53 citations · 129 across the 16 of their papers we have counts for
23 papers
Sampling recovery in and other norms
David Krieg, Kateryna Pozharska, Mario Ullrich +1
We study the recovery of functions in various norms, including with , based on function evaluations. We obtain worst case error bounds for general classes of…
Tractability of sampling recovery on unweighted function classes
David Krieg
It is well-known that the problem of sampling recovery in the -norm on unweighted Korobov spaces (Sobolev spaces with mixed smoothness) as well as classical smoothness classes…
New lower bounds for the integration of periodic functions
David Krieg, Jan Vybiral
We study the integration problem on Hilbert spaces of (multivariate) periodic functions. The standard technique to prove lower bounds for the error of quadrature rules uses bump fu…
Exponential tractability of -approximation with function values
David Krieg, Pawel Siedlecki, Mario Ullrich +1
We study the complexity of high-dimensional approximation in the -norm when different classes of information are available; we compare the power of function evaluations with t…
A sharp upper bound for sampling numbers in
Matthieu Dolbeault, David Krieg, Mario Ullrich
For a class of complex-valued functions on a set , we denote by its sampling numbers, i.e., the minimal worst-case error on , measured in , that can be achi…
Function recovery on manifolds using scattered data
David Krieg, Mathias Sonnleitner
We consider the task of recovering a Sobolev function on a connected compact Riemannian manifold when given a sample on a finite point set. We prove that the quality of the sam…
Recovery of Sobolev functions restricted to iid sampling
David Krieg, Erich Novak, Mathias Sonnleitner
We study -approximation and integration for functions from the Sobolev space and compare optimal randomized (Monte Carlo) algorithms with algorithms that can only u…
Lower bounds for integration and recovery in
Aicke Hinrichs, David Krieg, Erich Novak +1
Function values are, in some sense, "almost as good" as general linear information for -approximation (optimal recovery, data assimilation) of functions from a reproducing ker…
Random points are optimal for the approximation of Sobolev functions
David Krieg, Mathias Sonnleitner
We show that independent and uniformly distributed sampling points are as good as optimal sampling points for the approximation of functions from the Sobolev space on bo…
Lower Bounds for the Error of Quadrature Formulas for Hilbert Spaces
Aicke Hinrichs, David Krieg, Erich Novak +1
We prove lower bounds for the worst case error of quadrature formulas that use given sample points $\X_n = \{ x_1, \dots , x_n \}$. We are mainly interested in optimal point sets $…
Expected dispersion of uniformly distributed points
Aicke Hinrichs, David Krieg, Robert J. Kunsch +1
The dispersion of a point set in is the volume of the largest axis parallel box inside the unit cube that does not intersect with the point set. We study the expected dis…
Algorithms and Complexity for some Multivariate Problems
David Krieg
We study multivariate problems like function approximation, numerical integration, global optimization and dispersion. We obtain new results on the information complexity $n(\varep…
Function values are enough for -approximation
David Krieg, Mario Ullrich
We study the -approximation of functions from a Hilbert space and compare the sampling numbers with the approximation numbers. The sampling number is the minimal worst c…
On the power of random information
Aicke Hinrichs, David Krieg, Erich Novak +2
We study approximation and integration problems and compare the quality of optimal information with the quality of random information. For some problems random information is almos…
Random sections of ellipsoids and the power of random information
Aicke Hinrichs, David Krieg, Erich Novak +2
We study the circumradius of the intersection of an -dimensional ellipsoid with semi-axes with random subspaces of codimension . We find t…
Uniform recovery of high-dimensional -functions
David Krieg
We consider functions on the -dimensional unit cube whose partial derivatives up to order are bounded by one. It is known that the minimal number of function values that is…
Recovery algorithms for high-dimensional rank one tensors
David Krieg, Daniel Rudolf
We present deterministic algorithms for the uniform recovery of -variate rank one tensors from function values. These tensors are given as product of univariate functions wh…
On the Dispersion of Sparse Grids
David Krieg
For any natural number and positive number , we present a point set in the -dimensional unit cube that intersects every axis-aligned box of volume gre…
Optimal Monte Carlo Methods for -Approximation
David Krieg
We construct Monte Carlo methods for the -approximation in Hilbert spaces of multivariate functions sampling no more than function values of the target function. Their err…
Tensor power sequences and the approximation of tensor product operators
David Krieg
The approximation numbers of the -embedding of mixed order Sobolev functions on the -torus are well studied. They are given as the nonincreasing rearrangement of the -th…
On the Randomization of Frolov's Algorithm for Multivariate Integration
David Krieg
We are concerned with the numerical integration of functions from the Sobolev space of dominating mixed smoothness over the -dimensi…
A Universal Algorithm for Multivariate Integration
David Krieg, Erich Novak
We present an algorithm for multivariate integration over cubes that is unbiased and has optimal order of convergence (in the randomized sense as well as in the worst case setting)…
Rigidity of Circle Packings with Crosscuts
David Krieg, Elias Wegert
Circle packings with specified patterns of tangencies form a discrete counterpart of analytic functions. In this paper we study univalent packings (with a combinatorial closed disk…