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Partial differential equations (PDEs) have become an essential tool for modeling complex physical systems. Such equations are typically solved numerically via mesh-based methods, such as finite element methods, with solutions over the spatial domain. However, obtaining these solutions are often prohibitively costly, limiting the feasibility of exploring parameters in PDEs. In this article, we propose an efficient emulator that simultaneously predicts the solutions over the spatial domain, with theoretical justification of its uncertainty quantification. The novelty of the proposed method lies in the incorporation of the mesh node coordinates into the statistical model. In particular, the proposed method segments the mesh nodes into multiple clusters via a Dirichlet process prior and fits Gaussian process models with the same hyperparameters in each of them. Most importantly, by revealing the underlying clustering structures, the proposed method can provide valuable insights into qualitative features of the resulting dynamics that can be used to guide further investigations. Real examples are demonstrated to show that our proposed method has smaller prediction errors than its main competitors, with competitive computation time, and identifies interesting clusters of mesh nodes that possess physical significance, such as satisfying boundary conditions. An R package for the proposed methodology is provided in an open repository. 

Abstract The estimation of unknown parameters in simulations, also known as calibration, is crucial for practical management of epidemics and prediction of pandemic risk. A simple yet widely used approach is to estimate the parameters by minimising the sum of the squared distances between actual observations and simulation outputs. It is shown in this paper that this method is inefficient, particularly when the epidemic models are developed based on certain simplifications of reality, also known as imperfect models which are commonly used in practice. To address this issue, a new estimator is introduced that is asymptotically consistent, has a smaller estimation variance than the least-squares estimator, and achieves the semiparametric efficiency. Numerical studies are performed to examine the finite sample performance. The proposed method is applied to the analysis of the COVID-19 pandemic for 20 countries based on the susceptible-exposed-infectious-recovered model with both deterministic and stochastic simulations. The estimation of the parameters, including the basic reproduction number and the average incubation period, reveal the risk of disease outbreaks in each country and provide insights to the design of public health interventions. 

Abstract Model calibration is crucial for optimizing the performance of complex computer models across various disciplines. In the era of Industry 4.0, symbolizing rapid technological advancement through the integration of advanced digital technologies into industrial processes, model calibration plays a key role in advancing digital twin technology, ensuring alignment between digital representations and real‐world systems. This comprehensive review focuses on the Kennedy and O'Hagan (KOH) framework (Kennedy and O'Hagan, Journal of the Royal Statistical Society: Series B 2001; 63(3):425–464). In particular, we explore recent advancements addressing the challenges of the unidentifiability issue while accommodating model inadequacy within the KOH framework. In addition, we explore recent advancements in adapting the KOH framework to complex scenarios, including those involving multivariate outputs and functional calibration parameters. We also delve into experimental design strategies tailored to the unique demands of model calibration. By offering a comprehensive analysis of the KOH approach and its diverse applications, this review serves as a valuable resource for researchers and practitioners aiming to enhance the accuracy and reliability of their computer models. This article is categorized under:Statistical Models > Semiparametric ModelsStatistical Models > Simulation ModelsStatistical Models > Bayesian Models