Discontinuous information in the worst case and randomized settings

We believe that discontinuous linear information is never more powerful than continuous linear information for approximating continuous operators. We prove such a result in the worst case setting. In the randomized setting we consider compact linear operators defined between Hilbert spaces. In this case, the use of discontinuous linear information in the randomized setting cannot be much more powerful than continuous linear information in the worst case setting. These results can be applied when function evaluations are used even if function values are defined only almost everywhere

Optimal Algorithms for Numerical Integration: Recent Results and Open Problems

We present recent results on optimal algorithms for numerical integration and several open problems. The paper has six parts: 1. Introduction 2. Lower Bounds 3. Universality 4. General Domains 5. iid Information 6. Concluding RemarksComment: Survey written for the MCQMC conference in Linz, 26 pages. arXiv admin note: text overlap with arXiv:2108.0205

High-Dimensional Function Approximation: Breaking the Curse with Monte Carlo Methods

In this dissertation we study the tractability of the information-based complexity $n(\varepsilon,d)$ for $d$-variate function approximation problems. In the deterministic setting for many unweighted problems the curse of dimensionality holds, that means, for some fixed error tolerance $\varepsilon>0$ the complexity $n(\varepsilon,d)$ grows exponentially in $d$. For integration problems one can usually break the curse with the standard Monte Carlo method. For function approximation problems, however, similar effects of randomization have been unknown so far. The thesis contains results on three more or less stand-alone topics. For an extended five page abstract, see the section "Introduction and Results". Chapter 2 is concerned with lower bounds for the Monte Carlo error for general linear problems via Bernstein numbers. This technique is applied to the $L_{\infty}$-approximation of certain classes of $C^{\infty}$-functions, where it turns out that randomization does not affect the tractability classification of the problem. Chapter 3 studies the $L_{\infty}$-approximation of functions from Hilbert spaces with methods that may use arbitrary linear functionals as information. For certain classes of periodic functions from unweighted periodic tensor product spaces, in particular Korobov spaces, we observe the curse of dimensionality in the deterministic setting, while with randomized methods we achieve polynomial tractability. Chapter 4 deals with the $L_1$-approximation of monotone functions via function values. It is known that this problem suffers from the curse in the deterministic setting. An improved lower bound shows that the problem is still intractable in the randomized setting. However, Monte Carlo breaks the curse, in detail, for any fixed error tolerance $\varepsilon>0$ the complexity $n(\varepsilon,d)$ grows exponentially in $\sqrt{d}$ only.Comment: This is the author's submitted PhD thesis, still in the referee proces

Randomized Approximation of Sobolev Embeddings II

We study the approximation of Sobolev embeddings by linear randomized algorithms based on function values. Both the source and the target space are Sobolev spaces of non-negative smoothness order, defined on a bounded Lipschitz domain. The optimal order of convergence is determined. We also study the deterministic setting. Using interpolation, we extend the results to other classes of function spaces. In this context a problem posed by Novak and Wo´zniakowski is solved. Finally, we present an application to the complexity of general elliptic PDE