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 B-splines and compactly supported dual generators on ‫ޒ‬ 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 stable bases on each level. As a second step these initial complements are then projected into the desired complements spanned by compactly supported biorthogonal wavelets. Since all generators and wavelets on the primal and the dual sides have finitely supported masks, the corresponding decomposition and reconstruction algorithms are simple and efficient. The desired number of vanishing moments is implied by the polynomial exactness of the dual multiresolution. Again due to the polynomial exactness the primal and dual spaces satisfy corresponding Jackson estimates. In addition, Bernstein inequalities can be shown to hold for a range of Sobolev norms depending on the regularity of the primal and dual wavelets. Then it follows from general principles that the wavelets form Riesz bases for L 2 ([0, 1]) and that weighted sequence norms for the coefficients of such wavelet expansions characterize Sobolev spaces and their duals on [0, 1] within a range depending on the parameters in

Summary. This paper is concerned with multilevel techniques for preconditioning linear systems arising from Galerkin methods for elliptic boundary value problems. A general estimate is derived which is based on the characterization of Besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions. The result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers. In particular, our analysis indicates how to realize even uniformly bounded condition numbers. For example, the general results are used to show that the Bramble-Pasciak-Xu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform. Furthermore, they are applied to a general multivariate setting of refinable shift-invariant spaces, in particular, covering those induced by various types of wavelets. Classification (1991): 65F35, 65N30, 41A63, 41A17, 46E35 Mathematics Subject

Starting with Hermite cubic splines as the primal multigenerator, first a dual multigenerator on R is constructed that consists of continuous functions, has small support, and is exact of order 2. We then derive multiresolution sequences on the interval while retaining the polynomial exactness on the primal and dual sides. This guarantees moment conditions of the corresponding wavelets. The concept of stable completions [CDP] is then used to construct the corresponding primal and dual multiwavelets on the interval as follows. An appropriate variation of what is known as a hierarchical basis in finite element methods is shown to be an initial completion. This is then, in a second step, projected into the desired complements spanned by compactly supported biorthogonal multiwavelets. The masks of all multigenerators and multiwavelets are finite so that decomposition and reconstruction algorithms are simple and efficient. Furthermore, in addition to the Jackson estimates which follow from the exactness, one can also show Bernstein inequalities for the primal and dual multiresolutions. Consequently, sequence norms for the coefficients based on such multiwavelet expansions characterize Sobolev norms • H s ([0,1]) for s ∈ (−0.824926, 2.5). In particular, the multiwavelets form Riesz bases for L 2 ([0, 1]).

Abstract. This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic PDE. We address the (difficult) case that the state equation depends on a countable number of parameters i.e., on σ j with j ∈ N, and that the PDE operator may depend non-affinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [CDS1, CDS2], we show that the state and the control are analytic as functions depending on these parameters σ j . We establish sparsity of generalized polynomial chaos (gpc) expansions of both, state and control, in terms of the stochastic coordinate sequence σ = (σ j ) j≥1 of the random inputs, and prove convergence rates of best N -term truncations of these expansions. Such truncations are the key for subsequent computations since they do not assume that the stochastic input data has a finite expansion. In the follow-up paper [KS], we explain two methods how such best N -term truncations can practically be computed, by greedy-type algorithms as in [SG, G], or by multilevel Monte-Carlo methods as in [KSS]. The sparsity result allows in conjunction with adaptive wavelet Galerkin schemes for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [DK, GK, K], see [KS].

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We consider elliptic PDEs (partial differential equations) in the framework of isogeometric analysis, i.e., we treat the physical domain by means of a B-spline or Nurbs mapping which we assume to be regular. The numerical solution of the PDE is computed by means of tensor product B-splines mapped onto the physical domain. We construct additive multilevel preconditioners and show that they are asymptotically optimal, i.e., the spectral condition number of the resulting preconditioned stiffness matrix is independent of h. Together with a nested iteration scheme, this enables an iterative solution scheme of optimal linear complexity. The theoretical results are substantiated by numerical examples in two and three space dimensions.

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