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A Lanczos method for approximating composite functions
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DOI：10.1016/j.amc.2012.05.009.png)
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Abstract
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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. (C) 2012 Elsevier Inc. All rights reserved.
Keywords:
Dimension reduction
Lanczos' method
Orthogonal polynomials
Gaussian quadrature
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