TY - GEN
A1 - Griewank, Andreas
A1 - Streubel, Tom
A1 - Lehmann, Lutz
A1 - Hasenfelder, Richard
A1 - Radons, Manuel
T1 - Piecewise linear secant approximation via Algorithmic Piecewise Differentiation
N2 - It is shown how piecewise differentiable functions \(F: R^n → R^m\) that are defined by evaluation programs can be approximated locally by a piecewise linear model based on a pair of sample points x̌ and x̂. We show that the discrepancy between function and model at any point x is of the bilinear order O(||x − x̌|| ||x − x̂||). This is a little surprising since x ∈ R^n may vary over the whole Euclidean space, and we utilize only two function samples F̌ = F(x̌) and F̂ = F(x̂), as well as the intermediates computed during their evaluation. As an application of the piecewise linearization procedure we devise a generalized Newton’s method based on successive piecewise linearization and prove for it sufficient conditions for convergence and convergence rates equaling those of semismooth Newton. We conclude with the derivation of formulas for the numerically stable implementation of the aforedeveloped piecewise linearization methods.
T3 - ZIB-Report - 16-54
KW - Automatic differentiation
KW - Computational graph
KW - Lipschitz continuity
KW - Generalized Hermite interpolation
KW - ADOL-C
Y1 - 2016
UR - https://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/6164
UR - https://nbn-resolving.org/urn:nbn:de:0297-zib-61642
SN - 1438-0064
ER -