Sparse Pseudospectral Approximation Method
For researchers in uncertainty quantification and sensitivity analysis, this provides a principled way to use sparse grid integration for pseudospectral approximation, correcting a known deficiency.
The paper addresses the problem of inaccurate coefficients for higher-degree polynomials in sparse grid pseudospectral approximations. By reexamining Smolyak's algorithm and leveraging interpolation-projection connections, they construct a method that accurately reproduces coefficients corresponding to the sparse grid rule, with numerical results confirming its effectiveness.
Multivariate global polynomial approximations - such as polynomial chaos or stochastic collocation methods - are now in widespread use for sensitivity analysis and uncertainty quantification. The pseudospectral variety of these methods uses a numerical integration rule to approximate the Fourier-type coefficients of a truncated expansion in orthogonal polynomials. For problems in more than two or three dimensions, a sparse grid numerical integration rule offers accuracy with a smaller node set compared to tensor product approximation. However, when using a sparse rule to approximately integrate these coefficients, one often finds unacceptable errors in the coefficients associated with higher degree polynomials. By reexamining Smolyak's algorithm and exploiting the connections between interpolation and projection in tensor product spaces, we construct a sparse pseudospectral approximation method that accurately reproduces the coefficients of basis functions that naturally correspond to the sparse grid integration rule. The compelling numerical results show that this is the proper way to use sparse grid integration rules for pseudospectral approximation.