Related papers: On Quasi-Monte Carlo Methods in Weighted ANOVA Spa…
We study the convergence of these operators in a weighted space of functions on a positive semi-axis and estimate the approximation by using a new type of weighted modulus of continuity and error estimation.
We discuss the efficiency of Monte Carlo methods in solving continuum radiative transfer problems. The sampling of the radiation field and convergence of dust temperature calculations in the case of optically thick clouds are both studied.…
We study the application of a tailored quasi-Monte Carlo (QMC) method to a class of optimal control problems subject to parabolic partial differential equation (PDE) constraints under uncertainty: the state in our setting is the solution of…
We investigate the approximation of generalized Laguerre- or Laplace-weighted integrals over $\mathbb{R}^d_+$ or $\mathbb{R}^d$ of functions from generalized Laguerre- or Laplace-weighted Sobolev spaces of mixed smoothness, respectively. We…
Reinforcement learning constantly deals with hard integrals, for example when computing expectations in policy evaluation and policy iteration. These integrals are rarely analytically solvable and typically estimated with the Monte Carlo…
In this paper, we study the error in first order Sobolev norm in the approximation of solutions to linear parabolic PDEs. We use a Monte Carlo Euler scheme obtained from combining the Feynman--Kac representation with a Euler discretization…
Active subspaces can effectively reduce the dimension of high-dimensional parameter studies enabling otherwise infeasible experiments with expensive simulations. The key components of active subspace methods are the eigenvectors of a…
We analyze the convergence of higher order Quasi-Monte Carlo (QMC) quadratures of solution-functionals to countably-parametric, nonlinear operator equations with distributed uncertain parameters taking values in a separable Banach space $X$…
Monte Carlo methods are widely used for approximating complicated, multidimensional integrals for Bayesian inference. Population Monte Carlo (PMC) is an important class of Monte Carlo methods, which utilizes a population of proposals to…
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…
We consider the problem of estimating an expectation $ \mathbb{E}\left[ h(W)\right]$ by quasi-Monte Carlo (QMC) methods, where $ h $ is an unbounded smooth function on $ \mathbb{R}^d $ and $ W$ is a standard normal distributed random…
The purely numerical evaluation of multi-loop integrals and amplitudes can be a viable alternative to analytic approaches, in particular in the presence of several mass scales, provided sufficient accuracy can be achieved in an acceptable…
In a recent article by two of the present authors it turned out that Frolov's cubature formulae are optimal and universal for various settings (Besov-Triebel-Lizorkin spaces) of functions with dominating mixed smoothness. Those cubature…
We study the sample median of independently generated quasi-Monte Carlo estimators based on randomized digital nets and prove it approximates the target integral value at almost the optimal convergence rate for various function spaces. In…
We consider $\gamma$-weighted anchored and ANOVA spaces of functions with mixed first order partial derivatives bounded in a weighted $L_p$ norm with $1 \leq p \leq \infty$. The domain of the functions is $D^d$, where $D \subseteq…
We consider the problem of improving the efficiency of randomized Fourier feature maps to accelerate training and testing speed of kernel methods on large datasets. These approximate feature maps arise as Monte Carlo approximations to…
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…
This work studies numerical integration by the M\"obius-transformed trapezoidal rule, which combines the classical trapezoidal rule with a change of variables induced by a M\"obius transformation that maps the unit circle onto the real…
We analyse a multilevel Monte Carlo method for the approximation of distribution functions of univariate random variables. Since, by assumption, the target distribution is not known explicitly, approximations have to be used. We provide an…
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…