40 research outputs found
Metric entropy, n-widths, and sampling of functions on manifolds
We first investigate on the asymptotics of the Kolmogorov metric entropy and
nonlinear n-widths of approximation spaces on some function classes on
manifolds and quasi-metric measure spaces. Secondly, we develop constructive
algorithms to represent those functions within a prescribed accuracy. The
constructions can be based on either spectral information or scattered samples
of the target function. Our algorithmic scheme is asymptotically optimal in the
sense of nonlinear n-widths and asymptotically optimal up to a logarithmic
factor with respect to the metric entropy
Sampling recovery in uniform and other norms
We study the recovery of functions in the uniform norm based on function
evaluations. We obtain worst case error bounds for general classes of functions
in terms of the best -approximation from a given nested sequence of
subspaces combined with bounds on the the Christoffel function of these
subspaces.
Besides an explicit bound, we obtain that linear algorithms using samples
are optimal up to a factor among all algorithms using arbitrary
linear information. Moreover, our results imply that linear sampling algorithms
are optimal up to a constant factor for many reproducing kernel Hilbert spaces.
We also discuss results for approximation in more general seminorms, including
-approximation.Comment: change of title and substantial revision compared to v
Recommended from our members
Innovative Approaches to the Numerical Approximation of PDEs
This workshop was about the numerical solution of PDEs for which classical
approaches,
such as the finite element method, are not well suited or need further
(theoretical) underpinnings.
A prominent example of PDEs for which classical methods are not well
suited are PDEs posed in high space dimensions.
New results on low rank tensor approximation for those problems were
presented.
Other presentations dealt with regularity of PDEs, the numerical solution
of PDEs on surfaces,
PDEs of fractional order, numerical solvers for PDEs that converge with
exponential rates, and
the application of deep neural networks for solving PDEs
Recommended from our members
New Discretization Methods for the Numerical Approximation of PDEs
The construction and mathematical analysis of numerical methods for PDEs is a fundamental area of modern applied mathematics. Among the various techniques that have been proposed in the past, some – in particular, finite element methods, – have been exceptionally successful in a range of applications. There are however a number of important challenges that remain, including the optimal adaptive finite element approximation of solutions to transport-dominated diffusion problems, the efficient numerical approximation of parametrized families of PDEs, and the efficient numerical approximation of high-dimensional partial differential equations (that arise from stochastic analysis and statistical physics, for example, in the form of a backward Kolmogorov equation, which, unlike its formal adjoint, the forward Kolmogorov equation, is not in divergence form, and therefore not directly amenable to finite element approximation, even when the spatial dimension is low). In recent years several original and conceptionally new ideas have emerged in order to tackle these open problems.
The goal of this workshop was to discuss and compare a number of novel approaches, to study their potential and applicability, and to formulate the strategic goals and directions of research in this field for the next five years