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Output Least Squares Identifiability and Quadratically Wellposed NLS Problems

  • Guy ChaventEmail author
Chapter
Part of the Scientific Computation book series (SCIENTCOMP)

Abstract

We consider in this chapter the nonlinear least squares (NLS) problem (1.10), which we recall here for convenience:
$$\hat{x}\quad \mbox{ minimizes }\quad J(x) = \frac{1} {2}\|\varphi (x) - {z\|}_{F}^{2}\quad \mbox{ over }\quad C.$$
(4.1)
As we have seen in Chap. 1, this inverse problem describes the identification of the parameter xC from a measurement z of φ(x) in F. We suppose that the minimum set of hypothesis (1.12) of Chap. 1 holds:
$$\left \{\begin{array}{rcl} E\ & =&\ \mathrm{Banach\ space,\ with\ norm}{\quad \|\ \|}_{E}, \\ C\ & \subset &\ E\quad \mathrm{with}\ C\ \mathrm{convex\ and\ closed,} \\ F\ & =&\ \mathrm{Hilbert\ space,\ with\ norm}{\quad \|\ \|}_{F}, \\ z\ & \in &\ F \\ \varphi \ & : &\ C\ \rightsquigarrow \ F\ \mathrm{is\ differentiable\ along\ segments\ of\ C}, \\ \mathrm{and}& : &\exists \,{\alpha }_{M} \geq 0\ \mbox{ s.t. }\ \forall {x}_{0},{x}_{1} \in C,\ \forall t \in [0,1], \\ & &\|{D}_{t}\,\varphi {((1 - t){x}_{0} + t{x}_{1})\|}_{F} \leq \ {\alpha }_{M}\|{x}_{1} - {x{}_{0}\|}_{E},\end{array} \right.$$
(4.2)
and we recall the definition of stationary points

Keywords

Inverse Problem Singular Value Decomposition Linear Stability Nonlinear Little Square Deflection Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Ceremade, Université Paris-DauphineParis Cedex 16France
  2. 2.Inria-RocquencourtLe Chesnay CedexFrance

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