Prediction and Computer Model Calibration Using Outputs From Multifidelity Simulators

Abstract

Computer codes are widely used to describe physical processes in lieu of physical observations. In some cases, more than one computer simulator, each with different degrees of fidelity, can be used to explore the physical system. In this work, we combine field observations and model runs from deterministic multi-fidelity computer simulators to build a predictive model for the real process. The resulting model can be used to perform sensitivity analysis for the system, solve inverse problems and make predictions. Our approach is Bayesian and will be illustrated through a simple example, as well as a real application in predictive science at the Center for Radiative Shock Hydrodynamics at the University of Michigan.KEY WORDS: Computer Experiment; Gaussian process; Markov Chain Monte Carlo. arXiv:1208.2716v1 [stat.AP] 13 Aug 2012 occur, for example, because of the presence of reduced order physics in lower fidelity models, different levels of accuracy specified for numerical solvers or solutions obtained on finer grids. In these cases, a higher fidelity model is thought to better represent the physical process than a lower fidelity model, but also takes more computer time to produce an output than a lower fidelity model. So, combining relatively cheap lower fidelity model runs with more costly high fidelity runs to emulate the high fidelity model has been an significant problem of interest (Kennedy and O'Hagan, 2000;Qian et al., 2006 and.Another important application of computer models is that of calibration (e.g., Kennedy and O'Hagan, 2001;Higdon et al., 2004) where the aim is to combine simulator outputs with physical observations to build a predictive model and also estimate unknown parameters that govern the behaviour of the computer model. The latter endeavour amounts to solving a sort of inverse problem, while the former activity is a type of regression problem.Motivated by applications at the Center for Radiative Shock Hydrodynamics (CRASH) at the University of Michigan, the aim of this work is to develop new methodology to combine outputs from simulators with different levels of fidelity and field observations to make predictions of the physical system with associated measurements of uncertainty. In the spirit similar to Kennedy and O'Hagan (2000 and2001) and Higdon et al. (2004), we propose a predictive model that incorporates computer model outputs and field data, while attempting to find optimal values for some input parameters (i.e. calibration parameters). Different models are specified for each source of data (Kennedy and O'Hagan, 2000;Qian et al., 2006 and. The approach calibrates each computer model to the next highest level of fidelity model, and the simulator of the highest fidelity is then calibrated to the field measurements. All the response surfaces are Gaussian process (GP) models and the various sources of information that inform predictions of the physical system are combined with a Bayesian hierarchical model. The paper is organized as follows: In section 2, we will introduce the prop...

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

“…We provide a validation study of the proposed method with that of Goh et al [11] in a simple benchmark example (Sec. 3.1).…”

“…To validate the performance of our method, we pretend that we do not know which model is more accurate; although S p2q has higher fidelity than S p1q by construction. Moreover, we validate our method with respect to the multi-fidelity method of Goh et al [11] which is exclusively designed to address only cases with known fidelity order; hence for method of Goh et al [11] we use the extra information that S p2q is of higher fidelity than…”

“…We compare our method with the Bayesian multi-fidelity calibration (BMFC) procedure of Goh et al [11] in terms of predictive ability (Figure 3.3). For BMFC, we use the default Gaussian processes and prior model specifications suggested in [11], which actually resemble to those specified for our method.…”

“…In Section 2, we present the proposed method. In Section 3.1, we validate the proposed method with that of Goh et al [11] in a validation example. In Section 3.2, we assess the good performance of the method and compare it with that of Kennedy and O'Hagan [1] in a more challenging benchmark example.…”

laGP: Large-Scale Spatial Modeling via Local Approximate Gaussian Processes inR

“…Consider the following computer model test function used by Goh, Bingham, Holloway, Grosskopf, Kuranz, and Rutter (2013), which is an elaboration of one first described by Bastos and O'Hagan (2009) with N F = 2 * ny = 100, storing two replicates at each X F N F = X location. illustrated this example with ten replicates.…”