تجزیه و تحلیل حساسیت زمینه کامل از طریق کاهش ابعاد و مدل های جایگزین احتمالی

Computational mechanics models often are compromised by uncertainty in their governing parameters, especially when the operating environment is incompletely known. Computational sensitivity analysis of a spatially distributed process to its governing parameters therefore is an essential, but often costly, step in uncertainty quantification. A sensitivity analysis method is described which features probabilistic surrogate models developed through equitable sampling of the parameter space, proper orthogonal decomposition (POD) for compact representations of the process’ variability from an ensemble of realizations, and cluster-weighted models of the joint probability density function of each POD coefficient and the governing parameters. Full-field sensitivities, i.e. sensitivities at every point in the computational grid, are computed by analytically differentiating the conditional expected value function of each POD coefficient and projecting the sensitivities onto the POD basis. Statistics of the full-field sensitivities are estimated by sampling the surrogate model throughout the parameter space. Major benefits of this method are: (1) the sensitivities are computed analytically and efficiently from regularized surrogate models, and (2) the conditional variances of the surrogate models may be used to estimate the statistical uncertainty in the sensitivities, which provides a basis for pursuing more data to improve the model. Synthetic examples and a physical example involving near-ground sound propagation through a refracting atmosphere are presented to illustrate the properties of the surrogate models and how full-field sensitivities and their uncertainties are computed.

Computational models of complex physical systems are unavoidably compromised by assumptions, uncertainties, and errors. A natural consequence is that the analyst’s work may lack credibility, both because the problem at hand is complicated and the model building process lacks transparency. The concepts and method presented here support on-going computational modeling efforts to define models of systems and their environments in ways that promote increased confidence in predictions. We pursue this by developing tools that can provide (i) an informed basis for using measurements to support predictions, (ii) insight into the model building process, and (iii) insight into how limited knowledge of model parameters may compromise predictions. This paper describes a sampling-based method for developing probabilistic surrogate models in support of global, full-field sensitivity analysis, a concept which is defined below. The sensitivity calculations are fast and insensitive to the number of parameters because the sensitivities are computed by analytically differentiating the conditional expected value of the surrogate model rather than fitting a local response surface to samples of the surrogate model. Through the conditional variance of the surrogate model, the method also permits estimating the statistical uncertainty induced in the sensitivities by fitting the surrogate model to a given set of samples. Constructing and making the best use of parametric models for uncertainty quantification (UQ) requires knowledge of how the parameters may vary throughout the system’s lifetime and how these variations can alter the system’s state. Sensitivity analysis (SA) has been developed by many researchers and applications specialists to fulfill these needs. To continue enhancing the practicality and benefits of SA, we study what we call global, full-field sensitivity analysis, which we define in the following section. Our goal is to provide a framework that enables the use of SA to support UQ of realistic computational mechanics models. We see this as an essential step toward validating these models and enhancing their utility in decision making. The reader should note before proceeding that we do not consider non-probabilistic or generalized probabilistic methods, which commonly involve the introduction of fuzzy or non-additive measures; see the handbook edited by Nikolaidis, et al. [1] for authoritative introductions to the various methods and the supporting literature. We also do not pursue the even broader problem of assessing model validity. In the authors’ current field of study, predictions of near-ground sound propagation are compromised by statistical uncertainty and model errors in the atmosphere and terrain characterization. High quality physical and numerical representations are available, but imprecise knowledge of the heterogeneous propagation environment impedes attempts to achieve spatial and temporal accuracy in sound field predictions [2]. Embleton [3] summarizes many of these environmental factors, which include (i) the topography and acoustic impedance of the ground or lower boundary of the propagation domain, (ii) the interaction of turbulent and radiative exchanges with this surface, which alters the velocity and thermal gradients in the atmospheric surface layer (ASL), and (iii) spatio-temporal variability in the atmosphere. After describing the components and products of our SA method, we demonstrate it by studying the sensitivity of a near-ground sound propagation model throughout both the relevant parameter space and the physical domain. However, we emphasize that the method is independent of this application.

Sensitivity analysis of complex models often is facilitated through efficient representations of the response throughout the parameter space. Latin hypercube sampling (LHS) and proper orthogonal decomposition (POD) were employed to this end, in conjunction with cluster-weighted models (CWMs) of the POD coefficients, to approximate forecasted response fields and the spatial distributions of their sensitivities to governing parameters. Three example applications were shown: two were synthetic, single-parameter examples with various abrupt changes near key parameter values, and the third was a multiple-parameter computational model of two-dimensional, near-ground sound pressure level in a refracting atmosphere. The two single-parameter examples did not require POD because the response was a single scalar associated with each parameter sample. However, POD was essential to reducing the dimensionality of the discretized response in the acoustics example. The sensitivity analysis method computes sample statistics of the parametric sensitivities at every point in the spatial domain, so we refer to it as global, full-field sensitivity analysis. An apparently unique product of our method is an estimate of the variance in the forecast sensitivities. This information is helpful in assessing the credibility of forecast sensitivities in insufficiently-sampled regions of the parameter space, and it also may be useful in assessing the ability of a grid to adequately resolve localized features in a probabilistic computational mechanics model. However, the net estimate of forecast variance at each point in the parameter space and at each point in the spatial domain depends not only on the local sample density in the parameter space but also on the number of retained POD modes, the number of clusters in the CWM of each POD mode, the assumed form of local variations within each cluster, and how convergence of the each mode’s CWM is judged. We do not know a foolproof method for separating these effects, but we offer some reasonable conclusions here: (1) CWM variances in the coefficients of the lower-energy POD modes are lesser factors in the forecast variance of the net response than CWM variances in the stronger modes. (2) If an assumed local variation form (e.g. linear) suffices in the CWMs of the individual POD mode coefficients, the principle of parsimony suggests that little will be gained by employing a higher-order model of local variations. More complex local models could even result in greater statistical uncertainty because of the reduced regularization associated with the use of needless adjustable parameters in the local models. (3) If, according to cross-validation in the multidimensional parameter space, the number of clusters in each CWM represents the available data satisfactorily, then the number of clusters also should suffice for interpolating between available samples in this parameter space, even when those neighborhoods were not sampled directly in either the training or cross-validation set. This is likely to be the least generally applicable conclusion made here, as it assumes the response process varies smoothly with the governing parameters. In the sound propagation application, ensemble statistics of the sound pressure level sensitivities were computed to generate full-field contour plots of the mean and standard deviation of the sensitivities to two parameters. Pitfalls of sampling-based sensitivity analysis were described, the primary concern being the existence of voids in an ensemble due to sparse sampling of the parameter space. These voids can produce inconsistencies between the training and test ensembles and thus complicate comparisons. Relative values of the estimated maximum sensitivities indicated that uncertainty about both of the parameters considered here should be included in an uncertainty model if randomness or measurement imprecision are expected to affect the parameters. The importance of both randomness and measurement imprecision are too dependent on the details of a given application to permit further generality in this conclusion, but full-field sensitivity contour plots like those included here should help to guide future research and applications. These results suggest that the robustness and utility of our global FFSA method depend on sampling the full range of parameter variations. Latin hypercube sampling was expedient in meeting this objective, but perhaps would be surpassed by more recently developed approaches, e.g. Latinized Centroidal Voronoi Tesselation. However, the most effective use of the forecast variance estimate might be in conjunction with an incremental sampling method, the goal being to run more cases only in those regions of the parameter space that produce high forecast variances. More work is needed to determine the practical utility of implementing these changes. Although propagation parameters were modeled in this paper as independent random variables for sampling purposes, future efforts could extend this formulation to include random field models of the parameters with varying levels of correlation. The relative benefit of doing this would depend on whether the mechanics of the problem are sensitive to this correlation. The scatter plots presented above should be useful in making this decision, but physical insight likely will be the most important factor. We close by noting that we have demonstrated only first-order sensitivity analysis. The global FFSA method employed above herein offers at least two paths for quantifying nonlinear sensitivities — analytical derivatives and local nonlinear response surfaces — but these have not been attempted yet.