Assessing the Reliability of Complex Models Mathematical and Statistical Foundations of Verification, Validation, and Uncertainty Quantification (2012) / Chapter Skim
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4 Emulation, Reduced-Order Modeling, and Forward Propagation
4 Emulation, Reduced-Order Modeling, and Forward Propagation
Pages 37-51
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From page 37... ...
Finally, a few details should be noted before proceeding. The methods discussed -- such as emulation, reducedorder modeling, and polynomial chaos expansions -- use output produced from ensembles of simulations carried out at different input settings to capture the behavior of the computational model, the aim being to maximize the amount of information available for the uncertainty quantification (UQ)
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From page 38... ...
Both representations emulate the computer model output fairly well, but the GP has some obvious advantages. Notice that the fitted GP model passes through the observed points, thereby perfectly representing the deterministic computational model at the sampled inputs.
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From page 39... ...
Finding: Scalable methods do not exist for constructing emulators that reproduce the high-fidelity model results at each of the N training points, accurately capture the uncertainty away from the training points, and effectively exploit salient features of the response surface. Finally, most of the current technology for fitting response surfaces treats the computational model as a black box, ignoring features such as continuity or monotonicity that might be present in the physical system being modeled.
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From page 40... ...
. Extensions of these methods to handle nonlinear and para metrically varying problems have played a major role in moving model reduction from forward simulation and control to applications in optimization and UQ.
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From page 41... ...
The intrusive approach, however, synthesizes new equations that govern the behavior of the coefficients in the PC decomposition from the initial governing equations. Intrusive uncertainty propagation approaches involve a reformulation of the forward model, or its adjoint, to directly produce a probabilistic representation of uncertain model outputs induced by input uncertainty.
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From page 42... ...
There are several motivations for SA, including the following: enhancing the understanding of a complex model, finding aberrant model behavior, seeking out which inputs have a substantial effect on a particular output, exploring how combinations of inputs interact to affect outputs, seeking out regions of the input space that lead to rapid changes or extreme values in output, and gaining insight as to what additional information will improve the model's ability to predict. Even when a computational model is not adequate for reproducing physical system behavior, its sensitivities may still be useful for developing inferences about key features of the physical system.
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From page 43... ...
4.3.1 Global Sensitivity Analysis Global SA seeks to understand a complex function over a broad space of input settings, decomposing this function into a sum of increasingly complex components (see Box 4.1)
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From page 44... ...
4.3.2 Local Sensitivity Analysis Local sensitivity analysis is based on the partial derivatives of the forward model with respect to the inputs, evaluated at a nominal input setting. Hence, this sort of SA makes sense only for differentiable outputs.
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From page 45... ...
Still, when they do work, and when the intrusive methods described below are not feasible because of time constraints or lack of modularity of the forward code, AD can be a viable approach. If one has access to the underlying forward model, or if one is developing a local sensitivity capability from scratch, one can overcome many of the difficulties outlined above by using an intrusive method.
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From page 46... ...
For physical experiments, there are three broad principles for experimental design: randomization, replication, and blocking.2 For deterministic computer experiments, these issues do not apply -- replication, for example, is just wasted effort. In the absence of prior knowledge of the shape of the response surface, however, a simple rule of thumb worth following is that the design points should be spread out to explore as much of the input region as possible.
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From page 47... ...
Fueled by advances in both algorithms and computer hardware, Max well equation solvers have become indispensable in scientific and engineering disciplines ranging from remote sensing and biomedical imaging to antenna and circuit design, to name but a few. The application of VVUQ concepts to the statistical characterization of EMI phenomena described below leverages an integral equation-based Maxwell equation solver.
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From page 48... ...
as a function of the seven input parameters was constructed for this study. The emulator provides an accurate approximation of the received signal for all combinations of the parameters, yet can be evaluated in a fraction of the time required for executing the Maxwell equation solver.
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From page 49... ...
The relative accuracy of the ME-SC emulator with respect to the signal strength predicted by the Maxwell equation solver was below 0.1 percent for each of the 545 system configurations in the seven-dimensional input space. The cost of applying MC to the emulator is negligible compared to that of a single call to the Maxwell equation solver.
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From page 50... ...
2007. Global Sensitivity Analysis of Stochastic Computer Models with Generalized Additive Models.
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From page 51... ...
2004. Probabilistic Sensitivity Analysis of Complex Models: A Bayesian Approach.
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