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Abstract. This paper is concerned with the construction and analysis of wavelet-based 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 N-term 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 N-term 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.

This paper is concerned with the construction of biorthogonal wavelet bases defined on a union of parametric images of the unit d-cube. 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 pseudo-differential 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. Key Words: Biorthogonal wavelets, norm equivalences, boundary element methods, composite multiresolution, multiscale methods fo...

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...

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 non-homogeneous 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...

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 indenite or unsymmetric systems of operator equations. This is accomplished by rst introducing techniques (such as the least squares formulation developed in [DKS]) that transform the original (continuous) problem into an equivalent innite system of equations which is now well-posed in the Euclidean metric. It is then shown how to utilize adaptive techniques to solve the resulting innite 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 innite problem. This rests on an adaptive application of the innite dimensional operator to nite vectors representing elements from nite 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 wavelet-best N-term 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.

This paper is concerned with the construction of biorthogonal wavelet bases on n-dimensional cubes which provide Riesz bases for Sobolev and Besov spaces with homogeneous Dirichlet boundary conditions on part of the boundary. The essential point is that the primal and dual wavelets satisfy certain corresponding complementary boundary conditions. These results form the key ingredients of the construction of wavelet bases on manifolds [DS2] that have been developed for the treatment of operator equations of positive and negative order. Key Words: Topological isomorphisms, Sobolev and Besov spaces, biorthogonal wavelet bases, moment conditions, complementary boundary conditions. AMS Subject Classification: 46A20, 46E39, 46B15, 1 Introduction A number of recent investigations [CTU, DS1, DSt] aimed at extending the applicability of wavelet methods for the numerical treatment of operator equations to problems involving realistic domain geometries. In spite of the extremely promising pote...

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.

In this paper, we construct a class of locally supported wavelet bases for C 0 Lagrange finite element spaces on possibly non-uniform meshes on n- dimensional domains or manifolds. The wavelet bases are stable in the Sobolev spaces H s for jsj ! 3 2 (jsj 1 on Lipschitz' manifolds), and the wavelets can, in principal, be arranged to have any desired order of vanishing moments. As a consequence, these bases can be used e.g. for constructing an optimal solver of discretized H s -elliptic problems for s in above ranges. The construction of the wavelets consists of two parts: An implicit part involves some computations on a reference element which, for each type of finite element space, have to be performed only once. In addition there is an explicit part which takes care of the necessary adaptations of the wavelets to the actual mesh. The only condition we need for this construction to work is that the refinements of initial elements are uniform. We will show that the wavelet ba...

In this paper an adaptive wavelet scheme for saddle point problems is developed and analysed. Under the assumption that the underlying continuous problem satisfies the inf-sup condition it is shown in the first part under which circumstances the scheme exhibits asymptotically optimal complexity. This means that within a certain range the convergence rate which relates the achieved accuracy to the number of involved degrees of freedom is asymptotically the same as the best wavelet N-term approximation of the solution with respect to the relevant norms. Moreover, the computational work needed to compute the approximate solution stays proportional to the number of degrees of freedom. It is remarkable that compatibility constraints on the trial spaces such as the Ladyshenskaja-Babuska-Brezzi (LBB) condition do not arise. In the second part the general results are applied to the Stokes problem. Aside from the verification of those requirements on the algorithmic ingredients the theoretical analysis had been based upon, the regularity of the solutions in certain Besov scales is analyzed. These results reveal under which circumstances the work/accuracy balance of the adaptive scheme is even asymptotically better than that resulting from preassigned uniform refinements. This in turn is used to select and interpret some first numerical experiments that are to quantitatively complement the theoretical results for the Stokes problem.