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Scientific and engineering problems often involve parametric partial differential equations (PDEs), such as uncertainty quantification, optimizations, and inverse problems. However, solving these PDEs repeatedly can be prohibitively expensive, especially for large-scale complex applications. To address this issue, reduced order modeling (ROM) has emerged as an effective method to reduce computational costs. However, ROM often requires significant modifications to the existing code, which can be time-consuming and complex, particularly for large-scale legacy codes. Non-intrusive methods have gained attention as an alternative approach. However, most existing non-intrusive approaches are purely data-driven and may not respect the underlying physics laws during the online stage, resulting in less accurate approximations of the reduced solution. In this study, we propose a new non-intrusive bi-fidelity reduced basis method for time-independent parametric PDEs. Our algorithm utilizes the discrete operator, solutions, and right-hand sides obtained from the high-fidelity legacy solver. By leveraging a low-fidelity model, we efficiently construct the reduced operator and right-hand side for new parameter values during the online stage. Unlike other non-intrusive ROM methods, we enforce the reduced equation during the online stage. In addition, the non-intrusive nature of our algorithm makes it straightforward and applicable to general nonlinear time-independent problems. We demonstrate its performance through several benchmark examples, including nonlinear and multiscale PDEs.

Multiscale elliptic equations with scale separation are often approximated by the corresponding homogenized equations with slowly varying homogenized coefficients (the G-limit). The traditional homogenization techniques typically rely on the periodicity of the multiscale coefficients, thus finding the G-limits often requires sophisticated techniques in more general settings even when multiscale coefficient is known, if possible. Alternatively, we propose a simple approach to estimate the G-limits from (noisy-free or noisy) multiscale solution data, either from the existing forward multiscale solvers or sensor measurements. By casting this problem into an inverse problem, our approach adopts physics-informed neural networks (PINNs) algorithm to estimate the G-limits from the multiscale solution data by leveraging a priori knowledge of the underlying homogenized equations. Unlike the existing approaches, our approach does not rely on the periodicity assumption or the known multiscale coefficient during the learning stage, allowing us to estimate homogenized coefficients in more general settings beyond the periodic setting. We demonstrate that the proposed approach can deliver reasonable and accurate approximations to the G-limits as well as homogenized solutions through several benchmark problems.

The construction of efficient methods for uncertainty quantification in kinetic equations represents a challenge due to the high dimensionality of the models: often the computational costs involved become prohibitive. On the other hand, precisely because of the curse of dimensionality, the construction of simplified models capable of providing approximate solutions at a computationally reduced cost has always represented one of the main research strands in the field of kinetic equations. Approximations based on suitable closures of the moment equations or on simplified collisional models have been studied by many authors. In the context of uncertainty quantification, it is therefore natural to take advantage of such models in a multi-fidelity setting where the original kinetic equation represents the high-fidelity model, and the simplified models define the low-fidelity surrogate models. The scope of this article is to survey some recent results about multi-fidelity methods for kinetic equations that are able to accelerate the solution of the uncertainty quantification process by combining high-fidelity and low-fidelity model evaluations with particular attention to the case of compressible and incompressible hydrodynamic limits. We will focus essentially on two classes of strategies: multifidelity control variates methods and bi-fidelity stochastic collocation methods. The various approaches considered are analyzed in light of the different surrogate models used and the different numerical techniques adopted. Given the relevance of the specific choice of the surrogate model, an application-oriented approach has been chosen in the presentation., La construction de méthodes rapides pour la quantification del’incertitude dans les équations cinétiques représente un défi en raison de lagrande dimensionnalité des modèles qui impliquent souvent des coûts de calculprohibitifs. D’autre part, précisément à cause de fléau de la dimension,la construction de modèles simplifiés capables de fournir des solutions approchéesà un coût de calcul réduit a toujours représenté l’un des principaux axesde recherche dans le domaine des équations cinétiques. Des approximationsbasées sur des fermetures appropriées des équations des moments ou sur desmodèles collisionnels simplifiés ont été étudiées par de nombreux auteurs. Dansle cadre de la quantification de l’incertitude, il est donc naturel utiliser telsmodèles dans un cadre multi-fidélité où l’équation cinétique d’origine représentele modèle haute fidélité, et les modèles simplifiés définissent les modèlesà basse fidélité. Ce travail passe en revue quelques résultats récents sur les méthodesmulti-fidélité pour les équations cinétiques qui accélèrent la résolutiondu processus de quantification des incertitudes en combinant des évaluationsde modèles haute fidélité et basse fidélité avec une attention particulière aucas des limites hydrodynamiques compressibles et incompressibles. Nous nousconcentrerons essentiellement sur deux classes de stratégies : les méthodes desvariables de contrôle multi-fidélité et les méthodes de collocation stochastiquebi-fidélité. Les différentes approches sont analysées à la lumière des différentsmodèles simplifiés utilisés et des différentes techniques numériques adoptées.Compte tenu de la pertinence du choix spécifique du modèle de substitutionsimplifié, une approche orientée aux applications a été choisie dans la présentation.

In this paper, we consider the development of efficient numerical methods for linear transport equations with random parameters and under the diffusive scaling. We extend to the present case the bi-fidelity stochastic collocation method introduced in [33,50,51]. For the high-fidelity transport model, the asymptotic-preserving scheme [29] is used for each stochastic sample. We employ the simple two-velocity Goldstein-Taylor equation as low-fidelity model to accelerate the convergence of the uncertainty quantification process. The choice is motivated by the fact that both models, high fidelity and low fidelity, share the same diffusion limit. Speed-up is achieved by proper selection of the collocation points and relative approximation of the high-fidelity solution. Extensive numerical experiments are conducted to show the efficiency and accuracy of the proposed method, even in non diffusive regimes, with empirical error bound estimations as studied in [16].