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225
Adaptive wavelet methods for elliptic operator equations— convergence rates
 Math. Comput
, 2001
"... Abstract. This paper is concerned with the construction and analysis of waveletbased adaptive algorithms for the numerical solution of elliptic equations. These algorithms approximate the solution u of the equation by a linear combination of N wavelets. Therefore, a benchmark for their performance ..."
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Cited by 174 (33 self)
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Abstract. This paper is concerned with the construction and analysis of waveletbased adaptive algorithms for the numerical solution of elliptic equations. These algorithms approximate the solution u of the equation by a linear combination of N wavelets. Therefore, a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of N wavelets (so called Nterm approximation), which would be obtained by keeping the N largest wavelet coefficients of the real solution (which of course is unknown). The main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error O(N −s)in the energy norm, whenever such a rate is possible by Nterm approximation. The range of s>0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator. Moreover, it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to N. The adaptive algorithm applies to a wide class of elliptic problems and wavelet bases. The analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization. 1.
A Compressive Landweber Iteration for Solving IllPosed Inverse Problems
, 2008
"... In this paper we shall be concerned with the construction of an adaptive Landweber iteration for solving linear illposed and inverse problems. Classical Landweber iteration schemes provide in combination with suitable regularization parameter rules order optimal regularization schemes. However, for ..."
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Cited by 123 (4 self)
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In this paper we shall be concerned with the construction of an adaptive Landweber iteration for solving linear illposed and inverse problems. Classical Landweber iteration schemes provide in combination with suitable regularization parameter rules order optimal regularization schemes. However, for many applications the implementation of Landweber’s method is numerically very intensive. Therefore we propose an adaptive variant of Landweber’s iteration that significantly may reduce the computational expense, i.e. leading to a compressed version of Landweber’s iteration. We lend the concept of adaptivity that was primarily developed for wellposed operator equations (in particular, for elliptic PDE’s) essentially exploiting the concept of wavelets (frames), Besov regularity, best Nterm approximation and combine it with classical iterative regularization schemes. As the main result of this paper we define an adaptive variant of Landweber’s iteration. In combination with an adequate refinement/stopping rule (apriori as well as aposteriori principles) we prove that the proposed procedure is an regularization method which converges in norm for exact and noisy data. The proposed approach is verified in the field of computerized tomography imaging.
Solving chemical master equations by adaptive wavelet compression
, 2010
"... Solving chemical master equations numerically on a large state space is known to be a difficult problem because the huge number of unknowns is far beyond the capacity of traditional methods. We present an adaptive method which compresses the problem very efficiently by representing the solution in a ..."
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Cited by 112 (7 self)
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Solving chemical master equations numerically on a large state space is known to be a difficult problem because the huge number of unknowns is far beyond the capacity of traditional methods. We present an adaptive method which compresses the problem very efficiently by representing the solution in a sparse wavelet basis that is updated in each step. The stepsize is chosen adaptively according to estimates of the temporal and spatial approximation errors. Numerical examples demonstrate the reliability of the error estimation and show that the method can solve large problems with bimodal solution profiles.
Wavelets on Manifolds I: Construction and Domain Decomposition
 SIAM J. Math. Anal
, 1998
"... The potential of wavelets as a discretization tool for the numerical treatment of operator equations hinges on the validity of norm equivalences for Besov or Sobolev spaces in terms of weighted sequence norms of wavelet expansion coefficients and on certain cancellation properties. These features ar ..."
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Cited by 100 (22 self)
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The potential of wavelets as a discretization tool for the numerical treatment of operator equations hinges on the validity of norm equivalences for Besov or Sobolev spaces in terms of weighted sequence norms of wavelet expansion coefficients and on certain cancellation properties. These features are crucial for the construction of optimal preconditioners, for matrix compression based on sparse representations of functions and operators as well as for the design and analysis of adaptive solvers. However, for realistic domain geometries the relevant properties of wavelet bases could so far only be realized to a limited extent. This paper is concerned with concepts that aim at expanding the applicability of wavelet schemes in this sense. The central issue is to construct wavelet bases with the desired properties on manifolds which can be represented as the disjoint union of smooth parametric images of the standard cube. The approach considered here is conceptually different though from o...
Composite Wavelet Bases for Operator Equations
 MATH. COMP
, 1996
"... This paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit dcube. These bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains. This covers also elliptic boundary ..."
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Cited by 92 (22 self)
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This paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit dcube. These bases are to satisfy certain requirements imposed by applications to a class of operator equations acting on such domains. This covers also elliptic boundary value problems although this study is primarily motivated by our previous analysis of wavelet methods for pseudodifferential equations with special emphasis on boundary integral equations. In this case it is natural to model the boundary surface as a union of parametric images of the unit cube. It will be shown how to construct wavelet bases on the surface which are composed of wavelet bases defined on each surface patch. Here the relevant properties are the validity of norm equivalences in certain ranges of Sobolev scales as well as appropriate moment conditions.
Adaptive Wavelet Methods II  Beyond the Elliptic Case
 FOUND. COMPUT. MATH
, 2000
"... This paper is concerned with the design and analysis of adaptive wavelet methods for systems of operator equations. Its main accomplishment is to extend the range of applicability of the adaptive wavelet based method developed in [CDD] for symmetric positive denite problems to indefinite or unsymmet ..."
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Cited by 71 (18 self)
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This paper is concerned with the design and analysis of adaptive wavelet methods for systems of operator equations. Its main accomplishment is to extend the range of applicability of the adaptive wavelet based method developed in [CDD] for symmetric positive denite problems to indefinite or unsymmetric systems of operator equations. This is accomplished by first introducing techniques (such as the least squares formulation developed in [DKS]) that transform the original (continuous) problem into an equivalent infinite system of equations which is now wellposed in the Euclidean metric. It is then shown how to utilize adaptive techniques to solve the resulting infinite system of equations. This second step requires a signicant modication of the ideas from [CDD]. The main departure from [CDD] is to develop an iterative scheme that directly applies to the innite dimensional problem rather than nite subproblems derived from the infinite problem. This rests on an adaptive application of the innite dimensional operator to finite vectors representing elements from finite dimensional trial spaces. It is shown that for a wide range of problems, this new adaptive method performs with asymptotically optimal complexity, i.e., it recovers an approximate solution with desired accuracy at a computational expense that stays proportional to the number of terms in a corresponding waveletbest Nterm approximation. An important advantage of this adaptive approach is that it automatically stabilizes the numerical procedure so that, for instance, compatibility constraints on the choice of trial spaces like the LBB condition no longer arise.
Fully adaptive multiresolution finite volume schemes for conservation laws
 Math. Comp
, 2003
"... Abstract. The use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by A. Harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation. In this approach the solution is still represented at ..."
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Cited by 56 (14 self)
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Abstract. The use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by A. Harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation. In this approach the solution is still represented at each time step on the finest grid, resulting in an inherent limitation of the potential gain in memory space and computational time. The present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes, in which the solution is represented and computed in a dynamically evolved adaptive grid. A crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages. Several solutions to this problem are proposed, analyzed, and compared in terms of accuracy and complexity. 1.
Wavelet Adaptive Method for Second Order Elliptic Problems Boundary Conditions and Domain Decomposition
 Numer. Math
, 1998
"... Wavelet methods allow to combine high order accuracy, multilevel preconditioning techniques and adaptive approximation, in order to solve efficiently elliptic operator equations. One of the main difficulty in this context is the efficient treatment of nonhomogeneous boundary conditions. In this pap ..."
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Cited by 54 (6 self)
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Wavelet methods allow to combine high order accuracy, multilevel preconditioning techniques and adaptive approximation, in order to solve efficiently elliptic operator equations. One of the main difficulty in this context is the efficient treatment of nonhomogeneous boundary conditions. In this paper, we propose a strategy that allows to append such conditions in the setting of space refinement (i.e. adaptive) discretizations of second order problems. Our method is based on the use of compatible multiscale decompositions for both the domain\Omega and its boundary \Gamma, and on the possibility of characterizing various function spaces from the numerical properties of these decompositions. In particular, this allows the construction of a lifting operator which is stable for a certain range of smoothness classes, and preserves the compression of the solution in the wavelet basis. The analysis is first carried out for the tensor product domain ]0; 1[ 2 , a strategy is developed in orde...
Compression Techniques for Boundary Integral Equations  Optimal Complexity Estimates
, 2002
"... In this paper matrix compression techniques in the context of wavelet Galerkin schemes for boundary integral equations are developed and analyzed that exhibit optimal complexity in the following sense. The fully discrete scheme produces approximate solutions within discretization error accuracy offe ..."
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Cited by 48 (15 self)
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In this paper matrix compression techniques in the context of wavelet Galerkin schemes for boundary integral equations are developed and analyzed that exhibit optimal complexity in the following sense. The fully discrete scheme produces approximate solutions within discretization error accuracy offered by the underlying Galerkin method at a computational expense that is proven to stay proportional to the number of unknowns. Key issues are the second compression, that reduces the near field complexity significantly, and an additional aposteriori compression. The latter one is based on a general result concerning an optimal work balance, that applies, in particular, to the quadrature used to compute the compressed stiffness matrix with sufficient accuracy in linear time. The theoretical results are illustrated by a 3D example on a nontrivial domain.