Related papers: Adaptive Multidimensional Integration Based on Ran…
We study numerical integration over bounded regions in $\mathbb{R}^s, s\ge1$ with respect to some probability measure. We replace random sampling with quasi-Monte Carlo methods, where the underlying point set is derived from deterministic…
We investigate the applicability of Quasi-Monte Carlo methods to Euclidean lattice systems for quantum mechanics in order to improve the asymptotic error behavior of observables for such theories. In most cases the error of an observable…
We present an adaptive multilevel Monte Carlo (AMLMC) algorithm for approximating deterministic, real-valued, bounded linear functionals that depend on the solution of a linear elliptic PDE with a lognormal diffusivity coefficient and…
In this work, we consider the approximate reconstruction of high-dimensional periodic functions based on sampling values. As sampling schemes, we utilize so-called reconstructing multiple rank-1 lattices, which combine several preferable…
We develop algorithms for multivariate integration and approximation in the weighted half-period cosine space of smooth non-periodic functions. We use specially constructed tent-transformed rank-1 lattice points as cubature nodes for…
We investigate the possibility of fast, accurate and reliable computation of the Cauchy principal value integrals $\mathrm{P}\!\int_{a}^{b} f(x)(x-\tau)^{-1} dx$ $(a < \tau < b)$ using standard adaptive quadratures. In order to properly…
This paper studies randomized quasi-Monte Carlo (QMC) sampling for discontinuous integrands having singularities along the boundary of the unit cube $[0,1]^d$. Both discontinuities and singularities are extremely common in the pricing and…
The standard Kernel Quadrature method for numerical integration with random point sets (also called Bayesian Monte Carlo) is known to converge in root mean square error at a rate determined by the ratio $s/d$, where $s$ and $d$ encode the…
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…
Lattice rules are among the most prominently studied quasi-Monte Carlo methods to approximate multivariate integrals. A rank-1 lattice rule to approximate an $s$-dimensional integral is fully specified by its generating vector $\mathbf{z}…
This paper considers the problem of optimizing the average tracking error for an elliptic partial differential equation with an uncertain lognormal diffusion coefficient. In particular, the application of the multilevel quasi-Monte Carlo…
We investigate the application of randomized quasi-Monte Carlo (RQMC) methods in random feature approximations for kernel-based learning. Compared to the classical Monte Carlo (MC) approach \citep{rahimi2007random}, RQMC improves the…
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…
In this project we initiate an investigation of the applicability of Quasi-Monte Carlo methods to lattice field theories in order to improve the asymptotic error behavior of observables for such theories. In most cases the error of an…
The quasi-Monte Carlo method is widely used in computational finance, whose efficiency strongly depends on the smoothness and effective dimension of the integrand. In this work, we investigate the combination of importance sampling and the…
In Quasi-Monte Carlo integration, the integration error is believed to be generally smaller than in classical Monte Carlo with the same number of integration points. Using an appropriate definition of an ensemble of quasi-randompoint sets,…
Efficiently pricing multi-asset options is a challenging problem in quantitative finance. When the characteristic function is available, Fourier-based methods are competitive compared to alternative techniques because the integrand in the…
I consider the problem of integrating a function $f$ over the $d$-dimensional unit cube. I describe a multilevel Monte Carlo method that estimates the integral with variance at most $\epsilon^{2}$ in $O(d+\ln(d)d_{t}\epsilon^{-2})$ time,…
Continuous level Monte Carlo is an unbiased, continuous version of the celebrated multilevel Monte Carlo method. The approximation level is assumed to be continuous resulting in a stochastic process describing the quantity of interest.…
Due to the complexity of order statistics, the finite sample behaviour of robust statistics is generally not analytically solvable. While the Monte Carlo method can provide approximate solutions, its convergence rate is typically very slow,…