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) for all Sobolev spaces $H^{r, mix}([0,1]^d)$ and $H^s([0,1]^d)$ for $s>d/2$.
Aicke Hinrichs, David Krieg, Erich Novak et al.
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 almost optimal and for some other problems random information is much worse than optimal information. We prove new results and give a short survey of known results.
David Krieg
We construct Monte Carlo methods for the $L^2$-approximation in Hilbert spaces of multivariate functions sampling no more than $n$ function values of the target function. Their errors catch up with the rate of convergence and the preasymptotic behavior of the error of any algorithm sampling $n$ pieces of arbitrary linear information, including function values.
David Krieg
The approximation numbers of the $L_2$-embedding of mixed order Sobolev functions on the $d$-torus are well studied. They are given as the nonincreasing rearrangement of the $d$-th tensor power of the approximation number sequence in the univariate case. I present results on the asymptotic and preasymptotic behavior for tensor powers of arbitrary sequences of polynomial decay. This can be used to study the approximation numbers of many other tensor product operators, like the embedding of mixed order Sobolev functions on the $d$-cube into $L_2\left([0,1]^d\right)$ or the embedding of mixed order Jacobi functions on the $d$-cube into $L_2\left([0,1]^d,w_d\right)$ with Jacobi weight $w_d$.
David Krieg, Daniel Rudolf
We present deterministic algorithms for the uniform recovery of $d$-variate rank one tensors from function values. These tensors are given as product of $d$ univariate functions whose $r$th weak derivative is bounded by $M$. The recovery problem is known to suffer from the curse of dimensionality for $M\geq 2^r r!$. For smaller $M$, a randomized algorithm is known which breaks the curse. We construct a deterministic algorithm which is even less costly. In fact, we completely characterize the tractability of this problem by three different ranges of the parameter $M$.
David Krieg
We are concerned with the numerical integration of functions from the Sobolev space $H^{r,\text{mix}}([0,1]^d)$ of dominating mixed smoothness $r\in\mathbb{N}$ over the $d$-dimensional unit cube. In 1976, K. K. Frolov introduced a deterministic quadrature rule whose worst case error has the order $n^{-r} \, (\log n)^{(d-1)/2}$ with respect to the number $n$ of function evaluations. This is known to be optimal. 39 years later, Erich Novak and me introduced a randomized version of this algorithm using $d$ random dilations. We showed that its error is bounded above by a constant multiple of $n^{-r-1/2} \, (\log n)^{(d-1)/2}$ in expectation and by $n^{-r} \, (\log n)^{(d-1)/2}$ almost surely. The main term $n^{-r-1/2}$ is again optimal and it turns out that the very same algorithm is also optimal for the isotropic Sobolev space $H^s([0,1]^d)$ of smoothness $s>d/2$. We also added a random shift to this algorithm to make it unbiased. Just recently, Mario Ullrich proved that the expected error of the resulting algorithm on $H^{r,\text{mix}}([0,1]^d)$ is even bounded above by $n^{-r-1/2}$. This thesis is a review of the mentioned upper bounds and their proofs.
David Krieg, Mario Ullrich
We consider approximation or recovery of functions based on a finite number of function evaluations. This is a well-studied problem in optimal recovery, machine learning, and numerical analysis in general, but many fundamental insights were obtained only recently. We discuss different aspects of the information-theoretic limit that appears because of the limited amount of data available, as well as algorithms and sampling strategies that come as close to it as possible. We also discuss (optimal) sampling in a broader sense, allowing other types of measurements that may be nonlinear, adaptive and random, and present several relations between the different settings in the spirit of information-based complexity. We hope that this article provides both, a basic introduction to the subject and a contemporary summary of the current state of research.
David Krieg
We study multivariate problems like function approximation, numerical integration, global optimization and dispersion. We obtain new results on the information complexity $n(\varepsilon,d)$ of these problems. The information complexity is the amount of information (e.g. the number of function values) that is needed to solve the $d$-dimensional problem up to a prescribed error $\varepsilon>0$. We present optimal algorithms for some of these problems. An extended abstract can be found in the section "Introduction and Results".
David Krieg
We consider functions on the $d$-dimensional unit cube whose partial derivatives up to order $r$ are bounded by one. It is known that the minimal number of function values that is needed to approximate the integral of such functions up to the error $\varepsilon$ is of order $(d/ \varepsilon)^{d/r}$. Among other things, we show that the minimal number of function values that is needed to approximate such functions in the uniform norm is of order $(d^{r/2} /\varepsilon)^{d/r}$ whenever $r$ is even.
David Krieg
For any natural number $d$ and positive number $\varepsilon$, we present a point set in the $d$-dimensional unit cube $[0,1]^d$ that intersects every axis-aligned box of volume greater than $\varepsilon$. These point sets are very easy to handle and in a vast range for $\varepsilon$ and $d$, we do not know any smaller set with this property.