Optimal Numerical Integration

Asymptotically Optimal Weighted Numerical Integration

We study numerical integration of H older{type functions with respect to weights on the real line. Our study extends previous work by F. Curbera, [2] and relies on a connection between this problem and the approximation of distribution functions by empirical ones. The analysis is based on a lemma which is important within the theory of optimal designs for approximating stochastic processes. As...

Optimal algorithms for numerical integration: a survey

We discuss optimality of approximation algorithms and lower bounds on their complexity using Carl’s inequality, which states that for any two Banach spaces X and Y and any α > 0 there exists a constant γα > 0 such that sup 1≤k≤n kek(T ) ≤ γα sup 1≤k≤n kck(T ) holds for all linear and bounded operators T : X → Y . Here ek(T ) is the k-th entropy number of T and ck(T ) is the k-th Gelfand number ...

Optimal Stability Polynomials for Numerical Integration of Initial Value Problems

Geometric Numerical Integration of Nonholonomic Systems and Optimal Control Problems

A geometric derivation of numerical integrators for nonholonomic systems and optimal control problems is obtained. It is based in the classical technique of generating functions adapted to the special features of nonholonomic systems and optimal control problems.

Numerical integration using spline quasi-interpolants

In this paper, quadratic rules for obtaining approximate solution of definite integrals as well as single and double integrals using spline quasi-interpolants will be illustrated. The method is applied to a few test examples to illustrate the accuracy and the implementation of the method.