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54
Wavelet and Multiscale Methods for Operator Equations
 Acta Numerica
, 1997
"... this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of th ..."
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Cited by 171 (40 self)
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this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of these requirements as well as for their realization. This is also particularly important for understanding the severe obstructions, that keep us at present from readily materializing all the principally promising perspectives.
Overlapping Schwarz Methods On Unstructured Meshes Using NonMatching Coarse Grids
 Numer. Math
, 1996
"... . We consider two level overlapping Schwarz domain decomposition methods for solving the finite element problems that arise from discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Standard finite element interpolation from the coarse to the fine grid may ..."
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Cited by 49 (17 self)
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. We consider two level overlapping Schwarz domain decomposition methods for solving the finite element problems that arise from discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Standard finite element interpolation from the coarse to the fine grid may be used. Our theory requires no assumption on the substructures which constitute the whole domain, so each substructure can be of arbitrary shape and of different size. The global coarse mesh is allowed to be nonnested to the fine grid on which the discrete problem is to be solved and both the coarse meshes and the fine meshes need not be quasiuniform. In addition, the domains defined by the fine and coarse grid need not be identical. The one important constraint is that the closure of the coarse grid must cover any portion of the fine grid boundary for which Neumann boundary conditions are given. In this general setting, our algorithms have the same optimal convergence rate of the usual ...
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 36 (6 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 35 (10 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.
Weighted Max Norms, Splittings, and Overlapping Additive Schwarz Iterations
 NUMERISCHE MATHEMATIK
, 1998
"... Weighted maxnorm bounds are obtained for Algebraic Additive Schwarz Iterations with overlapping blocks for the solution of Ax = b, when the coefficient matrix A is an Mmatrix. The case of inexact local solvers is also covered. These bounds are analogous to those that exist using Anorms when the m ..."
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Cited by 31 (16 self)
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Weighted maxnorm bounds are obtained for Algebraic Additive Schwarz Iterations with overlapping blocks for the solution of Ax = b, when the coefficient matrix A is an Mmatrix. The case of inexact local solvers is also covered. These bounds are analogous to those that exist using Anorms when the matrix A is symmetric positive definite. A new theorem concerning P regular splittings is presented, which provides a useful tool for the Anorm bounds. Furthermore, a theory of splittings is developed to represent Algebraic Additive Schwarz Iterations. This representation makes a connection with multisplitting methods. With this representation, and using a comparison theorem, it is shown that a coarse grid correction improves the convergence of Additive Schwarz Iterations when measured in weighted max norm.
Stabilizing the Hierarchical Basis by Approximate Wavelets II: Implementation and Numerical Results
 I: Theory, Numer. Linear Alg. Appl., 4 Number
, 1998
"... This paper is the second part of a work on stabilizing the classical hierarchical basis HB by using waveletlike basis functions. Implementation techniques are of major concern for the multilevel preconditioners proposed by the authors in the first part of the work, which deals with algorithms and t ..."
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Cited by 27 (3 self)
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This paper is the second part of a work on stabilizing the classical hierarchical basis HB by using waveletlike basis functions. Implementation techniques are of major concern for the multilevel preconditioners proposed by the authors in the first part of the work, which deals with algorithms and their mathematical theory. Numerical results are presented to confirm the theory established there. A comparison of the performance of a number of multilevel methods is conducted for elliptic problems of three space variables. Key words. hierarchical basis, multilevel methods, preconditioning, finite element elliptic equations, approximate wavelets AMS subject classifications. 65F10, 65N20, 65N30 PII. S1064827596300668 1.
Algebraic theory of multiplicative Schwarz methods
 NUMER. MATH.
, 2001
"... The convergence of multiplicative Schwarztype methods for solving linear systems when the coefficient matrix is either a nonsingular Mmatrix or a symmetric positive definite matrix is studied using classical and new results from the theory of splittings. The effect on convergence of algorithmic ..."
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Cited by 26 (19 self)
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The convergence of multiplicative Schwarztype methods for solving linear systems when the coefficient matrix is either a nonsingular Mmatrix or a symmetric positive definite matrix is studied using classical and new results from the theory of splittings. The effect on convergence of algorithmic parameters such as the number of subdomains, the amount of overlap, the result of inexact local solves and of “coarse grid” corrections (global coarse solves) is analyzed in an algebraic setting. Results on algebraic additive Schwarz are also included.
Sparse grids and related approximation schemes for higher dimensional problems
"... The efficient numerical treatment of highdimensional problems is hampered by the curse of dimensionality. We review approximation techniques which overcome this problem to some extent. Here, we focus on methods stemming from Kolmogorov’s theorem, the ANOVA decomposition and the sparse grid approach ..."
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Cited by 25 (12 self)
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The efficient numerical treatment of highdimensional problems is hampered by the curse of dimensionality. We review approximation techniques which overcome this problem to some extent. Here, we focus on methods stemming from Kolmogorov’s theorem, the ANOVA decomposition and the sparse grid approach and discuss their prerequisites and properties. Moreover, we present energynorm based sparse grids and demonstrate that, for functions with bounded mixed derivatives on the unit hypercube, the associated approximation rate in terms of the involved degrees of freedom shows no dependence on the dimension at all, neither in the approximation order nor in the order constant.
Global Convergence of Subspace Correction Methods for Convex Optimization Problems
, 1998
"... A general space decomposition technique is used to solve nonlinear convex minimization problems. The dierential of the minimization functional is required to satisfy some growth conditions that are weaker than Lipschitz continuity and strong monotonicity. Optimal rate of convergence is proved. If th ..."
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Cited by 24 (6 self)
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A general space decomposition technique is used to solve nonlinear convex minimization problems. The dierential of the minimization functional is required to satisfy some growth conditions that are weaker than Lipschitz continuity and strong monotonicity. Optimal rate of convergence is proved. If the dierential is Lipschitz continuous and strongly monotone, then the algorithms have uniform rate of convergence. The algorithms can be used for domain decomposition and multigrid type of techniques. Applications to linear elliptic and some nonlinear degenerated partial dierential equation are considered. 1 Introduction Domain decomposition and multigrid methods have been intensively studied for linear partial dierential equations. Recent research, see for example [32], reveals that domain decomposition and multigrid methods can be analysed using a same framework, see also [3], [13], [23], [17]. The present work uses this framework to analyse the convergence of two algorithms for convex...
On Two Ways Of Stabilizing The Hierarchical Basis Multilevel Methods
 SIAM Review
, 1997
"... A survey of two approaches for stabilizing the hierarchical basis (HB) multilevel preconditioners, both additive and multiplicative, is presented. The first approach is based on the algebraic extension of the twolevel methods, exploiting recursive calls to coarser discretization levels. These recur ..."
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Cited by 23 (5 self)
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A survey of two approaches for stabilizing the hierarchical basis (HB) multilevel preconditioners, both additive and multiplicative, is presented. The first approach is based on the algebraic extension of the twolevel methods, exploiting recursive calls to coarser discretization levels. These recursive calls can be viewed as inner iterations (at a given discretization level), exploiting the already defined preconditioner at coarser levels in a polynomiallybased inner iteration method. The latter gives rise to hybridtype multilevel cycles. This is the socalled (hybrid) algebraic multilevel iteration (AMLI) method. The second approach is based on a different direct multilevel splitting of the finite element discretization space. This gives rise to the socalled wavelet multilevel decomposition based on L 2 projections, which in practice must be approximated. Both approachesthe AMLI one and the one based on approximate wavelet decompositionslead to optimal relative condition numbers of the multilevel preconditioners.