Nonlinear approximation of functions based on non-negative least squares solver

TL;DR

This paper introduces a nonlinear approximation method using non-negative least squares to efficiently approximate functions like power laws and stretched exponentials, with a focus on iterative parameter optimization.

Contribution

It presents a novel approach for nonlinear function approximation by integrating parameter estimation into the least squares framework, improving accuracy for specific function classes.

Findings

Effective approximation of $x^{-eta}$ for $0<eta<1$.

Successful approximation of stretched exponential functions.

Enhanced iterative algorithm for nonlinear least squares.

Abstract

In computational practice, most attention is paid to rational approximations of functions and approximations by the sum of exponents. We consider a wide enough class of nonlinear approximations characterized by a set of two required parameters. The approximating function is linear in the first parameter; these parameters are assumed to be positive. The individual terms of the approximating function represent a fixed function that depends nonlinearly on the second parameter. A numerical approximation minimizes the residual functional by approximating function values at individual points. The second parameter's value is set on a more extensive set of points of the interval of permissible values. The proposed approach's key feature consists in determining the first parameter on each separate iteration of the classical non-negative least squares method. The computational algorithm is used to rational approximate the function $x^{-\alpha}, \ 0 < \alpha < 1, \ x \geq 1$. The second example concerns the approximation of the stretching exponential function $\exp(- x^{\alpha} ), \ \ \quad 0 < \alpha < 1$ at $ x \geq 0$ by the sum of exponents.