  Sufficient dimension reduction (SDR) provides a framework for reducing the
predictor space dimension in regression problems. We consider SDR in the
context of deterministic functions of several variables such as those arising
in computer experiments. In this context, SDR serves as a methodology for
uncovering ridge structure in functions, and two primary algorithms for
SDR---sliced inverse regression (SIR) and sliced average variance estimation
(SAVE)---approximate matrices of integrals using a sliced mapping of the
response. We interpret this sliced approach as a Riemann sum approximation of
the particular integrals arising in each algorithm. We employ well-known tools
from numerical analysis---namely, multivariate tensor product Gauss-Christoffel
quadrature and orthogonal polynomials---to produce new algorithms that improve
upon the Riemann sum-based numerical integration in SIR and SAVE. We call the
new algorithms Lanczos-Stieltjes inverse regression (LSIR) and
Lanczos-Stieltjes average variance estimation (LSAVE) due to their connection
with Stieltjes' method---and Lanczos' related discretization---for generating a
sequence of polynomials that are orthogonal to a given measure. We show that
the quadrature-based approach approximates the desired integrals, and we study
the behavior of LSIR and LSAVE with three numerical examples. As expected in
high order numerical integration, the quadrature-based LSIR and LSAVE exhibit
exponential convergence in the integral approximations compared to the first
order convergence of the classical SIR and SAVE. The disadvantage of LSIR and
LSAVE is that the underlying tensor product quadrature suffers from the curse
of dimensionality---that is, the number of quadrature nodes grows exponentially
with the input space dimension. Therefore, the proposed approach is most
appropriate for deterministic functions with fewer than ten independent inputs.
