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

Abstract. We consider a space-time variational formulation for linear parabolic partial differential equations. We introduce an associated Petrov-Galerkin truth finite element discretization with favorable discrete inf-sup constant β δ , the inverse of which enters into error estimates: β δ is unity for the heat equation; β δ decreases only linearly in time for non-coercive (but asymptotically stable) convection operators. The latter in turn permits effective long-time a posteriori error bounds for reduced basis approximations, in sharp contrast to classical (pessimistic) exponentially growing energy estimates. The paper contains a full analysis and various extensions for the formulation introduced briefly by Urban and Patera (2012) as well as numerical results for a model reaction-convection-diffusion equation.

List of Algorithms xii Preface xiii Acknowledgements xv List of Figures xvii List of Tables xxvi 1 8 Wavelets on general domains 257 8.1 Multiresolution on the interval 8.1.1 Refinement matrices 8.

The Kolmogorov N -width d N (M) describes the rate of the worst-case error (w.r.t. a subset M ⊂ H of a normed space H) arising from a projection onto the best-possible linear subspace of H of dimension N ∈ N. Thus, d N (M) sets a limit to any projection-based approximation such as determined by the reduced basis method. While it is known that d N (M) decays exponentially fast for many linear coercive parametrized partial differential equations, i.e., d N (M) = O(e −βN ), we show in this note, that only d N (M) = O(N −1/2 ) for initial-boundaryvalue problems of the hyperbolic wave equation with discontinuous initial conditions. This is aligned with the known slow decay of d N (M) for the linear transport problem.2010 Mathematics Subject Classification. 41A46, 65D15.