The ROMES method for statistical modeling of reduced-order-model error

This work presents a technique for statistically modeling errors introduced by reduced-order models. The method employs Gaussian-process regression to construct a mapping from a small number of computationally inexpensive `error indicators' to a distribution over the true error. The variance of this distribution can be interpreted as the (epistemic) uncertainty introduced by the reduced-order model. To model normed errors, the method employs existing rigorous error bounds and residual norms as indicators; numerical experiments show that the method leads to a near-optimal expected effectivity in contrast to typical error bounds. To model errors in general outputs, the method uses dual-weighted residuals---which are amenable to uncertainty control---as indicators. Experiments illustrate that correcting the reduced-order-model output with this surrogate can improve prediction accuracy by an order of magnitude; this contrasts with existing `multifidelity correction' approaches, which often fail for reduced-order models and suffer from the curse of dimensionality. The proposed error surrogates also lead to a notion of `probabilistic rigor', i.e., the surrogate bounds the error with specified probability

Adaptive reduced basis methods for nonlinear convection-diffusion equations

evolution equations and depend on time-consuming parameter studies or need to ensure critical constraints on the simulation time. For both settings, model order reduction by the reduced basis methods is a suitable means to reduce computational time. In this proceedings, we show the applicability of the reduced basis framework to a finite volume scheme of a parametrized and highly nonlinear convectiondiffusion problem with discontinuous solutions. The complexity of the problem setting requires the use of several new techniques like parametrized empirical operator interpolation, efficient a posteriori error estimation and adaptive generation of reduced data. The latter is usually realized by an adaptive search for base functions in the parameter space. Common methods and effects are shortly revised in this presentation and supplemented by the analysis of a new strategy to adaptively search in the time domain for empirical interpolation data. Key words: finite volume methods, model reduction, reduced basis methods, empirical interpolation MSC2010: 65M08, 65J15, 65Y20

Reduced Basis Model Reduction of Parametrized Two-Phase Flow

Abstract: Many applications, e.g. in control theory and optimization depend on time– consuming parameter studies of parametrized evolution equations. Reduced basis methods are an approach to reduce the computation time of numerical simulations for these problems. The methods have gained popularity for model reduction of different numerical schemes with remarkable results preferably for scalar and linear problems with affine dependence on the parameter as in Patera and Rozza (2007). Over the last few years, the framework for the reduced basis methods has been continuously extended for non–linear discretizations, coupled problems and arbitrary dependence on the parameter, e.g. Grepl et al. (2007); Drohmann et al. (2010); Carlberg et al. (2011). In this presentation, we apply the framework developed in Drohmann et al. (2010) on a problem that combines all these difficulties. The considered problem models two–phase flow in a porous medium discretized by the finite volume method like in Michel (2004). For a first test, we do noth parametrize the problem and concentrate on the development of an efficient reduced basis scheme. This allows for separate approximations of the function spaces for the three physical unknowns and the non–linear terms in this numerical scheme. We shortly introduce the main aspects of the reduced basis method including the concept of offline/online decomposition, empirical operator interpolation method and reduced basis generation by greedy algorithms. We discuss how the coupling of the unknowns — saturation, velocity and pressure — must be reflected in the generated reduced spaces