Convexity results and sharp error estimates in approximate multivariate integration

Convexity results and sharp error estimates in approximate multivariate integration

An interesting property of the midpoint rule and trapezoidal rule, which is expressed by the so-called Hermite–Hadamard inequalities, is that they provide one-sided approximations to the integral of a convex function. We establish multivariate analogues of the Hermite–Hadamard inequalities and obtain access to multivariate integration formulae via convexity, in analogy to the univariate case. I...

Some Results on Convexity and Concavity of Multivariate Copulas

This paper provides some results on different types of convexity and concavity in the class of multivariate copulas. We also study their properties and provide several examples to illustrate our results.

Approximate Convexity and Submonotonicity in Locally Convex Spaces

Guaranteed and sharp a posteriori error estimates in isogeometric analysis

We present functional-type a posteriori error estimates in isogeometric analysis (IGA). These estimates, derived on functional grounds, provide guaranteed and sharp upper bounds of the true error in the energy norm. By exploiting the properties of non-uniform rational B-splines (NURBS), we present efficient computation of these error estimates. The numerical realization and the quality of the c...

On the Hardness of Approximate Multivariate Integration

We show that it is NP-hard to 2 k -approximate the integral of a positive, smooth, polynomialtime computable n-variate function, for any fixed integer k.