Development and application of reduced basis approach
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Full evaluation of expensive models for systems with characterization of uncertainty in the context of input parameters requires millions of evaluations of the partial differential equations governing the behavior of the system. This is often infeasible and we seek methods for rapid evaluation of the outputs of interest. We present an approach for rapid evaluation of the partial differential equations in the context of input-output relationship. The important components for this reduced basis approach are (a) availability of error estimators to guarantee the quality of the solution; (b) first principles representation of physics---better ability to provide a better response outside the calibration limits. First, we illustrate the reduced basis approach in a simple heat conduction and plane elasticity problem and compare the output errors with actual full-scale FEM solution. Next, we present the application of this method in metamodeling, where we introduce a hierarchical experimental design system using a combination of full-scale fine model and the reduced basis model. We explore the input parameter space for different outputs of interest by estimating the response surface error and the output error bounds. The computational efficiency of this method is shown by comparing the evaluation time with FEM method. Finally, we demonstrate the input data uncertainty evaluation using this approach and present the results.