material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

It has recently been demonstrated that locality of spatial supports in the parametrization of coefficients in elliptic PDEs can lead to improved convergence rates of sparse polynomial expansions of the corresponding parameter-dependent solutions. These results by themselves do not yield practically realizable approximations, since they do not cover the approximation of the arising expansion coefficients, which are functions of the spatial variable. In this work, we study the combined spatial and parametric approximability for elliptic PDEs with affine or lognormal parametrizations of the diffusion coefficients and corresponding Taylor, Jacobi, and Hermite expansions, to obtain fully discrete approximations. Our analysis yields convergence rates of the fully discrete approximation in terms of the total number of degrees of freedom. The main vehicle consists in ℓ p summability results for the coefficient sequences measured in higher-order Hilbertian Sobolev norms. We also discuss similar results for nonHilbertian Sobolev norms which arise naturally when using adaptive spatial discretizations.

In convex optimization, duality theory can sometimes lead to simpler solution methods than those resulting from direct primal analysis. In this paper, this principle is applied to a class of composite variational problems arising in particular in signal recovery. These problems are not easily amenable to solution by current methods but they feature Fenchel-Moreau-Rockafellar dual problems that can be solved by forward-backward splitting. The proposed algorithm produces simultaneously a sequence converging weakly to a dual solution, and a sequence converging strongly to the primal solution. Our framework is shown to capture and extend several existing duality-based signal recovery methods and to be applicable to a variety of new problems beyond their scope.

No abstract

We prove lower bounds for the error of optimal cubature formulae for d-variate functions from Besov spaces of mixed smoothness B α p,θ (G d ) in the case 0 < p, θ ≤ ∞ and α > 1/p, whereWe prove upper bounds for QMC methods of integration on the Fibonacci lattice for bivariate periodic functions from B α p,θ (T 2 ) in the case 1 ≤ p ≤ ∞, 0 < θ ≤ ∞, α > 1/p. A non-periodic modification of this classical formula yields upper bounds for B α p,θ (I 2 ) if 1/p < α < 1 + 1/p. In combination these results yield the correct asymptotic error of optimal cubature formulae for functions from B α p,θ (G 2 ) and indicate that a corresponding result is most likely also true in case d > 2. This is compared to the correct asymptotic of optimal cubature formulae on Smolyak grids which results in the observation that any cubature formula on Smolyak grids is never optimal for the general setting.