6,766 research outputs found
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 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 -dimensional manifolds and investigate some analytical properties
(such as, e.g., embeddings and best -term approximation rates) of the
resulting quasi-Banach spaces. In particular, we prove that different prominent
constructions of biorthogonal wavelet systems on domains or manifolds
which admit a decomposition into smooth patches actually generate the
same Besov-type function spaces , provided that
their univariate ingredients possess a sufficiently large order of cancellation
and regularity (compared to the smoothness parameter of the space).
For this purpose, a theory of almost diagonal matrices on related sequence
spaces 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 such as, e.g., boundaries of
polyhedral domains . Using suitable biorthogonal
wavelet bases , we introduce a new class of Besov-type spaces
of functions
. Special attention is paid on the
rate of convergence for best -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
into , ,
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 .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
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
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- 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 . 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- 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
for all , where 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
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 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- lattice rule
that obtains a rate of convergence arbitrarily close to
, where denotes the smoothness of our
function space and 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 is significantly improved
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
The Complexity of Linear Tensor Product Problems in (Anti-) Symmetric Hilbert Spaces
We study linear problems defined on tensor products of Hilbert spaces with an
additional (anti-) symmetry property. We construct a linear algorithm that uses
finitely many continuous linear functionals and show an explicit formula for
its worst case error in terms of the singular values of the univariate problem.
Moreover, we show that this algorithm is optimal with respect to a wide class
of algorithms and investigate its complexity. We clarify the influence of
different (anti-) symmetry conditions on the complexity, compared to the
classical unrestricted problem. In particular, for symmetric problems we give
characterizations for polynomial tractability and strong polynomial
tractability in terms of the amount of the assumed symmetry. Finally, we apply
our results to the approximation problem of solutions of the electronic
Schr\"odinger equation.Comment: Extended version (53 pages); corrected typos, added journal referenc
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