A Lanczos method for approximating composite functions

作者：Constantine Paul G^*; Phipps Eric T

来源：Applied Mathematics and Computation, 2012, 218(24): 11751-11762.

DOI：10.1016/j.amc.2012.05.009

• Dimension reduction
• Lanczos' method
• Orthogonal polynomials
• Gaussian quadrature

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.

• 出版日期2012-8-15