We investigate the use of numerical quadrature in the p-version of the finite element method. We describe a set of minimal conditions that the quadrature rules should satisfy, for various types of elements. Under sufficient assumptions of smoothness, we establish optimality of the asymptotic rate of convergence. Some computational results are presented, which illustrate under what conditions ov...

In recent years, Smolyak quadrature rules (also called quadratures on hyperbolic cross points or sparse grids) have gained interest as a possible competitor to number theoretic quadratures for high dimensional problems. A standard way of comparing the quality of multivariate quadra-ture formulas consists in computing their L 2-discrepancy. Especially for larger dimensions, such computations are...

Lower bounds for the error of quadrature formulas with positive weights are proved. We get intractability results for quasi-Monte Carlo methods and, more generally, for positive formulas. We consider general classes of functions but concentrate on lower bounds for relatively small classes of trigonometric polynomials. We also conjecture that similar lower bounds hold for arbitrary quadrature fo...

The problem of non-intrusive uncertainty quantification is studied, with a focus on two computational fluid dynamics cases. A collocation method using quadrature or cubature rules is applied, where the simulations are selected deterministically. A one-dimensional quadrature rule is proposed which is nested, symmetric, and has positive weights. The rule is based on the removal of nodes from an e...

This paper presents a modification of Krylov Subspace Spectral (KSS) Methods, which build on the work of Golub, Meurant and others pertaining to moments and Gaussian quadrature to produce high-order accurate approximate solutions to the variable-coefficient second-order wave equation. Whereas KSS methods currently use Lanczos iteration to compute the needed quadrature rules, the modification us...

We introduce and analyze a family of algorithms for an efficient numerical approximation of integrals of the form I = ∫ C(1) ∫ C(2) F (x, y, y−x)dydx where C, C are d-dimensional parallelotopes (i.e. affine images of d-hypercubes) and F has a singularity at y − x = 0. Such integrals appear in Galerkin discretization of integral operators in R. We construct a family of quadrature rules QN with N...

We consider the quadrature rules of “practical type” (with five knots) for approximately computation of the integral ∫ 2

Abstract. Many problems in science and engineering require the evaluation of functionals of the form Fu(A) = uT f(A)u, where A is a large symmetric matrix, u a vector, and f a nonlinear function. A popular and fairly inexpensive approach to determining upper and lower bounds for such functionals is based on first carrying out a few steps of the Lanczos procedure applied to A with initial vector...