Sparse Grid Quadrature Rules Based on Conformal Mappings

In this work, we demonstrate the extension of quadrature approximations, built from conformal mapping of interpolatory rules, to sparse grid quadrature in the multidimensional setting. In one dimension, computation of an integral involving an analytic function using these transformed quadrature rules can improve the convergence rate by a factor approaching π∕2 versus classical interpolatory quadrature (Hale and Trefethen, SIAM J Numer Anal 46:930–948, 2008). For the computation of high-dimensional integrals with analytic integrands, we implement the transformed quadrature rules in the sparse grid setting, and we show that in certain settings, the convergence improvement can be exponential with growing dimension. Numerical examples demonstrate the benefits and drawbacks of the approach, as predicted by the theory.

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