We discuss the numerical integration of polynomials times exponential weighting functions arising from multiscale finite element computations. The new rules are more accurate than standard quadratures and are better suited to existing codes than formulas computed by symbolic integration. We test our approach in a multiscale finite element method for the 2D reaction-diffusion equation. Standard ...

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...

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...

The classical theory of Gaussian quadrature assumes a positive weight function. We will show that in some cases Gaussian rules can be constructed with respect to an oscillatory weight, yielding methods with complex quadrature nodes and positive weights. These rules are well suited for highly oscillatory integrals because they attain optimal asymptotic order. We show that for the Fourier oscilla...

We describe the design and implementation of GAUSS Ð an online algorithm selection system for numerical quadrature. Given a quadrature problem and performance constraints on its solution, GAUSS selects the best (or nearly best) algorithm. GAUSS uses inductive logic programming to generalize a database of performance data; this produces high-level rules that correlate problem features with algor...

We present a discontinuous Galerkin (DG) scheme with suitable quadrature rules [15] for ideal compressible magnetohydrodynamic (MHD) equations on structural meshes. The semi-discrete scheme is analyzed to be entropy stable by using the symmetrizable version of the equations as introduced by Godunov [32], the entropy stable DG framework with suitable quadrature rules [15], the entropy conservati...