Reduced Basis Collocation Methods for Partial Differential Equations with Random Coefficients
Abstract
The sparse grid stochastic collocation method is a new method for
solving partial differential equations with random coefficients.
However, when the probability space has high dimensionality, the number
of points required for accurate collocation solutions can be large, and
it may be costly to construct the solution. We show that this process
can be made more efficient by combining collocation with reduced-basis
methods, in which a greedy algorithm is used to identify a reduced
problem to which the collocation method can be applied. Because the
reduced model is much smaller, costs are reduced significantly. We
demonstrate with numerical experiments that this is achieved with
essentially no loss of accuracy.