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

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 $B^\alpha_{\Psi,q}(L_p(\Gamma))$ 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 $d$-dimensional manifolds and investigate some analytical properties (such as, e.g., embeddings and best $n$-term approximation rates) of the resulting quasi-Banach spaces. In particular, we prove that different prominent constructions of biorthogonal wavelet systems $\Psi$ on domains or manifolds $\Gamma$ which admit a decomposition into smooth patches actually generate the same Besov-type function spaces $B^\alpha_{\Psi,q}(L_p(\Gamma))$, provided that their univariate ingredients possess a sufficiently large order of cancellation and regularity (compared to the smoothness parameter $\alpha$ of the space). For this purpose, a theory of almost diagonal matrices on related sequence spaces $b^\alpha_{p,q}(\nabla)$ 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

We study regularity properties of solutions to operator equations on patchwise smooth manifolds $\partial\Omega$ such as, e.g., boundaries of polyhedral domains $\Omega \subset \mathbb{R}^3$. Using suitable biorthogonal wavelet bases $\Psi$, we introduce a new class of Besov-type spaces $B_{\Psi,q}^\alpha(L_p(\partial \Omega))$ of functions $u\colon\partial\Omega\rightarrow\mathbb{C}$. Special attention is paid on the rate of convergence for best $n$-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 $\partial\Omega$ into $B_{\Psi,\tau}^\alpha(L_\tau(\partial \Omega))$, $1/\tau=\alpha/2 + 1/2$, 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 $\Omega$.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

A New Class of Cellular Automata for Reaction-Diffusion Systems

We introduce a new class of cellular automata to model reaction-diffusion systems in a quantitatively correct way. The construction of the CA from the reaction-diffusion equation relies on a moving average procedure to implement diffusion, and a probabilistic table-lookup for the reactive part. The applicability of the new CA is demonstrated using the Ginzburg-Landau equation.Comment: 4 pages, RevTeX 3.0 , 3 Figures 214972 bytes tar, compressed, uuencode

Construction of quasi-Monte Carlo rules for multivariate integration in spaces of permutation-invariant functions

We study multivariate integration of functions that are invariant under the permutation (of a subset) of their arguments. Recently, in Nuyens, Suryanarayana, and Weimar (Adv. Comput. Math. (2016), 42(1):55--84), the authors derived an upper estimate for the $n$th minimal worst case error for such problems, and showed that under certain conditions this upper bound only weakly depends on the dimension. We extend these results by proposing two (semi-) explicit construction schemes. We develop a component-by-component algorithm to find the generating vector for a shifted rank-$1$ lattice rule that obtains a rate of convergence arbitrarily close to $\mathcal{O}(n^{-\alpha})$, where $\alpha>1/2$ denotes the smoothness of our function space and $n$ is the number of cubature nodes. Further, we develop a semi-constructive algorithm that builds on point sets which can be used to approximate the integrands of interest with a small error; the cubature error is then bounded by the error of approximation. Here the same rate of convergence is achieved while the dependence of the error bounds on the dimension $d$ is significantly improved

Rank-1 lattice rules for multivariate integration in spaces of permutation-invariant functions: Error bounds and tractability

We study multivariate integration of functions that are invariant under permutations (of subsets) of their arguments. We find an upper bound for the $n$th minimal worst case error and show that under certain conditions, it can be bounded independent of the number of dimensions. In particular, we study the application of unshifted and randomly shifted rank-$1$ lattice rules in such a problem setting. We derive conditions under which multivariate integration is polynomially or strongly polynomially tractable with the Monte Carlo rate of convergence $O(n^{-1/2})$. Furthermore, we prove that those tractability results can be achieved with shifted lattice rules and that the shifts are indeed necessary. Finally, we show the existence of rank-$1$ lattice rules whose worst case error on the permutation- and shift-invariant spaces converge with (almost) optimal rate. That is, we derive error bounds of the form $O(n^{-\lambda/2})$ for all $1 \leq \lambda < 2 \alpha$, where $\alpha$ denotes the smoothness of the spaces. Keywords: Numerical integration, Quadrature, Cubature, Quasi-Monte Carlo methods, Rank-1 lattice rules.Comment: 26 pages; minor changes due to reviewer's comments; the final publication is available at link.springer.co

On lower bounds for integration of multivariate permutation-invariant functions

In this note we study multivariate integration for permutation-invariant functions from a certain Banach space E_{d,\alpha} of Korobov type in the worst case setting. We present a lower error bound which particularly implies that in dimension d every cubature rule which reduces the initial error necessarily uses at least d+1 function values. Since this holds independently of the number of permutation-invariant coordinates, this shows that the integration problem can never be strongly polynomially tractable in this setting. Our assertions generalize results due to Sloan and Wo\'zniakowski. Moreover, for large smoothness parameters \alpha our bound can not be improved. Finally, we extend our results to the case of permutation-invariant functions from Korobov-type spaces equipped with product weights. Keywords: Permutation-invariance, Integration, Information complexity, Tractability, Lower boundsComment: 16 page

Besov regularity of solutions to the p-Poisson equation

In this paper, we study the regularity of solutions to the $p$-Poisson equation for all $1<p<\infty$. In particular, we are interested in smoothness estimates in the adaptivity scale $B^\sigma_{\tau}(L_{\tau}(\Omega))$, $1/\tau = \sigma/d+1/p$, of Besov spaces. The regularity in this scale determines the order of approximation that can be achieved by adaptive and other nonlinear approximation methods. It turns out that, especially for solutions to $p$-Poisson equations with homogeneous Dirichlet boundary conditions on bounded polygonal domains, the Besov regularity is significantly higher than the Sobolev regularity which justifies the use of adaptive algorithms. This type of results is obtained by combining local H\"older with global Sobolev estimates. In particular, we prove that intersections of locally weighted H\"older spaces and Sobolev spaces can be continuously embedded into the specific scale of Besov spaces we are interested in. The proof of this embedding result is based on wavelet characterizations of Besov spaces.Comment: 45 pages, 2 figure

Adherence to secondary stroke prevention strategies - Results from the German stroke data bank

Only very limited data are available concerning patient adherence to antithrombotic medication intended to prevent a recurrent stroke. Reduced adherence and compliance could significantly influence the effects of any stroke prevention strategies. This study from a large stroke data bank provides representative data concerning the rate of stroke victims adhering to their recommended preventive medication. During a 2-year period beginning January 1, 1998, all patients with acute stroke or TIA in 23 neurological departments with an acute stroke unit were included in the German Stroke Data Bank. Data were collected prospectively, reviewed, validated and processed in a central data management unit. Only 12 centers with a follow-up rate of 80% or higher were included in this evaluation. 3,420 patients were followed up after 3 months, and 2,640 patients were followed up one year after their stroke. After one year, 96% of all patients reported still adhere to at least one medical stroke prevention strategy. Of the patients receiving aspirin at discharge, 92.6% reported to use that medication after 3 months and 84% after one year, while 81.6 and 61.6% were the respective figures for clopidogrel, and 85.2 and 77.4% for oral anticoagulation. Most patients who changed medication switched from aspirin to clopidogrel. Under the conditions of this observational study, adherence to stroke prevention strategies is excellent. The highest adherence rate is noticed for aspirin and oral anticoagulation. After one year, very few patients stopped taking stroke preventive medication. Copyright (C) 2003 S. Karger AG, Basel