Related papers: Higher order Quasi-Monte Carlo integration for Bay…
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 $X$…
We propose and analyze deterministic multilevel approximations for Bayesian inversion of operator equations with uncertain distributed parameters, subject to additive Gaussian measurement data. The algorithms use a multilevel (ML) approach…
This paper contributes to the study of optimal experimental design for Bayesian inverse problems governed by partial differential equations (PDEs). We derive estimates for the parametric regularity of multivariate double integration…
Monte Carlo (MC) and Quasi-Monte Carlo (QMC) methods are classical approaches for the numerical integration of functions $f$ over $[0,1]^d$. While QMC methods can achieve faster convergence rates than MC in moderate dimensions, their…
We review recent results on dimension-robust higher order convergence rates of Quasi-Monte Carlo Petrov-Galerkin approximations for response functionals of infinite-dimensional, parametric operator equations which arise in computational…
In this note, we study a concatenation of quasi-Monte Carlo and plain Monte Carlo rules for high-dimensional numerical integration in weighted function spaces. In particular, we consider approximating the integral of periodic functions…
There has been a surge of interest in uncertainty quantification for parametric partial differential equations (PDEs) with Gevrey regular inputs. The Gevrey class contains functions that are infinitely smooth with a growth condition on the…
Quasi-Monte Carlo (QMC) methods are applied to multi-level Finite Element (FE) discretizations of elliptic partial differential equations (PDEs) with a random coefficient, to estimate expected values of linear functionals of the solution.…
Quasi-Monte Carlo (QMC) integration of output functionals of solutions of the diffusion problem with a log-normal random coefficient is considered. The random coefficient is assumed to be given by an exponential of a Gaussian random field…
We study numerical integration of functions depending on an infinite number of variables. We provide lower error bounds for general deterministic linear algorithms and provide matching upper error bounds with the help of suitable multilevel…
We develop a convergence analysis of a multi-level algorithm combining higher order quasi-Monte Carlo (QMC) quadratures with general Petrov-Galerkin discretizations of countably affine parametric operator equations of elliptic and parabolic…
We study quasi-Monte Carlo (QMC) integration over the multi-dimensional unit cube in several weighted function spaces with different smoothness classes. We consider approximating the integral of a function by the median of several integral…
In this paper, we study quasi-Monte Carlo (QMC) integration in weighted Sobolev spaces. In contrast to many previous results the QMC algorithms considered here are of open type, i.e., they are extensible in the number of sample points…
We construct quasi-Monte Carlo methods to approximate the expected values of linear functionals of Galerkin discretizations of parametric operator equations which depend on a possibly infinite sequence of parameters. Such problems arise in…
By formulating the inverse problem of partial differential equations (PDEs) as a statistical inference problem, the Bayesian approach provides a general framework for quantifying uncertainties. In the inverse problem of PDEs, parameters are…
Quasi-Monte Carlo (QMC) integration over unbounded domains $\mathbb{R}^s$ remains challenging due to the high dimensionality of sampling space and the boundary growth of the integrand. In applications such as uncertainty quantification…
We propose a Bayesian methodology for estimating spiked covariance matrices with jointly sparse structure in high dimensions. The spiked covariance matrix is reparametrized in terms of the latent factor model, where the loading matrix is…
Quasi-Monte Carlo (QMC) methods for estimating integrals are attractive since the resulting estimators typically converge at a faster rate than pseudo-random Monte Carlo. However, they can be difficult to set up on arbitrary posterior…
We consider the application of a quasi-Monte Carlo cubature rule to Bayesian shape inversion subject to the Poisson equation under Gevrey regular parameterizations of domain uncertainty. We analyze the parametric regularity of the…
Sparse Bayesian factor models are routinely implemented for parsimonious dependence modeling and dimensionality reduction in high-dimensional applications. We provide theoretical understanding of such Bayesian procedures in terms of…