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82
Stability of Multiscale Transformations
 J. Fourier Anal. Appl
, 1996
"... After briefly reviewing the interrelation between Rieszbases, biorthogonality and a certain stability notion for multiscale basis transformations we establish a basic stability criterion for a general Hilbert space setting. An important tool in this context is a strengthened Cauchy inequality. It i ..."
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Cited by 92 (22 self)
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After briefly reviewing the interrelation between Rieszbases, biorthogonality and a certain stability notion for multiscale basis transformations we establish a basic stability criterion for a general Hilbert space setting. An important tool in this context is a strengthened Cauchy inequality. It is based on direct and inverse estimates for a certain scale of spaces induced by the underlying multiresolution sequence. Furthermore, we highlight some properties of these spaces pertaining to duality, interpolation, and applications to norm equivalences for Sobolev spaces. AMS Subject Classification: 41A17, 41A65, 46A35, 46B70, 46E35 Key Words: Riesz bases, biorthogonality, stability, projectors, interpolation theory, Kmethod, duality, Jackson, Bernstein inequalities 1 Background and Motivation A standard framework for approximately recapturing a function v in some infinite dimensional separable Hilbert space V , say, either from explicitly given data or as a solution of an operator equ...
Biorthogonal SplineWavelets on the Interval  Stability and Moment Conditions
 Appl. Comp. Harm. Anal
, 1997
"... This paper is concerned with the construction of biorthogonal multiresolution analyses on [0; 1] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal Bsplines and co ..."
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Cited by 89 (46 self)
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This paper is concerned with the construction of biorthogonal multiresolution analyses on [0; 1] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal Bsplines and compactly supported dual generators on IR developed by Cohen, Daubechies and Feauveau. In contrast to previous investigations we preserve the full degree of polynomial reproduction also for the dual multiresolution and prove in general that the corresponding modifications of dual generators near the end points of the interval still permit the biorthogonalization of the resulting bases. The subsequent construction of compactly supported biorthogonal wavelets is based on the concept of stable completions. As a first step we derive an initial decomposition of the spline spaces where the complement spaces between two successive levels are spanned by compactly supported splines which form uniformly...
Hierarchical Bases and the Finite Element Method
, 1997
"... CONTENTS 1 Introduction 1 2 Preliminaries 3 3 Fundamental TwoLevel Estimates 7 4 A Posteriori Error Estimates 16 5 TwoLevel Iterative Methods 23 6 Multilevel Cauchy Inequalities 30 7 Multilevel Iterative Methods 34 References 41 1. Introduction In this work we present a brief introduction to hie ..."
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Cited by 68 (4 self)
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CONTENTS 1 Introduction 1 2 Preliminaries 3 3 Fundamental TwoLevel Estimates 7 4 A Posteriori Error Estimates 16 5 TwoLevel Iterative Methods 23 6 Multilevel Cauchy Inequalities 30 7 Multilevel Iterative Methods 34 References 41 1. Introduction In this work we present a brief introduction to hierarchical bases, and the important part they play in contemporary finite element calculations. In particular, we examine their role in a posteriori error estimation, and in the Department of Mathematics, University of California at San Diego, La Jolla, CA 92093. The work of this author was supported by the Office of Naval Research under contract N0001489J1440. 2 Randolph E. Bank formulation of iterative methods for solving the large sparse sets of linear equations arising from the finite element discretization. Our goal is that the development should be largely selfcontained, but at the same time accessible and interest
Some Nonoverlapping Domain Decomposition Methods
, 1998
"... . The purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving secondorder elliptic problems in two and three dimensions. The methods under scrutiny fall into two major categories: the substructuringtype methods and the Neumann ..."
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Cited by 47 (8 self)
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. The purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving secondorder elliptic problems in two and three dimensions. The methods under scrutiny fall into two major categories: the substructuringtype methods and the NeumannNeumanntype methods. The basic framework used for analysis is the parallel subspace correction method or additive Schwarz method, and other technical tools include localglobal and globallocal techniques. The analyses for both two and threedimensional cases are carried out simultaneously. Some internal relationships between various algorithms are observed and several new variants of the algorithms are also derived. Key words. nonoverlapping domain decomposition, Schur complement, localglobal and globallocal techniques, jumps in coe#cients, substructuring, NeumannNeumann, balancing methods AMS subject classifications. 65N30, 65N55, 65F10 PII. S0036144596306800 1. Introduction. T...
A nonoverlapping domain decomposition method for Maxwell’s equations in three dimensions
 SIAM J. Numer. Anal
"... Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with muc ..."
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Cited by 44 (16 self)
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Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with much simpler coarse solvers. Though the condition number of the preconditioned system may not have a good bound, we are able to show that the convergence rate of the PCG method with such substructuring preconditioner is nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficient in the elliptic equation. 1.
The method of alternating projections and the method of subspace corrections in Hilbert space
 J. Am. Math. Soc
"... Abstract. A new identity is given in this paper for estimating the norm of the product of nonexpansive operators in Hilbert space. This identity can be applied for the design and analysis of the method of alternating projections and the method of subspace corrections. The method of alternating proje ..."
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Cited by 42 (7 self)
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Abstract. A new identity is given in this paper for estimating the norm of the product of nonexpansive operators in Hilbert space. This identity can be applied for the design and analysis of the method of alternating projections and the method of subspace corrections. The method of alternating projections is an iterative algorithm for determining the best approximation to any given point in a Hilbert space from the intersection of a nite number of subspaces by alternatively computing the best approximations from the individual subspaces which make up the intersection. The method of subspace corrections is an iterative algorithm for nding the solution of a linear equation in a Hilbert space by approximately solving equations restricted on a number of closed subspaces which make up the entire space. The new identity given in the paper provides a sharpest possible estimate for the rate of convergence of these algorithms. It is also proved in the paper that the method of alternating projections is essentially equivalent to the method of subspace corrections. Some simple examples of multigrid and domain decomposition methods are given to illustrate the application of the new identity. 1.
Monotone Multigrid Methods for Elliptic Variational Inequalities I
 I. Numer. Math
, 1993
"... . We derive fast solvers for discrete elliptic variational inequalities of the first kind (obstacle problems) as resulting from the approximation of related continuous problems by piecewise linear finite elements. Using basic ideas of successive subspace correction, we modify wellknown relaxation ..."
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Cited by 37 (13 self)
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. We derive fast solvers for discrete elliptic variational inequalities of the first kind (obstacle problems) as resulting from the approximation of related continuous problems by piecewise linear finite elements. Using basic ideas of successive subspace correction, we modify wellknown relaxation methods by extending the set of search directions. Extended underrelaxations are called monotone multigrid methods, if they are quasioptimal in a certain sense. By construction, all monotone multigrid methods are globally convergent. We take a closer look at two natural variants, the standard monotone multigrid method and a truncated version. For the considered model problems, the asymptotic convergence rates resulting from the standard approach suffer from insufficient coarsegrid transport, while the truncated monotone multigrid method provides the same efficiency as in the unconstrained case. Key words: obstacle problems, adaptive finite element methods, multigrid methods AMS (MOS) subje...
Multilevel methods for elliptic problems on domains not resolved by the coarse grid
 Contemporary Mathematics
, 1994
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Adaptive Multilevel Methods for Edge Element Discretizations of Maxwell's Equations
, 1997
"... . The focus of this paper is on the efficient solution of boundary value problems involving the doublecurl operator. Those arise in the computation of electromagnetic fields in various settings, for instance when solving the electric or magnetic wave equation with implicit timestepping, when tackl ..."
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Cited by 23 (10 self)
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. The focus of this paper is on the efficient solution of boundary value problems involving the doublecurl operator. Those arise in the computation of electromagnetic fields in various settings, for instance when solving the electric or magnetic wave equation with implicit timestepping, when tackling timeharmonic problems or in the context of eddycurrent computations. Their discretization is based on on N'ed'elec's H(curl;\Omega\Gamma7131/59948 edge elements on unstructured grids. In order to capture local effects and to guarantee a prescribed accuracy of the approximate solution adaptive refinement of the grid controlled by a posteriori error estimators is employed. The hierarchy of meshes created through adaptive refinement forms the foundation for the fast iterative solution of the resulting linear systems by a multigrid method. The guiding principle underlying the design of both the error estimators and the multigrid method is the separate treatment of the kernel of the cu...