Rank-1 numerical index of some families of norms on the plane

We compute the rank-1 numerical index of a family of hexagonal norms and two families of octagonal norms on the real plane.     

A stabilized method for a secondary consolidation Biot's model

This work deals with the numerical solution of a secondary consolidation Biot's model. A family of finite difference methods on staggered grids in both time and spatial variables is considered. These numerical methods use a weighted two‐level discretization in time and the classical central difference discretization in space. A priori estimates and convergence results for displacements and pressure in discrete energy norms are obtained. Numerical examples illustrate the convergence properties of the proposed numerical schemes, showing also a non‐oscillatory behavior of the pressure approximation. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Family of finite-difference schemes with approximate transparent boundary conditions for the generalized nonstationary Schrödinger equation in a semi-infinite strip

An initial-boundary value problem for the generalized Schrödinger equation in a semi-infinite strip is solved. A new family of two-level finite-difference schemes with averaging over spatial variables on a finite mesh is constructed, which covers a set of finite-difference schemes built using various methods. For the family, an abstract approximate transparent boundary condition (TBC) is formulated and the solutions are proved to be absolutely stable in two norms with respect to both initial data and free terms. A discrete TBC is derived, and the stability of the family of schemes with this TBC is proved. The implementation of schemes with the discrete TBC is discussed, and numerical results are presented.     

Finite-difference analysis of fully dynamic problems for saturated porous media

Finite-difference methods, using staggered grids in space, are considered for the numerical approximation of fully dynamic poroelasticity problems. First, a family of second-order schemes in time is analyzed. A priori estimates for displacements in discrete energy norms are obtained and the corresponding convergence results are proved. Numerical examples are given to illustrate the convergence properties of these methods. As in the case of an incompressible fluid and small permeability, these schemes suffer from spurious oscillations in time, a first order scheme is proposed and analyzed. For this new scheme a priori estimates and convergence results are also given. Finally, numerical examples in one and two dimensions are presented to show the good monotonicity properties of this method.     

Finite-difference analysis of fully dynamic problems for saturated porous media

Finite-difference methods, using staggered grids in space, are considered for the numerical approximation of fully dynamic poroelasticity problems. First, a family of second-order schemes in time is analyzed. A priori estimates for displacements in discrete energy norms are obtained and the corresponding convergence results are proved. Numerical examples are given to illustrate the convergence properties of these methods. As in the case of an incompressible fluid and small permeability, these schemes suffer from spurious oscillations in time, a first order scheme is proposed and analyzed. For this new scheme a priori estimates and convergence results are also given. Finally, numerical examples in one and two dimensions are presented to show the good monotonicity properties of this method.     

The boundedness of the Cauchy singular integral operator in weighted Besov type spaces with uniform norms

The mapping properties of the Cauchy singular integral operator with constant coefficients are studied in couples of spaces equipped with weighted uniform norms. Recently weighted Besov type spaces got more and more interest in approximation theory and, in particular, in the numerical analysis of polynomial approximation methods for Cauchy singular integral equations on an interval. In a scale of pairs of weighted Besov spaces the authors state the boundedness and the invertibility of the Cauchy singular integral operator. Such result was not expected for a long time and it will affect further investigations essentially. The technique of the paper is based on properties of the de la Vallée Poussin operator constructed with respect to some Jacobi polynomials.     

On a general class of matrix nearness problems

The problem is considered of finding the nearest rank-deficient matrix to a given rectangular matrix. For a wide class of matrix norms, and arbitrary sparsity imposed on the matrix of perturbations, it is shown that when a solution exists, it can be calculated from the solution of a simpler problem involving fewer variables and only vector norms. Some numerical results are given for a special case involving the Frobenius norm.Communicated by Charles Micchelli.     

Induced norms, states, and numerical ranges

It is shown that two induced norms are the same if and only if the corresponding norm numerical ranges or radii are the same, which in turn is equivalent to the vector states and mixed states arising from the norms being the same. The proofs depend on an auxiliary result of independent interest which concerns when two closed convex sets in a topological vector space are multiples of each other.

     

Analysis of the advection-diffusion operator using fractional order norms

Summary. In this paper we obtain a family of optimal estimates for the linear advection-diffusion operator. More precisely we define norms on the domain of the operator, and norms on its image, such that it behaves as an isomorphism: it stays bounded as well as its inverse does, uniformly with respect to the diffusion parameter. The analysis makes use of the interpolation theory between function spaces. One motivation of the present work is our interest in the theoretical properties of stable numerical methods for this kind of problem: we will only give some hints here and we will take a deeper look in a further paper.Mathematics Subject Classification (2000):65N30     

Samuel multiplicity and the structure of semi-Fredholm operators

Two numerical invariants refining the Fredholm index are introduced for any semi-Fredholm operator in such a way that their difference calculates the Fredholm index. These two invariants are inspired by Samuel multiplicity in commutative algebra, and can be regarded as the stabilized dimension of the kernel and cokernel. A geometric interpretation of these invariants leads naturally to a 4×4 uptriangular matrix model for any semi-Fredholm operator on a separable Hilbert space. This model can be regarded as a refined, local version of the Apostol's 3×3 triangular representation for arbitrary operators. Some classical results, such as Gohberg's punctured neighborhood theorem, can be read off directly from our matrix model. Banach space operators are also considered.     

An enhanced‐physics‐based scheme for the NS‐α turbulence model

We study a new enhanced‐physics‐based numerical scheme for the NS‐alpha turbulence model that conserves both energy and helicity. Although most turbulence models (in the continuous case) conserve only energy, NS‐alpha is one of only a very few that also conserve helicity. This is one reason why it is becoming accepted as the most physically accurate turbulence model. However, no numerical scheme for NS‐alpha, until now, conserved both energy and helicity, and thus the advantage gained in physical accuracy by modeling with NS‐alpha could be lost in a computation. This report presents a finite element numerical scheme, and gives a rigorous analysis of its conservation properties, stability, solution existence, and convergence. A key feature of the analysis is the identification of the discrete energy and energy dissipation norms, and proofs that these norms are equivalent (provided a careful choice of filtering radius) in the discrete space to the usual energy and energy dissipation norms. Numerical experiments are given to demonstrate the effectiveness of the scheme over usual (helicity‐ignoring) schemes. A generalization of this scheme to a family of high‐order NS‐alpha‐deconvolution models, which combine the attractive physical properties of NS‐alpha with the high accuracy gained by combining α‐filtering with van Cittert approximate deconvolution. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Finite difference method combined with differential quadrature method for numerical computation of the modified equal width wave equation

The aim of this study is to improve the numerical solution of the modified equal width wave equation. For this purpose, finite difference method combined with differential quadrature method with Rubin and Graves linearizing technique has been used. Modified cubic B‐spline base functions are used as base function. By the combination of two numerical methods and effective linearizing technique high accurate numerical algorithm is obtained. Three main test problems are solved for various values of the coefficients. To observe the performance of the present method, the error norms of the single soliton problem which has analytical solution are calculated. Besides these error norms, three invariants are reported. Comparison of the results displays that our algorithm produces superior results than those given in the literature.     

On some norm equalities in pre-Hilbert C*-modules

We characterize when the norm of the sum of two elements in a pre-Hilbert C*-module equals the sum of their norms. We also give the necessary and sufficient conditions for two orthogonal elements of a pre-Hilbert C*-module to satisfy Pythagoras’ equality.     

Zero-preserving iso-spectral flows based on parallel sums

Driessel [K.R. Driessel, Computing canonical forms using flows, Linear Algebra Appl 379 (2004) 353-379] introduced the notion of quasi-projection onto the range of a linear transformation from one inner product space into another inner product space. Here we introduce the notion of quasi-projection onto the intersection of the ranges of two linear transformations from two inner product spaces into a third inner product space. As an application, we design a new family of iso-spectral flows on the space of symmetric matrices that preserves zero patterns. We discuss the equilibrium points of these flows. We conjecture that these flows generically converge to diagonal matrices. We perform some numerical experiments with these flows which support this conjecture. We also compare our zero-preserving flows with the Toda flow.     

Preconditioning techniques for the iterative solution of scattering problems

We consider a time-harmonic electromagnetic scattering problem for an inhomogeneous medium. Some symmetry hypotheses on the refractive index of the medium and on the electromagnetic fields allow to reduce this problem to a two-dimensional scattering problem. This boundary value problem is defined on an unbounded domain, so its numerical solution cannot be obtained by a straightforward application of usual methods, such as for example finite difference methods, and finite element methods. A possible way to overcome this difficulty is given by an equivalent integral formulation of this problem, where the scattered field can be computed from the solution of a Fredholm integral equation of second kind. The numerical approximation of this problem usually produces large dense linear systems. We consider usual iterative methods for the solution of such linear systems, and we study some preconditioning techniques to improve the efficiency of these methods. We show some numerical results obtained with two well known Krylov subspace methods, i.e., Bi-CGSTAB and GMRES.     

On the periodic points of a two-parameter family of maps of the plane

We study the dynamics of a two-parameter family of noninvertible maps of the plane, derived from a model in population dynamics. We prove that, as one parameter varies with the other held fixed, the nonwandering set changes from the empty set to an unstable Cantor set on which the map is topologically equivalent to the shift endomorphism on two symbols. With the help of some numerical work, we trace the genealogies of the periodic points of the family of period 5, and describe their stability types and bifurcations. Among our results we find that the family has a fixed point which undergoes fold, flip and Hopf bifurcations, and that certain families of period five points are interconnected through a codimension-two cusp bifurcation.     

A family of estimators for multivariate kurtosis in a nonnormal linear regression model

In this paper, we propose a new estimator for a kurtosis in a multivariate nonnormal linear regression model. Usually, an estimator is constructed from an arithmetic mean of the second power of the squared sample Mahalanobis distances between observations and their estimated values. The estimator gives an underestimation and has a large bias, even if the sample size is not small. We replace this squared distance with a transformed squared norm of the Studentized residual using a monotonic increasing function. Our proposed estimator is defined by an arithmetic mean of the second power of these squared transformed squared norms with a correction term and a tuning parameter. The correction term adjusts our estimator to an unbiased estimator under normality, and the tuning parameter controls the sizes of the squared norms of the residuals. The family of our estimators includes estimators based on ordinary least squares and predicted residuals. We verify that the bias of our new estimator is smaller than usual by constructing numerical experiments.     

Numerical Methods for Stiff Index-3 DAEs

Space semidiscretization of PDAEs, i.e. coupled systems of PDEs and algebraic equations, give raise to stiff DAEs and thus the standard theory of numerical methods for DAEs is not valid. As the study of numerical methods for stiff ODEs is done in terms of logarithmic norms, it seems natural to use also logarithmic norms for stiff DAEs. In this paper we show how the standard conditions imposed on the PDAE and the semidiscretized problem are formally the same if they are expressed in terms of logarithmic norms. To study the mathematical problem and their numerical approximations, this link between the standard conditions and logarithmic norms allow us to use for stiff DAEs techniques similar to the ones used for stiff ODEs. The analysis is done for problems which appear in the context of elastic multibody systems, but once the tools, i.e., logarithmic norms, are developed, they can also be used for the analysis of other PDAEs/DAEs.     

Eulerian quasisymmetric functions

We introduce a family of quasisymmetric functions called Eulerian quasisymmetric functions, which specialize to enumerators for the joint distribution of the permutation statistics, major index and excedance number on permutations of fixed cycle type. This family is analogous to a family of quasisymmetric functions that Gessel and Reutenauer used to study the joint distribution of major index and descent number on permutations of fixed cycle type. Our central result is a formula for the generating function for the Eulerian quasisymmetric functions, which specializes to a new and surprising q-analog of a classical formula of Euler for the exponential generating function of the Eulerian polynomials. This q-analog computes the joint distribution of excedance number and major index, the only of the four important Euler-Mahonian distributions that had not yet been computed. Our study of the Eulerian quasisymmetric functions also yields results that include the descent statistic and refine results of Gessel and Reutenauer. We also obtain q-analogs, (q,p)-analogs and quasisymmetric function analogs of classical results on the symmetry and unimodality of the Eulerian polynomials. Our Eulerian quasisymmetric functions refine symmetric functions that have occurred in various representation theoretic and enumerative contexts including MacMahon's study of multiset derangements, work of Procesi and Stanley on toric varieties of Coxeter complexes, Stanley's work on chromatic symmetric functions, and the work of the authors on the homology of a certain poset introduced by Björner and Welker.     

Admissibility on the half line for evolution families

For an arbitrary evolution family, we consider the notion of an exponential dichotomy with respect to a family of norms and characterize it completely in terms of the admissibility of bounded solutions, that is, the existence of a unique bounded solution for each bounded perturbation. In particular, by considering a family of Lyapunov norms, we recover the notion of a nonuniform exponential dichotomy. As a nontrivial application of the characterization, we establish the robustness of the notion of an exponential dichotomy with respect to a family of norms under sufficiently small Lipschitz and C ¹ parameterized perturbations. Moreover, we establish the optimal regularity of the dependence on the parameter of the projections onto the stable spaces of the perturbation.