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Many of the input-parameter-to-output-quantity-of-interest maps that arise incomputational science admit a surprising low-dimensional structure, where theoutputs vary primarily along a handful of directions in the high-dimensionalinput space. This type of structure is well modeled by a ridge function, whichis a composition of a low-dimensional linear transformation with a nonlinearfunction. If the goal is to compute statistics of the output (e.g., as inuncertainty quantification or robust design) then one should exploit thislow-dimensional structure, when present, to accelerate computations. We developGaussian quadrature and the associated polynomial approximation forone-dimensional ridge functions. The key elements of our method are (i)approximating the univariate density of the given linear combination of inputsby repeated convolutions and (ii) a Lanczos-Stieltjes method for constructingorthogonal polynomials and Gaussian quadrature.

A Lanczos-Stieltjes method for one-dimensional ridge function approximation and integration