Variational formulation with error estimates for uncertainty quantification via collocation, regression, and sprectral projection

Many real world problems need simplifications in such a way that computing is reduced for answering specific questions, for example, to quantify uncertainties. Therefore so‐called metamodels or surrogate models are developed which are based on interpolation or approximation methods. In this paper we transform the usual approximation or interpolation problem into a variational form such as it is known from the Finite Element method (FEM). With this variational framework it is possible to derive error estimators, which can be used later on for adaptivity. To compute the coefficients of the metamodel one needs some quadrature rules, which should be related to the given data. A numerical example shows the advantages of our proposed methods. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)