasticity Element Using The Matrix Bubble, Jay Gopalakrishnan, Johnny Guzmán

Mathematics and Statistics Faculty Publications and Presentations

We presented a family of finite elements that use a polynomial space augmented by certain matrix bubbles in Cockburn et al. (2010) A new elasticity element made for enforcing weak stress symmetry. Math. Comput., 79, 1331–1349 . In this sequel we exhibit a second family of elements that use the same matrix bubble. This second element uses a stress space smaller than the first while maintaining the same space for rotations (which are the Lagrange multipliers corresponding to a weak symmetry constraint). The space of displacements is of one degree less than the first method. The analysis, while similar to …

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Analysis Of The Dpg Method For The Poisson Equation, Leszek Demkowicz, Jay Gopalakrishnan Jan 2011

Analysis Of The Dpg Method For The Poisson Equation, Leszek Demkowicz, Jay Gopalakrishnan

Mathematics and Statistics Faculty Publications and Presentations

We give an error analysis of the recently developed DPG method applied to solve the Poisson equation and a convection-dffusion problem. We prove that the method is quasioptimal. Error estimates in terms of both the mesh size h and the polynomial degree p (for various element shapes) can be derived from our results. Results of extensive numerical experiments are also presented.

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Polynomial Extension Operators. Part Iii, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl Jan 2011

Polynomial Extension Operators. Part Iii, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl

Mathematics and Statistics Faculty Publications and Presentations

In this concluding part of a series of papers on tetrahedral polynomial extension operators, the existence of a polynomial extension operator in the Sobolev space H(div) is proven constructively. Specifically, on any tetrahedron K, given a function w on the boundary ∂K that is a polynomial on each face, the extension operator applied to w gives a vector function whose components are polynomials of at most the same degree in the tetrahedron. The vector function is an extension in the sense that the trace of its normal component on the boundary ∂K coincides with w. Furthermore, the extension operator is …

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Commuting Smoothed Projectors In Weighted Norms With An Application To Axisymmetric Maxwell Equations, Jay Gopalakrishnan, Minah Oh Jan 2011

Commuting Smoothed Projectors In Weighted Norms With An Application To Axisymmetric Maxwell Equations, Jay Gopalakrishnan, Minah Oh

Mathematics and Statistics Faculty Publications and Presentations

We construct finite element projectors that can be applied to functions with low regularity. These projectors are continuous in a weighted norm arising naturally when modeling devices with axial symmetry. They have important commuting diagram properties needed for finite element analysis. As an application, we use the projectors to prove quasioptimal convergence for the edge finite element approximation of the axisymmetric time-harmonic Maxwell equations on nonsmooth domains. Supplementary numerical investigations on convergence deterioration at high wavenumbers and near Maxwell eigenvalues and are also reported.

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Symmetric Nonconforming Mixed Finite Elements For Linear Elasticity, Jay Gopalakrishnan, Johnny Guzmán Jan 2011

Symmetric Nonconforming Mixed Finite Elements For Linear Elasticity, Jay Gopalakrishnan, Johnny Guzmán

Mathematics and Statistics Faculty Publications and Presentations

We present a family of mixed methods for linear elasticity that yield exactly symmetric, but only weakly conforming, stress approximations. The method is presented in both two and three dimensions (on triangular and tetrahedral meshes). The method is efficiently implementable by hybridization. The degrees of freedom of the Lagrange multipliers, which approximate the displacements at the faces, solve a symmetric positive-definite system. The design and analysis of this method is motivated by a new set of unisolvent degrees of freedom for symmetric polynomial matrices. These new degrees of freedom are also used to give a new simple calculation of the …

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Analysis Of Hdg Methods For Stokes Flow, Bernardo Cockburn, Jay Gopalakrishnan, Ngoc Cuong Nguyen, Jaume Peraire, Francisco-Javier Sayas Jan 2011

Analysis Of Hdg Methods For Stokes Flow, Bernardo Cockburn, Jay Gopalakrishnan, Ngoc Cuong Nguyen, Jaume Peraire, Francisco-Javier Sayas

Mathematics and Statistics Faculty Publications and Presentations

In this paper, we analyze a hybridizable discontinuous Galerkin method for numerically solving the Stokes equations. The method uses polynomials of degree $ k$ for all the components of the approximate solution of the gradient-velocity-pressure formulation. The novelty of the analysis is the use of a new projection tailored to the very structure of the numerical traces of the method. It renders the analysis of the projection of the errors very concise and allows us to see that the projection of the error in the velocity superconverges. As a consequence, we prove that the approximations of the velocity gradient, the …

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Balance Systems And The Variational Bicomplex, Serge Preston Jan 2011

Balance Systems And The Variational Bicomplex, Serge Preston

Mathematics and Statistics Faculty Publications and Presentations

In this work we show that the systems of balance equations (balance systems) of continuum thermodynamics occupy a natural place in the variational bicomplex formalism. We apply the vertical homotopy decomposition to get a local splitting (in a convenient domain) of a general balance system as the sum of a Lagrangian part and a complemental "pure non-Lagrangian" balance system. In the case when derivatives of the dynamical fields do not enter the constitutive relations of the balance system, the "pure non-Lagrangian" systems coincide with the systems introduced by S. Godunov [Soviet Math. Dokl. 2 (1961), 947–948] and, later, asserted as …

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