Recovering Mesh Geometry from a Stiffness Matrix

We introduce the following class of mesh recovery problems: Given a stiffness matrix A and a PDE, construct a mesh M such that the finite-element formulation of the PDE over M is A. We show, under certain assumptions, that it is possible to reconstruct the original mesh for the special case of the Laplace operator discretized on an unstructured mesh of triangular elements with linear basis functions. The reconstruction is achieved through a series of techniques from graph theory and numerical analysis, some of which are new and can find application in other scientific areas. Finally, we discuss extensions to other operators and some open questions related to this class of problems.     

Numerical ranges and dilations ∗

Denote by W(A) the numerical range of a bounded linear operator A. For two operators A and B (which may act on different Hilbert spaces), we study the relation between the inclusion relation W(A)?W(B) and the condition that A can be dilated to an operator of the form B?I. We also investigate the possibilities of dilating an operator A to operators with simple structure under the assumption that W(A) is included in a special region.     

Sup and Max Properties for the Numerical Radius of Operators in Banach Spaces

The article deals with the following problem: given a bounded linear operator A in a Banach space X, how can multiplication of A by an operator of norm one (contraction) affect the numerical radius of A? The approach used in this work is close to that employed by Vieira and Kubrusly in 2007 for their study concerning spectral radius. It turns out that this study is closely related to the study of V-operators conducted in 2005 by Khatskevich, Ostrovskii, and Shulman; the results of this article demonstrate that in certain cases the obtained property of an operator implies that it is a V-operator, while in some other cases the converse is true.     

Least squares and bounded variation regularization with nondifferentiable functionals

Let A be an operator from a real Banach space into a real Hilbert space. In this paper we study least squares regularization methods for the ill-posed operator equation A(u) = f using nonlinear nondifferentiable penalty functionals. We introduce a notion of distributional approximation, and use constructs of distributional approximations to establish convergence and stability of approximations of bounded variation solutions of the operator equation. We also show that the results provide a framework for a rigorous analysis of numerical methods based on Euler-Lagrange equations to solve the minimization problem. This justifies many of the numerical implementation schemes of bounded variation minimization that have been recently proposed.     

Ergodic properties of operators in some semi-Hilbertian spaces

This article deals with linear operators T on a complex Hilbert space ?, which are bounded with respect to the seminorm induced by a positive operator A on ?. The A-adjoint and A ^1/2-adjoint of T are considered to obtain some ergodic conditions for T with respect to A. These operators are also employed to investigate the class of orthogonally mean ergodic operators as well as that of A-power bounded operators. Some classes of orthogonally mean ergodic or A-ergodic operators, which come from the theory of generalized Toeplitz operators are considered. In particular, we give an example of an A-ergodic operator (with an injective A) which is not Cesàro ergodic, such that T ^?* is not a quasiaffine transform of an orthogonally mean ergodic operator.     

Why Gaussian quadrature in the complex plane?

This paper synthesizes formally orthogonal polynomials, Gaussian quadrature in the complex plane and the bi-conjugate gradient method together with an application. Classical Gaussian quadrature approximates an integral over (a region of) the real line. We present an extension of Gaussian quadrature over an arc in the complex plane, which we call complex Gaussian quadrature. Since there has not been any particular interest in the numerical evaluation of integrals over the long history of complex function theory, complex Gaussian quadrature is in need of motivation. Gaussian quadrature in the complex plane yields approximations of certain sums connected with the bi-conjugate gradient method. The scattering amplitude c ^T A ^–1 b is an example where A is a discretization of a differential–integral operator corresponding to the scattering problem and b and c are given vectors. The usual method to estimate this is to use c ^T x ^(k). A result of Warnick is that this is identically equal to the complex Gaussian quadrature estimate of 1/. Complex Gaussian quadrature thereby replaces this particular inner product in the estimate of the scattering amplitude.     

Study of generalized eigenfunctions of a perturbed isotropic elastic half‐space

We consider the self‐adjoint operator governing the propagation of elastic waves in a perturbed isotropic half‐space (perturbation with compact support of a homogeneous isotropic half‐space) with a free boundary condition. We propose a method to obtain, numerical values included, a complete set of generalized eigenfunctions that diagonalize this operator. The first step gives an explicit representation of these functions using a perturbative method. The unbounded boundary is a new difficulty compared with the method used by Wilcox [25], who set the problem in the complement of bounded open set. The second step is based on a boundary integral equations method which allows us to compute these functions. For this, we need to determine explicitly the Green's function of (A₀ – ω²), where A₀ is the self‐adjoint operator describing elastic waves in a homogeneous isotropic half‐space. Copyright © 2000 John Wiley & Sons, Ltd.     

Closed Projections and Peak Interpolation for Operator Algebras

The closed one-sided ideals of a C ^*-algebra are exactly the closed subspaces supported by the orthogonal complement of a closed projection. Let A be a (not necessarily selfadjoint) subalgebra of a unital C ^*-algebra B which contains the unit of B. Here we characterize the right ideals of A with left contractive approximate identity as those subspaces of A supported by the orthogonal complement of a closed projection in B ^** which also lies in . Although this seems quite natural, the proof requires a set of new techniques which may be viewed as a noncommutative version of the subject of peak interpolation from the theory of function spaces. Thus, the right ideals with left approximate identity are closely related to a type of peaking phenomena in the algebra. In this direction, we introduce a class of closed projections which generalizes the notion of a peak set in the theory of uniform algebras to the world of operator algebras and operator spaces.     

On the convexity of numerical range in quaternionic hilbert spaces

Let A be a bounded linear operator in a quaternionic Hilberi space (H,(·,·)). The numerical range of A is defined to be the set W (A)={( Au, u): u ε H, (u,u) = 1}. Quite different from the complex caseW (A) may not be convex. In this note the author proves that W (A) is convex if and only if R ∩ W (A) = {Req : q ε W(A)}. where R is the real field and Re q denotes the real part of the quaternion q. For a normal operator A in a finite dimensional space H, the author gives a characterization on the convexity of W(A) in terms of ihe eigenvalues of A and also proves that the generalized numerical range of A is convex if and only if A is Hermitian.     

Operator trigonometry

The purpose of the present paper is to survey, from a historical perspective and including some new results, a theory which I will call operator trigonometry. This theory, which is little known, is closely associated with the numerical range W(A) of an operator A. Among the new results is a beautiful connection to numerical linear algebra, in which gradient descent and conjugate gradient convergence rates are shown to be trigonometric.     

Spectral inclusion for unbounded block operator matrices

In this paper we establish a new analytic enclosure for the spectrum of unbounded linear operators A admitting a block operator matrix representation. For diagonally dominant and off-diagonally dominant block operator matrices, we show that the recently introduced quadratic numerical range W²(A) contains the eigenvalues of A and that the approximate point spectrum of A is contained in the closure of W²(A). This provides a new method to enclose the spectrum of unbounded block operator matrices by means of the non-convex set W²(A). Several examples illustrate that this spectral inclusion may be considerably tighter than the one by the usual numerical range or by perturbation theorems, both in the non-self-adjoint case and in the self-adjoint case. Applications to Dirac operators and to two-channel Hamiltonians are given.     

Properties of a quasi-uniformly monotone operator and its application to the electromagnetic p-curl systems

In this paper we propose a new concept of quasi-uniform monotonicity weaker than the uniform monotonicity which has been developed in the study of nonlinear operator equation Au = b. We prove that if A is a quasi-uniformly monotone and hemi-continuous operator, then A^?1 is strictly monotone, bounded and continuous, and thus the Galerkin approximations converge. Also we show an application of a quasi-uniformly monotone and hemi-continuous operator to the proof of the well-posedness and convergence of Galerkin approximations to the solution of steady-state electromagnetic p-curl systems.

     

Eigenvalue Inequalities in Terms of Schatten Norm Bounds on Differences of Semigroups, and Application to Schrödinger Operators

We develop a new method for obtaining bounds on the negative eigenvalues of self-adjoint operators B in terms of a Schatten norm of the difference of the semigroups generated by A and B, where A is an operator with non-negative spectrum. Our method is based on the application of the Jensen identity of complex function theory to a suitably constructed holomorphic function, whose zeros are in one-to-one correspondence with the negative eigenvalues of B. Applying our abstract results, together with bounds on Schatten norms of semigroup differences obtained by Demuth and Van Casteren, to Schr?dinger operators, we obtain inequalities on moments of the sequence of negative eigenvalues, which are different from the Lieb–Thirring inequalities. Guy Katriel: Partially supported by the Minerva Foundation (Germany). Submitted: September 4, 2007. Accepted: December 11, 2007.     

Galerkin method for a nonstationary equation with a monotone operator

In the present paper, we consider the Galerkin method for a quasilinear differentialoperator equation with a leading self-adjoint operator A(t) and a subordinate monotone operator K. For the projection subspaces we take linear spans of eigenelements of an operator similar to the leading operator A(t). We obtain new estimates for the Galerkin method and consider applications to an initial-boundary value problem for a parabolic equation of higher order.     

Nonnegative perturbations of selfadjoint operators

Let 4 be a selfadjoint operator on a Hilbert space H. The results in this paper provide necessary and sufficient conditions on A in order that there exist a nontrivial nonnegative operator D and a unitary operator U with UA = (A − D)U. In one case considered, it is required that the least subspace reducing A, U and containing the range of D is the full Hilbert space. In this case the operators U, D exist if and only if the operator A is not a scalar multiple of the identity and the maximum and minimum of the spectrum of A are not eigenvalues of finite multip icity. This result is used to complete a characterization of the absolute value of a completely nonnormal hyponormal operator.     

On a Class of Operators of Finite Type

This paper studies some class of pure operators A with finite rank self-commutators satisfying the condition that there is a finite dimensional subspace containing the image of the self-commutator and invariant with respect to A*. Besides, in this class the spectrum of operator A is covered by the projection of a union of quadrature domains in some Riemann surfaces. In this paper the analytic model, the mosaic and some kernel related to the eigenfunctions are introduced which are the analogue of those objects in the theory of subnormal operators.     

Closed operator ideals and interpolation

We consider closed operator ideals, which mean operator ideals A whose components A(E, F) are closed subspaces of the space L(E, F). Using interpolation techniques, we obtain general results on products of closed ideals. Furthermore, we investigate which closed ideals A possess the factorization property, i.e., each operator of A factors through a space with the related property “A”. Applications of these results yield the answer to some open questions in ideal theory.     

Asymptotic behavior of a differential operator with discontinuities at two points

In this paper, we study a Sturm–Liouville operator with eigenparameter‐dependent boundary conditions and transmission conditions at two interior points. By establishing a new operator A associated with the problem, we prove that the operator A is self‐adjoint in an appropriate space H, discuss completeness of its eigenfunctions in H, and obtain its Green function. Copyright © 2010 John Wiley & Sons, Ltd.     

On the non-round points of the boundary of the numerical range ∗

Let A be a bounded linear operator acting on a Hilbert space. It is well known (Donoghue, 1957) that comer points of the numerical range W(A) are eigenvalues of A. Recently (1995), this result was generalized by Hiibner who showed that points of infinite curvature on the boundary of W(A) lie in the spectrum of A. Hübner also conjectured that all such points are either corner points or lie in the essential spectrum of A. In this paper, we give a short proof of this conjecture.     

Semi-Conjugate Direction Methods for Real Positive Definite Systems

In this preliminary work, left and right conjugate direction vectors are defined for nonsymmetric, nonsingular matrices A and some properties of these vectors are studied. A left conjugate direction (LCD) method for solving nonsymmetric systems of linear equations is proposed. The method has no breakdown for real positive definite systems. The method reduces to the usual conjugate gradient method when A is symmetric positive definite. A finite termination property of the semi-conjugate direction method is shown, providing a new simple proof of the finite termination property of conjugate gradient methods. The new method is well defined for all nonsingular M-matrices. Some techniques for overcoming breakdown are suggested for general nonsymmetric A. The connection between the semi-conjugate direction method and LU decomposition is established. The semi-conjugate direction method is successfully applied to solve some sample linear systems arising from linear partial differential equations, with attractive convergence rates. Some numerical experiments show the benefits of this method in comparison to well-known methods. This revised version was published online in July 2006 with corrections to the Cover Date.