A Lanczos Method for Approximating Composite Functions
Abstract
We seek to approximate a composite function h(x) = g(f(x)) with a global polynomial. The standard approach chooses points x in the domain of f and computes h(x) at each point, which requires an evaluation of f and an evaluation of g. We present a Lanczos-based procedure that implicitly approximates g with a polynomial of f. By constructing a quadrature rule for the density function of f, we can approximate h(x) using many fewer evaluations of g. The savings is particularly dramatic when g is much more expensive than f or the dimension of x is large. We demonstrate this procedure with two numerical examples: (i) an exponential function composed with a rational function and (ii) a Navier-Stokes model of fluid flow with a scalar input parameter that depends on multiple physical quantities.
Cite
@article{arxiv.1110.0058,
title = {A Lanczos Method for Approximating Composite Functions},
author = {Paul G. Constantine and Eric T. Phipps},
journal= {arXiv preprint arXiv:1110.0058},
year = {2013}
}