A Lanczos Method for Approximating Composite Functions

TL;DR

This paper introduces a Lanczos-based method to efficiently approximate composite functions by reducing the number of expensive evaluations of g, especially useful when g is costly or the input dimension is high.

Contribution

The paper presents a novel Lanczos-based approach that constructs a polynomial approximation of g(f(x)) with fewer evaluations, improving efficiency over standard methods.

Findings

Significant reduction in g evaluations in numerical examples

Effective approximation for high-dimensional inputs

Applicable to complex models like Navier-Stokes equations

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.