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