2,073 research outputs found

    Almost diagonal matrices and Besov-type spaces based on wavelet expansions

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    This paper is concerned with problems in the context of the theoretical foundation of adaptive (wavelet) algorithms for the numerical treatment of operator equations. It is well-known that the analysis of such schemes naturally leads to function spaces of Besov type. But, especially when dealing with equations on non-smooth manifolds, the definition of these spaces is not straightforward. Nevertheless, motivated by applications, recently Besov-type spaces BΚ,qα(Lp(Γ))B^\alpha_{\Psi,q}(L_p(\Gamma)) on certain two-dimensional, patchwise smooth surfaces were defined and employed successfully. In the present paper, we extend this definition (based on wavelet expansions) to a quite general class of dd-dimensional manifolds and investigate some analytical properties (such as, e.g., embeddings and best nn-term approximation rates) of the resulting quasi-Banach spaces. In particular, we prove that different prominent constructions of biorthogonal wavelet systems Κ\Psi on domains or manifolds Γ\Gamma which admit a decomposition into smooth patches actually generate the same Besov-type function spaces BΚ,qα(Lp(Γ))B^\alpha_{\Psi,q}(L_p(\Gamma)), provided that their univariate ingredients possess a sufficiently large order of cancellation and regularity (compared to the smoothness parameter α\alpha of the space). For this purpose, a theory of almost diagonal matrices on related sequence spaces bp,qα(∇)b^\alpha_{p,q}(\nabla) of Besov type is developed. Keywords: Besov spaces, wavelets, localization, sequence spaces, adaptive methods, non-linear approximation, manifolds, domain decomposition.Comment: 38 pages, 2 figure

    Besov regularity for operator equations on patchwise smooth manifolds

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    We study regularity properties of solutions to operator equations on patchwise smooth manifolds ∂Ω\partial\Omega such as, e.g., boundaries of polyhedral domains Ω⊂R3\Omega \subset \mathbb{R}^3. Using suitable biorthogonal wavelet bases Κ\Psi, we introduce a new class of Besov-type spaces BΚ,qα(Lp(∂Ω))B_{\Psi,q}^\alpha(L_p(\partial \Omega)) of functions u ⁣:∂Ω→Cu\colon\partial\Omega\rightarrow\mathbb{C}. Special attention is paid on the rate of convergence for best nn-term wavelet approximation to functions in these scales since this determines the performance of adaptive numerical schemes. We show embeddings of (weighted) Sobolev spaces on ∂Ω\partial\Omega into BΚ,τα(Lτ(∂Ω))B_{\Psi,\tau}^\alpha(L_\tau(\partial \Omega)), 1/τ=α/2+1/21/\tau=\alpha/2 + 1/2, which lead us to regularity assertions for the equations under consideration. Finally, we apply our results to a boundary integral equation of the second kind which arises from the double layer ansatz for Dirichlet problems for Laplace's equation in Ω\Omega.Comment: 42 pages, 3 figures, updated after peer review. Preprint: Bericht Mathematik Nr. 2013-03 des Fachbereichs Mathematik und Informatik, Universit\"at Marburg. To appear in J. Found. Comput. Mat

    Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs

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    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 σj\sigma_j with j∈Nj\in\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 σj\sigma_j. We establish sparsity of generalized polynomial chaos (gpc) expansions of both, state and control, in terms of the stochastic coordinate sequence σ=(σj)j≄1\sigma = (\sigma_j)_{j\ge 1} of the random inputs, and prove convergence rates of best NN-term truncations of these expansions. Such truncations are the key for subsequent computations since they do {\em not} assume that the stochastic input data has a finite expansion. In the follow-up paper [KS2], we explain two methods how such best NN-term truncations can practically be computed, by greedy-type algorithms as in [SG, Gi1], 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 [KS2]

    An adaptive Uzawa FEM for the Stokes problem: Convergence without the inf-sup condition

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    We introduce and study an adaptive finite element method (FEM) for the Stokes system based on an Uzawa outer iteration to update the pressure and an elliptic adaptive inner iteration for velocity. We show linear convergence in terms of the outer iteration counter for the pairs of spaces consisting of continuous finite elements of degree k for velocity, whereas for pressure the elements can be either discontinuous of degree k - 1 or continuous of degree k -1 and k. The popular Taylor-Hood family is the sole example of stable elements included in the theory, which in turn relies on the stability of the continuous problem and thus makes no use of the discrete inf-sup condition. We discuss the realization and complexity of the elliptic adaptive inner solver and provide consistent computational evidence that the resulting meshes are quasi-optimal.Fil: BÀnsch, Eberhard. Freie UniversitÀt Berlin;Fil: Morin, Pedro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemåtica Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemåtica Aplicada del Litoral; ArgentinaFil: Nochetto, Ricardo Horacio. University of Maryland; Estados Unido
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