Results 1  10
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22
Anisotropic Finite Elements: Local Estimates and Applications
, 1999
"... The solution of elliptic boundary value problems my have... ..."
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Cited by 63 (3 self)
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The solution of elliptic boundary value problems my have...
Adaptive wavelet scheme for nonlinear variational problems with convergence rates
 SIAM J Numer Anal
"... Abstract. We develop and analyze wavelet based adaptive schemes for nonlinear variational problems. We derive estimates for convergence rates and corresponding work counts that turn out to be asymptotically optimal. Our approach is based on a new paradigm that has been put forward recently for a cla ..."
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Cited by 22 (10 self)
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Abstract. We develop and analyze wavelet based adaptive schemes for nonlinear variational problems. We derive estimates for convergence rates and corresponding work counts that turn out to be asymptotically optimal. Our approach is based on a new paradigm that has been put forward recently for a class of linear problems. The original problem is transformed first into an equivalent one which is well posed in the Euclidean metric ℓ2. Then conceptually one seeks iteration schemes for the infinite dimensional problem that exhibits at least a fixed error reduction per step. This iteration is then realized approximately through an adaptive application of the involved operators with suitable dynamically updated accuracy tolerances. The main conceptual ingredients center around nonlinear tree approximation and the sparse evaluation of nonlinear mappings of wavelet expansions. We prove asymptotically optimal complexity for adaptive realizations of first order iterations and of Newton’s method. Key words. variational problems, wavelet representations, semilinear equations, mapping properties, gradient iteration, convergence rates, adaptive application of operators, sparse evaluation of nonlinear mappings of wavelet expansions, tree approximation, Newton’s scheme
Wavelet Methods for PDEs  Some Recent Developments
 J. Comput. Appl. Math
, 1999
"... this article will be on recent developments in the last area. Rather than trying to give an exhaustive account of the state of the art I would like to bring out some mechanisms which are in my opinion important for the application of wavelets to operator equations. To accomplish this I found it nece ..."
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Cited by 13 (4 self)
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this article will be on recent developments in the last area. Rather than trying to give an exhaustive account of the state of the art I would like to bring out some mechanisms which are in my opinion important for the application of wavelets to operator equations. To accomplish this I found it necessary to address some of the pivotal issues in more detail than others. Nevertheless, such a selected `zoom in' supported by an extensive list of references should provide a sound footing for conveying also a good idea about many other related branches that will only be briey touched upon. Of course, the selection of material is biased by my personal experience and therefore is not meant to reect any objective measure of importance. The paper is organized around two essential issues namely adaptivity and the development of concepts for coping with a major obstruction in this context namely practically relevant domain geometries
Structured Jordan Canonical Forms for Structured Matrices that are Hermitian, Skew Hermitian or Unitary with Respect to Indefinite Inner Products
, 1999
"... For inner products defined by a symmetric indefinite matrix \Sigma p;q , we study canonical forms for real or complex \Sigma p;q Hermitian matrices, \Sigma p;q skew Hermitian matrices and \Sigma p;q unitary matrices under equivalence transformations which keep the class invariant. ..."
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Cited by 12 (3 self)
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For inner products defined by a symmetric indefinite matrix \Sigma p;q , we study canonical forms for real or complex \Sigma p;q Hermitian matrices, \Sigma p;q skew Hermitian matrices and \Sigma p;q unitary matrices under equivalence transformations which keep the class invariant.
Adaptive Wavelet Solvers for the Unsteady Incompressible NavierStokes Equations
 ADVANCED MATHEMATICAL THEORIES IN FLUID MECHANICS
, 2000
"... In this paper we describe adaptive waveletbased solvers for the NavierStokes equations. Our approach employs a PetrovGalerkin scheme with tensor products of Interpolet wavelets as ansatz functions. We present the fundamental algorithms for the adaptive evaluation of differential operators and non ..."
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Cited by 11 (3 self)
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In this paper we describe adaptive waveletbased solvers for the NavierStokes equations. Our approach employs a PetrovGalerkin scheme with tensor products of Interpolet wavelets as ansatz functions. We present the fundamental algorithms for the adaptive evaluation of differential operators and nonlinear terms. Furthermore, a simple but efficient preconditioning technique for the resulting linear systems is introduced. For the NavierStokes equations a Chorintype projection method with a stabilized pressure discretization is used. Numerical examples demonstrate the efficiency of our approach.
Adaptive wavelet techniques in numerical simulation
 Encyclopedia of Computational Mechanics
, 2004
"... This chapter highlights recent developments concerning adaptive wavelet methods for time dependent and stationary problems. The first problem class focusses on hyperbolic conservation laws where wavelet concepts exploit sparse representations of the conserved variables. Regarding the second problem ..."
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Cited by 10 (0 self)
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This chapter highlights recent developments concerning adaptive wavelet methods for time dependent and stationary problems. The first problem class focusses on hyperbolic conservation laws where wavelet concepts exploit sparse representations of the conserved variables. Regarding the second problem class, we begin with matrix compression in the context of boundary integral equations where the key issue is now to obtain sparse representations of (global) operators like singular integral operators in wavelet coordinates. In the remainder of the chapter a new fully adaptive algorithmic paradigm along with some analysis concepts are outlined which, in particular, works for nonlinear problems and where the sparsity of both, functions and operators, is exploited. key words: Conservation laws, boundary integral equations, elliptic problems, saddle point problems, mixed formulations, nonlinear problems, matrix compression, adaptive application of operators, best Nterm approximation, tree approximation, convergence rates, complexity estimates 1.
Inverse Inequalities on NonQuasiuniform Meshes and Application to the Mortar Element Method
 MATH. COMP
, 2001
"... We present a range of meshdependent inequalities for piecewise constant and continuous piecewise linear finite element functions u defined on locally refined shaperegular (but possibly nonquasiuniform) meshes. These inequalities involve norms of the form kh ff uk W s;p(\Omega\Gamma for positi ..."
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Cited by 9 (2 self)
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We present a range of meshdependent inequalities for piecewise constant and continuous piecewise linear finite element functions u defined on locally refined shaperegular (but possibly nonquasiuniform) meshes. These inequalities involve norms of the form kh ff uk W s;p(\Omega\Gamma for positive and negative s and ff, where h is a function which reflects the local mesh diameter in an appropriate way. The only global parameter involved is N , the total number of degrees of freedom in the finite element space, and we avoid estimates involving either the global maximum or minimum mesh diameter. Our inequalities include new variants of inverse inequalities as well as trace and extension theorems. They can be used in several areas of finite element analysis to extend results  previously known only for quasiuniform meshes  to the locally refined case. Here we describe applications to: (i) the theory of nonlinear approximation and (ii) the stability of the mortar element method for locally refined meshes.
Adaptive Wavelet Galerkin BEM
"... The wavelet Galerkin scheme for the fast solution of boundary integral equations produces approximate solutions within discretization error accuracy o#ered by the underlying Galerkin method at a computational expense that stays proportional to the number of unknowns. In this paper we present an ..."
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Cited by 8 (1 self)
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The wavelet Galerkin scheme for the fast solution of boundary integral equations produces approximate solutions within discretization error accuracy o#ered by the underlying Galerkin method at a computational expense that stays proportional to the number of unknowns. In this paper we present an adaptive version of the scheme which preserves the superconvergence of the Galerkin method.
Sparse evaluation of compositions of functions using multiscale expansions
 SIAM J. Math. Anal
"... Abstract. This paper is concerned with the estimation and evaluation of wavelet coefficients of the composition F◦u of two functions F and u from the wavelet coefficients of u. Our main objective is to show that certain sequence spaces that can be used to measure the sparsity of the arrays of wavele ..."
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Cited by 8 (3 self)
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Abstract. This paper is concerned with the estimation and evaluation of wavelet coefficients of the composition F◦u of two functions F and u from the wavelet coefficients of u. Our main objective is to show that certain sequence spaces that can be used to measure the sparsity of the arrays of wavelet coefficients are stable under a class of nonlinear mappings F that occur naturally, e.g., in nonlinear PDEs. We indicate how these results can be used to facilitate the sparse evaluation of arrays of wavelet coefficients of compositions at asymptotically optimal computational cost. Furthermore, the basic requirements are verified for several concrete choices of nonlinear mappings. These results are generalized to compositions by a multivariate map F of several functions u1,...,un and their derivatives, i.e., F(Dα1u1,...,Dαnun).
Fast Iterative Solution of Saddle Point Problems in Optimal Control Based on Wavelets
 COMPUT. OPTIM. APPL
, 2000
"... For the numerical solution of an elliptic boundary value problem with boundary control, the problem is formulated as minimizing a quadratic functional involving natural norms of the state and the control. Firstly the constraint, the elliptic boundary value problem, is formulated in an appropriate we ..."
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Cited by 6 (4 self)
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For the numerical solution of an elliptic boundary value problem with boundary control, the problem is formulated as minimizing a quadratic functional involving natural norms of the state and the control. Firstly the constraint, the elliptic boundary value problem, is formulated in an appropriate weak form that allows to handle varying boundary conditions explicitly. Namely, the boundary conditions are appended by Lagrange multipliers, leading to a saddle point problem. This is combined with a fictitious domain approach in order to cover also more complicated boundaries. From the optimality conditions for the minimization problem, a second saddle point problem is derived. It is shown by standard techniques that the resulting weakly coupled system of the two saddle point problems admits a unique solution. For its discretization, (biorthogonal) wavelets are used which allows to formulate the matrix of the coupled system as an ` 2operator, initially in an infinitedimensional space. Using...