We consider multivariate integration in the weighted spaces of functions with mixed first derivatives bounded in Lp norms and the weighted coefficients introduced via `q norms, where p, q ∈ [1,∞]. The integration domain may be bounded or unbounded. The worst-case error and randomized error are investigated for quasi-Monte Carlo quadrature rules. For the worst-case setting the quadrature rule us...

We study a semi-discrete Galerkin method for solving the single-layer equation Vu = f with an approximating subspace of piecewise constant functions. Error bounds in Sobolev norms kkk s with ?1 s < 1 2 are proven and are of the same order as for the original Galerkin method. The distinctive features of the present work are that we handle irregular meshes and do not rely on Fourier methods. The ...

A Sine function approach is used to derive a new Hunter type quadrature rule for the evaluation of Cauchy principal value integrals of certain analytic functions. Integration over a general arc in the complex plane is considered. Special treatment is given to integrals over the interval (-1, 1). It is shown that the quadrature error is of order 0(e~ ), where N is the number of nodes used, and w...

Cranley and Patterson put forward the following randomization as the basis for the estimation of the error of a lattice rule for an integral of a one-periodic function over the unit cube in s dimensions. The lattice rule is randomized using independent random shifts in each coordinate direction that are uniformly distributed in the interval [0, 1]. This randomized lattice rule results in an unb...

Let d and k be positive integers. Let μ be a positive Borel measure on R2 possessing moments up to degree 2d − 1. If the support of μ is contained in an algebraic curve of degree k, then we show that there exists a quadrature rule for μ with at most dk many nodes all placed on the curve (and positive weights) that is exact on all polynomials of degree at most 2d − 1. This generalizes both Gauss...

This paper presents convergence analysis of kernel-based quadrature rules in misspecified settings, focusing on deterministic quadrature in Sobolev spaces. In particular, we deal with misspecified settings where a test integrand is less smooth than a Sobolev RKHS based on which a quadrature rule is constructed. We provide convergence guarantees based on two different assumptions on a quadrature...

Abstract: For finite positive Borel measures supported on the real line we consider a new type of quadrature rule with maximal algebraic degree of exactness, which involves function derivatives. We prove the existence of such quadrature rules and describe their basic properties. Also, we give an application of these quadrature rules to the solution of a Cauchy problem without solving it directl...

Convergence results are proved for a class of quadrature methods for integral equations of the form y(t) = fit) + /ô° k(t, s)y(s) ds. An important special case is the Nystrom method, in which the integral term is approximated by an ordinary quadrature rule. For all of the methods considered here, the rate of convergence is the same, apart from a constant factor, as that of the quadrature approx...

Szegő quadrature rules are discretization methods for approximating integrals of the form ∫ π −π f(e it)dμ(t). This paper presents a new class of discretization methods, which we refer to as anti-Szegő quadrature rules. AntiSzegő rules can be used to estimate the error in Szegő quadrature rules: under suitable conditions, pairs of associated Szegő and anti-Szegő quadrature rules provide upper a...

A quadrature finite element Galerkin scheme for a Dirichlet boundary value problem for the biharmonic equation is analyzed for a solution existence, uniqueness, and convergence. Conforming finite element space of Bogner-Fox-Schmit rectangles and an integration rule based on the two-point Gaussian quadrature are used to formulate the discrete problem. An H2-norm error estimate is obtained for th...