Related papers: A Node Elimination Algorithm for Cubature of High-…
Iterative methods with certified convergence for the computation of Gauss--Jacobi quadratures are described. The methods do not require a priori estimations of the nodes to guarantee its fourth-order convergence. They are shown to be…
A method of deriving quadrature rules has been developed which gives nodes and weights for a Gaussian-type rule which integrates functions of the form: f(x,y,t) = a(x,y,t)/((x-t)^2+y^2) + b(x,y,t)/([(x-t)^2+y^2]^{1/2}) +…
We study numerical integration of smooth functions defined over the $s$-dimensional unit cube. A recent work by Dick et al. (2019) has introduced so-called extrapolated polynomial lattice rules, which achieve the almost optimal rate of…
The modeling of electric machines and power transformers typically involves systems of nonlinear magnetostatics or -quasistatics, and their efficient and accurate simulation is required for the reliable design, control, and optimization of…
We study the fundamental problem of polytope membership aiming at large convex polytopes, i.e. in high dimension and with many facets, given as an intersection of halfspaces. Standard data-structures as well as brute force methods cannot…
Inferring probabilistic networks from data is a notoriously difficult task. Under various goodness-of-fit measures, finding an optimal network is NP-hard, even if restricted to polytrees of bounded in-degree. Polynomial-time algorithms are…
A quadrature rule of a measure $\mu$ on the real line represents a convex combination of finitely many evaluations at points, called nodes, that agrees with integration against $\mu$ for all polynomials up to some fixed degree. In this…
Particle methods are widely used because they can provide accurate descriptions of evolving measures. Recently it has become clear that by stepping outside the Monte Carlo paradigm these methods can be of higher order with effective and…
When modeling scientific and industrial problems, geometries are typically modeled by explicit boundary representations obtained from computer-aided design software. Unfitted (also known as embedded or immersed) finite element methods offer…
Lattice rules and polynomial lattice rules are quadrature rules for approximating integrals over the $s$-dimensional unit cube. Since no explicit constructions of such quadrature methods are known for dimensions $s > 2$, one usually has to…
We introduce the multivariate decomposition finite element method (MDFEM) for solving elliptic PDEs with uniform random diffusion coefficients. We show that the MDFEM can be used to reduce the computational complexity of estimating the…
Node embedding methods find latent lower-dimensional representations which are used as features in machine learning models. In the last few years, these methods have become extremely popular as a replacement for manual feature engineering.…
Optimization problems in disciplines such as machine learning are commonly solved with iterative methods. Gradient descent algorithms find local minima by moving along the direction of steepest descent while Newton's method takes into…
Cluster deletion is an NP-hard graph clustering objective with applications in computational biology and social network analysis, where the goal is to delete a minimum number of edges to partition a graph into cliques. We first provide a…
A new algebraic cubature formula of degree $2n+1$ for the product Chebyshev measure in the $d$-cube with $\approx n^d/2^{d-1}$ nodes is established. The new formula is then applied to polynomial hyperinterpolation of degree $n$ in three…
A novel mathematical framework is derived for the addition of nodes to univariate and interpolatory quadrature rules. The framework is based on the geometrical interpretation of the Vandermonde matrix describing the relation between the…
We show how to obtain a fast component-by-component construction algorithm for higher order polynomial lattice rules. Such rules are useful for multivariate quadrature of high-dimensional smooth functions over the unit cube as they achieve…
We describe an algorithm for controlling the relative error in the numerical evaluation of a bivariate integral, without prior knowledge of the magnitude of the integral. In the event that the magnitude of the integral is less than unity,…
In numerical integration, cubature methods are effective, especially when the integrands can be well-approximated by known test functions, such as polynomials. However, the construction of cubature formulas has not generally been known, and…
We discuss a recursive family of iterative methods for the numerical approximation of roots of nonlinear functions in one variable. These methods are based on Newton-Cotes closed quadrature rules. We prove that when a quadrature rule with…