Related papers: Optimal confidence for Monte Carlo integration of …
We prove upper bounds on the order of convergence of Frolov's cubature formula for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study…
We propose a multi-index algorithm for the Monte Carlo (MC) discretization of a linear, elliptic PDE with affine-parametric input. We prove an error vs. work analysis which allows a multi-level finite-element approximation in the physical…
We consider the problem of constructing honest and adaptive confidence sets in Lp-loss (with p>=1 and p < infinity) over sets of Sobolev-type classes, in the setting of non-parametric Gaussian regression. The objective is to adapt the…
The main purpose of the paper is to study sharp estimates of approximation of periodic functions in the H\"older spaces $H_p^{r,\alpha}$ for all $0<p\le\infty$ and $0<\alpha\le r$. By using modifications of the classical moduli of…
QMC rules are equal weight quadrature rules for approximating integrals over $[0,1]^s$. One line of research studies the integration error of functions in the unit ball of so-called Korobov spaces, which are Hilbert spaces of periodic…
Hamiltonian Monte Carlo can provide powerful inference in complex statistical problems, but ultimately its performance is sensitive to various tuning parameters. In this paper we use the underlying geometry of Hamiltonian Monte Carlo to…
Using tools from the theory of operator ideals and s-numbers, we develop a general approach to transfer estimates for $L_2$ -approximation of Sobolev functions into estimates for $L_\infty$-approximation, with precise control of all…
Numerical (and experimental) data analysis often requires the restoration of a smooth function from a set of sampled integrals over finite bins. We present the bin hierarchy method that efficiently computes the maximally smooth function…
In this paper we study the sampling recovery problem for certain relevant multivariate function classes which are not compactly embedded into $L_\infty$. Recent tools relating the sampling numbers to the Kolmogorov widths in the uniform…
The problem of estimating the probability p=P(g(X<0) is considered when X represents a multivariate stochastic input of a monotone function g. First, a heuristic method to bound p is formally described, involving a specialized design of…
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…
We propose confidence regions for the parameters of incomplete models with exact coverage of the true parameter in finite samples. Our confidence region inverts a test, which generalizes Monte Carlo tests to incomplete models. The test…
In the context of global optimization of mixed-integer nonlinear optimization formulations, we consider smoothing univariate functions $f$ that satisfy $f(0)=0$, $f$ is increasing and concave on $[0,+\infty)$, $f$ is twice differentiable on…
In the framework of uncertainty quantification, we consider a quantity of interest which depends non-smoothly on the high-dimensional parameter representing the uncertainty. We show that, in this situation, the multilevel Monte Carlo…
We mainly study numerical integration of real valued functions defined on the $d$-dimensional unit cube with all partial derivatives up to some finite order $r\ge1$ bounded by one. It is well known that optimal algorithms that use $n$…
Multifidelity Monte Carlo methods rely on a hierarchy of possibly less accurate but statistically correlated simplified or reduced models, in order to accelerate the estimation of statistics of high-fidelity models without compromising the…
We study integration and $L_2$-approximation on countable tensor products of function spaces of increasing smoothness. We obtain upper and lower bounds for the minimal errors, which are sharp in many cases including, e.g., Korobov, Walsh,…
We study the recovery of multivariate functions from reproducing kernel Hilbert spaces in the uniform norm. Our main interest is to obtain preasymptotic estimates for the corresponding sampling numbers. We obtain results in terms of the…
In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly…
Probabilistic prediction of sequences from images and other high-dimensional data is a key challenge, particularly in risk-sensitive applications. In these settings, it is often desirable to quantify the uncertainty associated with the…