Effectively Subsampled Quadratures For Least Squares Polynomial
Approximations
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by
Pranay Seshadri, Akil Narayan, Sankaran Mahadevan
2016
Abstract
This paper proposes a new deterministic sampling strategy for constructing
polynomial chaos approximations for expensive physics simulation models. The
proposed approach, effectively subsampled quadratures involves sparsely
subsampling an existing tensor grid using QR column pivoting. For polynomial
interpolation using hyperbolic or total order sets, we then solve the following
square least squares problem. For polynomial approximation, we use a column
pruning heuristic that removes columns based on the highest total orders and
then solves the tall least squares problem. While we provide bounds on the
condition number of such tall submatrices, it is difficult to ascertain how
column pruning effects solution accuracy as this is problem specific. We
conclude with numerical experiments on an analytical function and a model
piston problem that show the efficacy of our approach compared with randomized
subsampling. We also show an example where this method fails.
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