6 research outputs found
Algorithms and Complexity for Functions on General Domains
Error bounds and complexity bounds in numerical analysis and
information-based complexity are often proved for functions that are defined on
very simple domains, such as a cube, a torus, or a sphere. We study optimal
error bounds for the approximation or integration of functions defined on and only assume that is a bounded Lipschitz domain. Some
results are even more general. We study three different concepts to measure the
complexity: order of convergence, asymptotic constant, and explicit uniform
bounds, i.e., bounds that hold for all (number of pieces of information)
and all (normalized) domains. It is known for many problems that the order of
convergence of optimal algorithms does not depend on the domain . We present examples for which the following statements are true:
1) Also the asymptotic constant does not depend on the shape of or the
imposed boundary values, it only depends on the volume of the domain.
2) There are explicit and uniform lower (or upper, respectively) bounds for
the error that are only slightly smaller (or larger, respectively) than the
asymptotic error bound.Comment: minor revision; to appear in Journal of Complexit
Counting via entropy: new preasymptotics for the approximation numbers of Sobolev embeddings
In this paper, we reveal a new connection between approximation numbers of
periodic Sobolev type spaces, where the smoothness weights on the Fourier
coefficients are induced by a (quasi-)norm on , and
entropy numbers of the embedding . This connection yields preasymptotic error bounds for
approximation numbers of isotropic Sobolev spaces, spaces of analytic
functions, and spaces of Gevrey type in and , which find application
in the context of Galerkin methods. Moreover, we observe that approximation
numbers of certain Gevrey type spaces behave preasymptotically almost identical
to approximation numbers of spaces of dominating mixed smoothness. This
observation can be exploited, for instance, for Galerkin schemes for the
electronic Schr\"odinger equation, where mixed regularity is present
Tight error bounds for rank-1 lattice sampling in spaces of hybrid mixed smoothness
We consider the approximate recovery of multivariate periodic functions from
a discrete set of function values taken on a rank- integration lattice. The
main result is the fact that any (non-)linear reconstruction algorithm taking
function values on a rank- lattice of size has a dimension-independent
lower bound of when considering the optimal
worst-case error with respect to function spaces of (hybrid) mixed smoothness
on the -torus. We complement this lower bound with upper bounds
that coincide up to logarithmic terms. These upper bounds are obtained by a
detailed analysis of a rank-1 lattice sampling strategy, where the rank-1
lattices are constructed by a component-by-component (CBC) method. This
improves on earlier results obtained in [25] and [27]. The lattice (group)
structure allows for an efficient approximation of the underlying function from
its sampled values using a single one-dimensional fast Fourier transform. This
is one reason why these algorithms keep attracting significant interest. We
compare our results to recent (almost) optimal methods based upon samples on
sparse grids
-Numbers of Embeddings of Weighted Wiener Algebras
In this paper we study the asymptotic behavior of Kolmogorov, approximation,
Bernstein and Weyl numbers of embeddings and , where
is a weighted Wiener algebra of
mixed smoothness and is the Wiener algebra
itself, both defined on the -dimensional torus . Our main
interest consists in the calculation of the associated asymptotic constants.Comment: 25 page
How anisotropic mixed smoothness affects the decay of singular numbers of Sobolev embeddings
We continue the research on the asymptotic and preasymptotic decay of
singular numbers for tensor product Hilbert-Sobolev type embeddings in high
dimensions with special emphasis on the influence of the underlying dimension
. The main focus in this paper lies on tensor products involving univariate
Sobolev type spaces with different smoothness. We study the embeddings into
and . In other words, we investigate the worst-case approximation
error measured in and when only linear samples of the function
are available. Recent progress in the field shows that accurate bounds on the
singular numbers are essential for recovery bounds using only function values.
The asymptotic bounds in our setting are known for a long time. In this paper
we contribute the correct asymptotic constant and explicit bounds in the
preasymptotic range for . We complement and improve on several results in
the literature. In addition, we refine the error bounds coming from the setting
where the smoothness vector is moderately increasing, which has been already
studied by Papageorgiou and Wo{\'z}niakowski
Hyperbolic Cross Approximation
Hyperbolic cross approximation is a special type of multivariate
approximation. Recently, driven by applications in engineering, biology,
medicine and other areas of science new challenging problems have appeared. The
common feature of these problems is high dimensions. We present here a survey
on classical methods developed in multivariate approximation theory, which are
known to work very well for moderate dimensions and which have potential for
applications in really high dimensions. The theory of hyperbolic cross
approximation and related theory of functions with mixed smoothness are under
detailed study for more than 50 years. It is now well understood that this
theory is important both for theoretical study and for practical applications.
It is also understood that both theoretical analysis and construction of
practical algorithms are very difficult problems. This explains why many
fundamental problems in this area are still unsolved. Only a few survey papers
and monographs on the topic are published. This and recently discovered deep
connections between the hyperbolic cross approximation (and related sparse
grids) and other areas of mathematics such as probability, discrepancy, and
numerical integration motivated us to write this survey. We try to put emphases
on the development of ideas and methods rather than list all the known results
in the area. We formulate many problems, which, to our knowledge, are open
problems. We also include some very recent results on the topic, which
sometimes highlight new interesting directions of research. We hope that this
survey will stimulate further active research in this fascinating and
challenging area of approximation theory and numerical analysis.Comment: 185 pages, 24 figure