Adaptive error estimation for linear functionals approximation and applications

Abstract

While concerning numerical approximation to bounded linear functionals, the approximate errors are generally discussed by derivatives of the highest orders regarding the exactness degrees of the approximate functionals. This approach, however, is only valid for functions with sufficient smoothness. For less smooth functions, the Peano/Sard kernels theorems are known to be useful tools. Actually, the Peano/Sard theorems supply error representations of full orders, including the highest order representation. Therefore, depending on the smoothness of the underlying functions, there can be more alternatives to represent an approximation error. It thus makes adaptive local error estimation possible. These important mathematical results have no more only the theoretical values, by automatic differetiation and interval arithmetic, the kernels method finally can be realized in numerical computation on computers. One critical task prior to their applications is the nontrivial calculation of Peano Sard constants. The talk will present all these aspects and their applications in numerical integration.