On the Predictability of Computer simulations: Advances in Verification and Validation

Abstract

We will present recent advances on the topics of Verification and Validation in order to assess the reliability and predictability of computer simulations. The first part of the talk will focus on goal-oriented error estimation for nonlinear boundary-value problems and nonlinear quantities of interest, in which case the error representation consists of two contributions: 1) a first contribution, involving the residual and the solution of the linearized adjoint problem, which quantifies the discretization or modeling error; and 2) a second contribution, combining higher-order terms that describe the linearization error. The linearization error contribution is in general neglected with respect to the discretization or modeling error. However, when nonlinear effects are significant, it is unclear whether ignoring linearization effects may produce poor convergence of the adaptive process. The objective will be to show how both contributions can be estimated and employed in an adaptive scheme that simultaneously controls the two errors in a balanced manner. In the second part of the talk, we will present novel approach for calibration of model parameters. The proposed inverse problem not only involves the minimization of the misfit between experimental observables and their theoretical estimates, but also an objective function that takes into account some design goals on specific design scenarios. The method can be viewed as a regularization approach of the inverse problem, one, however, that best respects some design goals for which mathematical models are intended. The inverse problem is solved by a Bayesian method to account for uncertainties in the data. We will show that it shares the same structure as the deterministic problem that one would obtain by multi-objective optimization theory. The method is illustrated on an example of heat transfer in a two-dimensional fin. The proposed approach has the main benefit that it increases the confidence in predictive capabilities of mathematical models.