Selection of polynomial chaos bases via Bayesian model uncertainty methods with applications to sparse approximation of PDEs with stochastic inputs

Georgios Karagiannis, Guang Lin
2014 Journal of Computational Physics  
2014) 'Selection of polynomial chaos bases via Bayesian model uncertainty methods with applications to sparse approximation of PDEs with stochastic inputs.', Journal of computational physics., 259 . pp. 114-134. Additional information: Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference
more » ... made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details. Abstract For many real systems, several computer models may exist with different physics and predictive abilities. To achieve more accurate simulations/predictions, it is desirable for these models to be properly combined and calibrated. We propose the Bayesian calibration of computer model mixture method which relies on the idea of representing the real system output as a mixture of the available computer model outputs with unknown input dependent weight functions. The method builds a fully Bayesian predictive model as an emulator for the real system output by combining, weighting, and calibrating the available models in the Bayesian framework. Moreover, it fits a mixture of calibrated computer models that can be used by the domain scientist as a mean to combine the available computer models, in a flexible and principled manner, and perform reliable simulations. It can address realistic cases where one model may be more accurate than the others at different input values because the mixture weights, indicating the contribution of each model, are functions of the input. Inference on the calibration parameters can consider multiple computer models associated with different physics. The method does not require knowledge of the fidelity order of the models. We provide a technique able to mitigate the computational overhead due to the consideration of multiple computer models that is suitable to the mixture model framework. We implement the proposed method in a real world application involving the Weather Research and Forecasting large-scale climate model. represent the real system accurately. Often, calibration of a computer model is performed in the presence of experimental data in order to find optimal values for the unknown calibration parameters. In cases that the computer models are expensive to run, there is interest in building inexpensive predictive statistical models. Kennedy and O'Hagan [1] proposed an effective Bayesian computer model calibration to address such cases. Briefly, the experimental observations are represented as a sum of three functional terms: the computer model output, a systematic discrepancy, and an observational error. These functional terms are modeled as Gaussian processes [1, 2, 3, 4], because computer models are often computationally expensive, and available training data are limited. Literature includes several variations of computer model calibration which can handle different issues; e.g. discontinuity/non-stationarity in the outputs [5], discrete inputs [6], calibration in the frequentest context [7], high-dimensional outputs [8], dynamic discrepancy [9], large number of inputs and outputs [10], etc. However, these works are restricted in cases where a single computer model is available. Nowadays, there is a plethora of computer models that aim at simulating the same real system. These models may differ either in precision (multi-fidelity case) of the solvers involved, or in the theories based on which they are designed (multi-physics case). Recently, Goh et al. [11] proposed a procedure to perform Bayesian calibration of computer models available at different levels of fidelity. It combines the models in a nested structure according to a given fidelity order. However, this approach is restricted to address only multi-fidelity cases where the fidelity order of the computer models is known. Often, there are available several computer models, based on different theories, that represent the same real system. Each single computer model may have its own unique properties and predictive capabilities in representing the real system. Therefore, there is not a commonly acceptable way to order such models. Possible reasons for example can be: (i) incomplete knowledge of the complex real system, (ii) different computational capabilities of research groups, (iii) different scientific theories or perspectives describing the same real system, etc. In such cases, using only a single computer model may lead to misleading inferences and predictions and ignore the physics considered by other computer models only. Furthermore, traditional multi-fidelity calibration methods, such as [11] , are not suitable to address such cases because the fidelity order of the models is not available a priori, or because nesting one model to another could possibly impose unrealistic relations among the models. Moreover, in the presence of moderately large number of models, the direct implementation of standard multi-fidelity calibration method becomes very expensive. Here, the question of interest is how to properly combine and calibrate such computer models in order to represent the real system output accurately. In this study, the motivation for addressing the aforesaid problem raises from the Weather Research and Forecasting (WRF) regional climate model [12] . WRF allows for different configurations (sub-models), e.g. different parametrization suits, physics schemes, or resolutions, which in principle can constitute different models. Briefly, here the available computer models consist of different combinations of radiation schemes,
doi:10.1016/j.jcp.2013.11.016 fatcat:6w3nha6i6bdgllc5mc5tun3vpq