Accurate, high-order representation of complex three-dimensional surfaces via Fourier continuation analysis

We present a new method for construction of high-order parametrizations of surfaces: starting from point clouds, the method we propose can be used to produce full surface parametrizations (by sets of local charts, each one representing a large surface patch – which, typically, contains thousands of the points in the original point-cloud) for complex surfaces of scientific and engineering relevance. The proposed approach accurately renders both smooth and non-smooth portions of a surface: it yields super-algebraically convergent Fourier series approximations to a given surface up to and including all points of geometric singularity, such as corners, edges, conical points, etc. In view of their C^∞ smoothness (except at true geometric singularities) and their properties of high-order approximation, the surfaces produced by this method are suitable for use in conjunction with high-order numerical methods for boundary value problems in domains with complex boundaries, including PDE solvers, integral equation solvers, etc. Our approach is based on a very simple concept: use of Fourier analysis to continue smooth portions of a piecewise smooth function into new functions which, defined on larger domains, are both smooth and periodic. The “continuation functions” arising from a function f converge super-algebraically to f in its domain of definition as discretizations are refined. We demonstrate the capabilities of the proposed approach for a number of surfaces of engineering relevance.     

Convergence of mann iteration processes for nonexpansive operators in equi-connected metric spaces

The paper deals with various conditions implying the convergence of a Mann type iteration process constructed for a non-expansive operator in an equi-connected space (i. e. metric space equipped with a connecting function; so the iterates are taken along certain curves). Coefficients of the iterates do not have to be separated from 0 or 1. 1].     

The relativistic consistent angular-momentum projected shell model study of the <Emphasis Type="Italic">N</Emphasis>=<Emphasis Type="Italic">Z</Emphasis> nucleus <Superscript>52</Superscript>Fe

The relativistic consistent angular-momentum projected shell model(ReCAPS) is used in the study of the structure and electromagnetic transitions of the low-lying states in the N=Z nucleus 52Fe.The model calculations show a reasonably good agreement with the data.The backbending at 12+ is reproduced and the energy level structure suggests that neutron-proton interactions play important roles.     

Defining the size and location of the slit in rainbow holography

A simple method is described for defining the size and location of the slit in a rainbow holography optical system in order to guarantee the optimal observation conditions of the reconstructed image. A brief explanation of the basic principles of rainbow holography is given and the projection of viewing slits and pseudoscopic, or orthoscopic, images is considered. An application of the method for afocal systems used in rainbow holography is presented.     

A Mann type iterative scheme for variational inequalities in noncompact subsets of Banach spaces

In this paper, we study the existence of solutions and approximation of the solutions by using a Mann type iterative scheme for variational inequalities in noncompact subsets of Banach spaces. The results presented in this paper generalize the corresponding results of J. Li [J. Li, On the existence of solutions of variational inequalities in Banach spaces, J. Math. Anal. Appl. 295 (2004) 115-126].     

VC‐dimension on manifolds: a first approach

The Vapnik–Chervonenkis‐dimension is an index of the capacity of a learning machine. It has been computed in several cases, but always in a Euclidean context. This paper extends the notion to classifiers acting in the more general environment of a manifold. General properties are proved, and some examples of simple classifiers on elementary manifolds are given. A large part of the research is directed toward a still open problem on product manifolds. Copyright © 2007 John Wiley & Sons, Ltd.     

A CSP and tabulation-based adaptive chemistry model

We demonstrate the feasibility of a new strategy for the construction of an adaptive chemistry model that is based on an explicit integrator stabilized by an approximation of the Computational Singular Perturbation (CSP)-slow-manifold projector. We examine the effectiveness and accuracy of this technique first using a model problem with variable stiffness. We assess the effect of using an approximation of the CSP-slow-manifold by either reusing the CSP vectors calculated in previous steps or from a pre-built tabulation. We find that while accuracy is preserved, the associated CPU cost was reduced substantially by this method. We used two ignition simulations – hydrogen–air and heptane–air mixtures – to demonstrate the feasibility of using the new method to handle realistic kinetic mechanisms. We test the effect of utilizing an approximation of the CSP-slow-manifold and find that its use preserves the order of the explicit integrator, produces no degradation in accuracy, and results in a scheme that is competitive with traditional implicit integration. Further analysis on the performance data demonstrates that the tabulation of the CSP-slow-manifold provides an increasing level of efficiency as the size of the mechanism increases. From the software engineering perspective, all the machinery developed is Common Component Architecture compliant, giving the software a distinct advantage in the ease of maintainability and flexibility in its utilization. Extension of this algorithm is underway to implement an automated tabulation of the CSP-slow-manifold for a detailed chemical kinetic system either off-line, or on-line with a reactive flow simulation code.     

NUMERICAL LOCALIZATION OF ELECTROMAGNETIC IMPERFECTIONS FROM A PERTURBATION FORMULA IN THREE DIMENSIONS

This work deals with the numerical localization of small electromagnetic inhomogeneities. The underlying inverse problem considers, in a three-dimensional bounded domain, the time-harmonic Maxwell equations formulated in electric field. Typically, the domain contains a finite number of unknown inhomogeneities of small volume and the inverse problem attempts to localize these inhomogeneities from a finite number of boundary measurements. Our localization approach is based on a recent framework that uses an asymptotic expansion for the perturbations in the tangential boundary trace of the curl of the electric field. We present three numerical localization procedures resulting from the combination of this asymptotic expansion with each of the following inversion algorithms： the Current Projection method, the MUltiple Signal Classification （MUSIC） algorithm, and an Inverse Fourier method. We perform a numerical study of the asymptotic expansion and compare the numerical results obtained from the three localization procedures in different settings.     

Mortar element methods for parabolic problems

In this article a standard mortar finite element method and a mortar element method with Lagrange multiplier are used for spatial discretization of a class of parabolic initial‐boundary value problems. Optimal error estimates in L^∞(L²) and L^∞(H¹)‐norms for semidiscrete methods for both the cases are established. The key feature that we have adopted here is to introduce a modified elliptic projection. In the standard mortar element method, a completely discrete scheme using backward Euler scheme is discussed and optimal error estimates are derived. The results of numerical experiments support the theoretical results obtained in this article. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Contractive projections on Banach algebras

In this paper, we explore the properties of projections of norm one on general Banach algebras, in particular the relation with conditional expectations for algebras which arise in harmonic analysis.