Howard Elman, University of Maryland, Department of Computer Science, United States
Joint work with Virginia Forstall and Qifeng Liao
The reduced basis methodology is an efficient approach to solve parameterized discrete partial differential equations when the solution is needed at many parameter values. An offline step approximates the solution space and an online step utilizes this approximation, the reduced basis, to solve a smaller reduced problem, which provides an accurate estimate of the solution. We explore two issues for this methodology: 1. In the case where the parameters are random quantities, we show that reducedbasis methods can be combined with stochastic collocation methods to dramatically reduce the cost of using collocation in cases where a large number of collocation points are required. 2. Traditionally, the reduced problems are solved using direct methods. However, the size of the reduced system needed to produce solutions of a given accuracy depends on the characteristics of the problem, and it may happen that the size is significantly smaller than that of the original discrete problem but large enough to make direct solution costly. In this scenario, it may be more effective to use iterative methods to solve the reduced problem. We construct preconditioners for reduced iterative methods that are derived from preconditioners for the full problem and show that this approach permits reduced-basis methods to be practical for larger bases than direct methods allow.

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