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We present goal-oriented a posteriori error estimates for the automatic variationally stable finite element (AVS-FE) method (Calo et al., 2020) for scalar-valued convection–diffusion problems. The AVS-FE method is a Petrov–Galerkin method in which the test space is broken, whereas the trial space consists of classical FE basis functions, e.g., C 0 or Raviart–Thomas functions. We employ the concept of optimal test functions of the discontinuous Petrov–Galerkin (DPG) method by Demkowicz and Gopalakrishnan (Demkowicz and Gopalakrishnan, 2010; Carstensen et al., 2014; Demkowicz and Gopalakrishnan, 2011a; Demkowicz and Gopalakrishnan, 2011b; Demkowicz and Gopalakrishnan, 2012), leading to unconditionally stable FE approximations. Remarkably, by using C 0 or Raviart–Thomas trial spaces, the optimal discontinuous test functions can be computed in a completely decoupled element-by-element fashion. To establish the error estimators we present two approaches: (i) following the classical approach of Becker and Rannacher (Becker and Rannacher, 2001), i.e., the dual solution is sought in the (broken) test space, and (ii) introducing an alternative approach in which we seek C 0 , or Raviart–Thomas, AVS-FE approximations of the dual solution by using the underlying strong form of the dual boundary value problem (BVP). Various numerical verifications for 2D convection-dominated diffusion BVPs show that the estimates of the approximation error by the new alternative method are highly accurate, while the classical approach leads to error estimates of poor quality. Lastly, we present an algorithm for h -adaptive processes based on control of the numerical approximation error via the new alternative approach. Numerical verifications show that the estimator maintains high accuracy as the error converges to zero. [ABSTRACT FROM AUTHOR]

We consider a linear elliptic partial differential equation (PDE) with a generic uniformly bounded parametric coefficient. The solution to this PDE problem is approximated in the framework of stochastic Galerkin finite element methods. We perform a posteriori error analysis of Galerkin approximations and derive a reliable and efficient estimate for the energy error in these approximations. Practical versions of this error estimate are discussed and tested numerically for a model problem with non-affine parametric representation of the coefficient. Furthermore, we use the error reduction indicators derived from spatial and parametric error estimators to guide an adaptive solution algorithm for the given parametric PDE problem. The performance of the adaptive algorithm is tested numerically for model problems with two different non-affine parametric representations of the coefficient. [ABSTRACT FROM AUTHOR]

Abstract: Many learning algorithms use hypothesis spaces which are trained from samples, but little theoretical work has been devoted to the study of these algorithms. In this paper we show that mathematical analysis for these algorithms is essentially different from that for algorithms with hypothesis spaces independent of the sample or depending only on the sample size. The difficulty lies in the lack of a proper characterization of approximation error. To overcome this difficulty, we propose an idea of using a larger function class (not necessarily linear space) containing the union of all possible hypothesis spaces (varying with the sample) to measure the approximation ability of the algorithm. We show how this idea provides error analysis for two particular classes of learning algorithms in kernel methods: learning the kernel via regularization and coefficient based regularization. We demonstrate the power of this approach by its wide applicability. [Copyright &y& Elsevier]