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Nonlinear least-squares method via an isomorphic geometrical setup
Abstract The resolution of a nonlinear parametric adjustment model is addressed through an isomorphic geometrical setup with tensor structure and notation, represented by a u-dimensional “model surface” embedded in a flat n-dimensional “observational space”. Then observations correspond to the observational-space coordinates of the pointQ, theu initial parameters correspond to the model-surface coordinates of the “initial” pointP, and theu adjusted parameters correspond to the model-surface coordinates of the “least-squares” point$$\bar P$$. The least-squares criterion results in a minimum-distance property implying that the vector$$\bar P$$Q must be orthogonal to the model surface. The geometrical setup leads to the solution of modified normal equations, characterized by a positive-definite matrix. The latter contains second-order and, optionally, thirdorder partial derivatives of the observables with respect to the parameters. This approach significantly shortens the convergence process as compared to the standard (linearized) method.
Nonlinear least-squares method via an isomorphic geometrical setup
Abstract The resolution of a nonlinear parametric adjustment model is addressed through an isomorphic geometrical setup with tensor structure and notation, represented by a u-dimensional “model surface” embedded in a flat n-dimensional “observational space”. Then observations correspond to the observational-space coordinates of the pointQ, theu initial parameters correspond to the model-surface coordinates of the “initial” pointP, and theu adjusted parameters correspond to the model-surface coordinates of the “least-squares” point$$\bar P$$. The least-squares criterion results in a minimum-distance property implying that the vector$$\bar P$$Q must be orthogonal to the model surface. The geometrical setup leads to the solution of modified normal equations, characterized by a positive-definite matrix. The latter contains second-order and, optionally, thirdorder partial derivatives of the observables with respect to the parameters. This approach significantly shortens the convergence process as compared to the standard (linearized) method.
Nonlinear least-squares method via an isomorphic geometrical setup
Blaha, Georges (author) / Bessette, Robert P. (author)
Bulletin géodésique ; 63
1989
Article (Journal)
English
Geodäsie , Geometrie , Geodynamik , Zeitschrift , Mathematik , Mineralogie
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