Fitting parametric curves and surfaces by l∞ distance regression

Alistair Watson; I. Al-Subaihi

doi:10.1007/s10543-005-0018-z

Fitting parametric curves and surfaces by l∞ distance regression

Alistair Watson
, I. Al-Subaihi

Research output: Contribution to journal › Article › peer-review

19 Citations (Scopus)

Abstract

For fitting curves or surfaces to observed or measured data, a common criterion is orthogonal distance regression. We consider here a natural generalization of a particular formulation of that problem which involves the replacement of least squares by the Chebyshev norm. For example, this criterion may be a more appropriate one in the context of accept/reject decisions for manufactured parts. The resulting problem has some interesting features: it has much structure which can be exploited, but generally the solution is not unique. We consider a method of Gauss-Newton type and show that if the non-uniqueness is resolved in a way which is consistent with a particular way of exploiting the structure in the linear subproblem, this can not only allow the method to be properly defined, but can permit a second order rate of convergence. Numerical examples are given to illustrate this.

Original language	English
Pages (from-to)	443-461
Number of pages	19
Journal	BIT Numerical Mathematics
Volume	45
Issue number	3
DOIs	https://doi.org/10.1007/s10543-005-0018-z
Publication status	Published - 2005

Keywords

l∞ norm
Curve fitting
Non-unique solution
Gauss-Newton method
Simplex method

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10.1007/s10543-005-0018-z

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abstract = "For fitting curves or surfaces to observed or measured data, a common criterion is orthogonal distance regression. We consider here a natural generalization of a particular formulation of that problem which involves the replacement of least squares by the Chebyshev norm. For example, this criterion may be a more appropriate one in the context of accept/reject decisions for manufactured parts. The resulting problem has some interesting features: it has much structure which can be exploited, but generally the solution is not unique. We consider a method of Gauss-Newton type and show that if the non-uniqueness is resolved in a way which is consistent with a particular way of exploiting the structure in the linear subproblem, this can not only allow the method to be properly defined, but can permit a second order rate of convergence. Numerical examples are given to illustrate this.",

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T1 - Fitting parametric curves and surfaces by l∞ distance regression

AU - Watson, Alistair

AU - Al-Subaihi, I.

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N2 - For fitting curves or surfaces to observed or measured data, a common criterion is orthogonal distance regression. We consider here a natural generalization of a particular formulation of that problem which involves the replacement of least squares by the Chebyshev norm. For example, this criterion may be a more appropriate one in the context of accept/reject decisions for manufactured parts. The resulting problem has some interesting features: it has much structure which can be exploited, but generally the solution is not unique. We consider a method of Gauss-Newton type and show that if the non-uniqueness is resolved in a way which is consistent with a particular way of exploiting the structure in the linear subproblem, this can not only allow the method to be properly defined, but can permit a second order rate of convergence. Numerical examples are given to illustrate this.

AB - For fitting curves or surfaces to observed or measured data, a common criterion is orthogonal distance regression. We consider here a natural generalization of a particular formulation of that problem which involves the replacement of least squares by the Chebyshev norm. For example, this criterion may be a more appropriate one in the context of accept/reject decisions for manufactured parts. The resulting problem has some interesting features: it has much structure which can be exploited, but generally the solution is not unique. We consider a method of Gauss-Newton type and show that if the non-uniqueness is resolved in a way which is consistent with a particular way of exploiting the structure in the linear subproblem, this can not only allow the method to be properly defined, but can permit a second order rate of convergence. Numerical examples are given to illustrate this.

KW - l∞ norm

KW - Curve fitting

KW - Non-unique solution

KW - Gauss-Newton method

KW - Simplex method

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DO - 10.1007/s10543-005-0018-z

M3 - Article

SN - 0006-3835

VL - 45

SP - 443

EP - 461

JO - BIT Numerical Mathematics

JF - BIT Numerical Mathematics

IS - 3

ER -

Fitting parametric curves and surfaces by l∞ distance regression

Abstract

Keywords

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