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Maintained by Bernadette Watts and Deanna Ceballos
Analyzing Sparse-Grid Uncertainty Quantification Techniques for Verification
within Predictive Science
Clayton Webster
Computer Science Research Institute
Sandia National Laboratories
Our modern treatment of predicting the behavior of physical and engineering problems relies on mathematical modeling followed by computer simulation. The modeling process may describe the solution in terms of high dimensional spaces, particularly in the case when the input data (coefficients, forcing terms, boundary conditions, geometry, etc) are affected by a large amount of uncertainty. Therefore, the goal of the mathematical and computational analysis becomes the prediction of statistical moments (mean value, variance, covariance, etc.) or even the whole probability distribution of some responses of the system (quantities of physical interest), given the probability distribution of the input random data. By using an ensemble-based approach such as Monte Carlo (MC) or Stochastic Collocation (SC), these problems can be equivalently represented as numerical integration or interpolation problems, in an N-dimensional hypercube. For higher accuracy, the computer simulation must increase the number of random variables (dimensions), and expend more effort approximating the quantity of interest in each individual dimension. The resulting explosion in computational effort is a symptom of the curse of dimensionality. Sparse grid techniques yield methods to discretize these higher dimensional problems with a feasible amount of unknowns leading to usable methods. It is the aim of this talk to survey the fundamentals and analysis of both the standard isotropic sparse grid method and our novel dimensional adaptive (anisotropic) sparse grid approach within the context of uncertainty quantification. These methods have proven to have dramatic impact on several application areas, including statistical mechanics, financial mathematics, bioinformatics, and other fields that must properly predict certain model behaviors. Both theoretical and computational evidence will be presented to show the effectiveness of the sparse grid methods when compared to other approaches, including tensor products and Monte Carlo.