Robert J. Kunsch

Robert J. Kunsch, Erich Novak, Daniel Rudolf

We compute the integral of a function or the expectation of a random variable with minimal cost and use, for our new algorithm and for upper bounds of the complexity, i.i.d. samples. Under certain assumptions it is possible to select a sample size based on a variance estimation, or -- more generally -- based on an estimation of a (central absolute) $p$-moment. That way one can guarantee a small absolute error with high probability, the problem is thus called solvable. The expected cost of the method depends on the $p$-moment of the random variable, which can be arbitrarily large. In order to prove the optimality of our algorithm we also provide lower bounds. These bounds apply not only to methods based on i.i.d. samples but also to general randomized algorithms. They show that -- up to constants -- the cost of the algorithm is optimal in terms of accuracy, confidence level, and norm of the particular input random variable. Since the considered classes of random variables or integrands are very large, the worst case cost would be infinite. Nevertheless one can define adaptive stopping rules such that for each input the expected cost is finite. We contrast these positive results with examples of integration problems that are not solvable.

Robert J. Kunsch

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.

Robert J. Kunsch

We study the $L_{\infty}$-approximation of $d$-variate functions from Hilbert spaces via linear functionals as information. It is a common phenomenon in tractability studies that unweighted problems (with each dimension being equally important) suffer from the curse of dimensionality in the deterministic setting, that is, the number $n(\varepsilon,d)$ of information needed in order to solve a problem to within a given accuracy $\varepsilon > 0$ grows exponentially in $d$. We show that for certain approximation problems in periodic tensor product spaces, in particular Korobov spaces with smoothness $r > 1/2$, switching to the randomized setting can break the curse of dimensionality, now having polynomial tractability, namely $n(\varepsilon,d) \preceq \varepsilon^{-2} \, d \, (1 + \log d)$. Similar benefits of Monte Carlo methods in terms of tractability have only been known for integration problems so far.

Robert J. Kunsch

We study the $L_1$-approximation of $d$-variate monotone functions based on information from $n$ function evaluations. It is known that this problem suffers from the curse of dimensionality in the deterministic setting, that is, the number $n(\varepsilon,d)$ of function evaluations needed in order to approximate an unknown monotone function within a given error threshold $\varepsilon$ grows at least exponentially in $d$. This is not the case in the randomized setting (Monte Carlo setting) where the complexity $n(\varepsilon,d)$ grows exponentially in $\sqrt{d}$ (modulo logarithmic terms) only. An algorithm exhibiting this complexity is presented. Still, the problem remains difficult as best known methods are deterministic if $\varepsilon$ is comparably small, namely $\varepsilon \preceq 1/\sqrt{d}$. This inherent difficulty is confirmed by lower complexity bounds which reveal a joint $(\varepsilon,d)$-dependency and from which we deduce that the problem is not weakly tractable.

Robert J. Kunsch, Daniel Rudolf

We study the complexity of approximating integrals of smooth functions at absolute precision $\varepsilon > 0$ with confidence level $1 - δ\in (0,1)$. The optimal error rate for multivariate functions from classical isotropic Sobolev spaces $W_p^r(G)$ with sufficient smoothness on bounded Lipschitz domains $G \subset \mathbb{R}^d$ is determined. It turns out that the integrability index $p$ has an effect on the influence of the uncertainty $δ$ in the complexity. In the limiting case $p = 1$ we see that deterministic methods cannot be improved by randomization. In general, higher smoothness reduces the additional effort for diminishing the uncertainty. Finally, we add a discussion about this problem for function spaces with mixed smoothness.

Glenn Byrenheid, Robert J. Kunsch, Van Kien Nguyen

We consider the order of convergence for linear and nonlinear Monte Carlo approximation of compact embeddings from Sobolev spaces of dominating mixed smoothness defined on the torus $\mathbb{T}^d$ into the space $L_{\infty}(\mathbb{T}^d)$ via methods that use arbitrary linear information. These cases are interesting because we can gain a speedup of up to $1/2$ in the main rate compared to the worst case approximation. In doing so we determine the rate for some cases that have been left open by Fang and Duan.