Equivalent operator preconditioning for elliptic problems

The numerical solution of linear elliptic partial differential equations most often involves a finite element or finite difference discretization. To preserve sparsity, the arising system is normally solved using an iterative solution method, commonly a preconditioned conjugate gradient method. Preconditioning is a crucial part of such a solution process. In order to enable the solution of very large-scale systems, it is desirable that the total computational cost will be of optimal order, i.e. proportional to the degrees of freedom of the approximation used, which also induces mesh independent convergence of the iteration. This paper surveys the equivalent operator approach, which has proven to provide an efficient general framework to construct such preconditioners. Hereby one first approximates the given differential operator by some simpler differential operator, and then chooses as preconditioner the discretization of this operator for the same mesh. In this survey we give a uniform presentation of this approach, including theoretical foundation and several practically important applications for both symmetric and nonsymmetric equations and systems, and some nonlinear examples in the context of Newton linearization. Dedicated to the memory of Gene Golub for his friendly manner and for his broad interest and significant impact on numerical analysis.     

Aggregation of ordinal information

Our interest is with the fusion of information which has an ordinal structure. Information fusion in this environment requires the availability of ordinal aggregation operations. Basic ordinal operations are first introduced. Next we investigate conjunctive and disjunction aggregations of ordinal information. The idea of a pseudo-log in the ordinal environment is presented. We discuss the introduction of a zero like point on an ordinal scale along with the related ideas of bipolarity (positive and negative values) and uni-norm aggregation operators. We introduce mean like aggregation operators as well weighted averages on a ordinal scale. The problem of selecting between ordinal models is considered.     

Bound states of discrete Schrödinger operators with super-critical inverse square potentials

We consider discrete one-dimensional Schrödinger operators whose potentials decay asymptotically like an inverse square. In the super-critical case, where there are infinitely many discrete eigenvalues, we compute precise asymptotics of the number of eigenvalues below a given energy as this energy tends to the bottom of the essential spectrum.

     

Powers and roots of Toeplitz operators

We study the commutativity of two Toeplitz operators whose symbols are quasihomogeneous functions. We give a relationship between this commutativity and the roots (or powers) of the Toeplitz operators. We use this to characterize Toeplitz operators with symbols in which commute with Toeplitz operators whose symbols are of the form .

     

A policy iteration algorithm for fixed point problems with nonexpansive operators

The aim of this paper is to solve the fixed point problems:

Weighted weak type inequalities for modified Hardy operators and geometric means operators in dimensions one and greater

We characterize the pairs of weights (u,v) such that the geometric mean operator G₁, defined for positive functions f on (0,∞) by

     

C0-estimates and smoothness of solutions to the parabolic equation defined by Kimura operators

Kimura diffusions serve as a stochastic model for the evolution of gene frequencies in population genetics. Their infinitesimal generator is an elliptic differential operator whose second-order coefficients matrix degenerates on the boundary of the domain. In this article, we consider the inhomogeneous initial-value problem defined by generators of Kimura diffusions, and we establish $C^0$-estimates, which allows us to prove that solutions to the inhomogeneous initial-value problem are smooth up to the boundary of the domain where the operator degenerates, even when the initial data is only assumed to be continuous.     

The preservation of Sahlqvist equations in completions of Boolean algebras with operators

Monk [1970] extended the notion of the completion of a Boolean algebra to Boolean algebras with operators. Under the assumption that the operators of such an algebra are completely additive, he showed that the completion of always exists and is unique up to isomorphisms over . Moreover, strictly positive equations are preserved under completions a strictly positive equation that holds in must hold in the completion of . In this paper we extend Monk’s preservation theorem by proving that certain kinds of Sahlqvist equations (as well as some other types of equations and implications) are preserved under completions. An example is given that shows that arbitrary Sahlqvist equations need not be preserved. Received May 3, 1998; accepted in final form October 7, 1998.