The distinction between a priori and a posteriori knowledge has been the subject of an enormous amount of discussion, but the literature is biased against recognizing the intimate relationship between these forms of knowledge. For instance, it seems to be almost impossible to find a sample of pure a priori or a posteriori knowledge. In this paper, it will be suggested that distinguishing betwee...

‎in this paper‎, ‎we study spectral element approximation for a constrained‎ ‎optimal control problem in one dimension‎. ‎the equivalent a posteriori error estimators are derived for‎ ‎the control‎, ‎the state and the adjoint state approximation‎. ‎such estimators can be used to‎ ‎construct adaptive spectral elements for the control problems.

A least-squares mixed finite element (LSMFE) method for the numerical solution of fourth-order elliptic equations is analyzed and developed in this paper. The a posteriori error estimator which is needed in the adaptive refinement algorithm is proposed. The local evaluation of the least-squares functional serves as a posteriori error estimator. The posteriori errors are effectively estimated.

The a posteriori error estimation of finite element approximations of elliptic boundary value problems has reached some state of maturity, as it is documented by a variety of monographs on this subject (cf., e.g., [1, 2, 3, 4, 5]). There are different concepts such as • residual type a posteriori error estimators, • hierarchical type a posteriori error estimators, • error estimators based on lo...

This paper considers the model problem in two-dimensional domain. The a posteriori error indicator for p-version Finite Element Methods (FEM) is discussed, and the reliable property of this a posteriori error indicator is investigated. Specially, we reformulate the a posteriori error indicator by orthogonal polynomials, which is easily used in practical applications.

This paper presents new results on the complexity of graph-theoretical models that represent probabilities (Bayesian networks) and that represent interval and set valued probabilities (credal networks). We define a new class of networks with bounded width, and introduce a new decision problem for Bayesian networks, the maximin a posteriori. We present new links between the Bayesian and credal n...

In this paper, we present some results on a posteriori error analysis of finite element methods for solving linear nonlocal diffusion and bond-based peridynamic models. In particular, we aim to propose a general abstract frame work for a posteriori error analysis of the peridynamic problems. A posteriori error estimators are consequently prompted, the reliability and efficiency of the estimator...

We derive a posteriori error bounds for a quasilinear parabolic problem, which is approximated by the hp-version interior penalty discontinuous Galerkin method (IPDG). The error is measured in the energy norm. The theory is developed for the semidiscrete case for simplicity, allowing to focus on the challenges of a posteriori error control of IPDG space-discretizations of strictly monotone quas...