Output Least Squares Identifiability and Quadratically Wellposed NLS Problems

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


We consider in this chapter the nonlinear least squares (NLS) problem (1.10), which we recall here for convenience:
$$\hat{x}\quad \mbox{ minimizes }\quad J(x) = \frac{1} {2}\|\varphi (x) - {z\|}_{F}^{2}\quad \mbox{ over }\quad C.$$
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:
$$\left \{\begin{array}{rcl} E\ & =&\ \mathrm{Banach\ space,\ with\ norm}{\quad \|\ \|}_{E}, \\ C\ & \subset &\ E\quad \mathrm{with}\ C\ \mathrm{convex\ and\ closed,} \\ F\ & =&\ \mathrm{Hilbert\ space,\ with\ norm}{\quad \|\ \|}_{F}, \\ z\ & \in &\ F \\ \varphi \ & : &\ C\ \rightsquigarrow \ F\ \mathrm{is\ differentiable\ along\ segments\ of\ C}, \\ \mathrm{and}& : &\exists \,{\alpha }_{M} \geq 0\ \mbox{ s.t. }\ \forall {x}_{0},{x}_{1} \in C,\ \forall t \in [0,1], \\ & &\|{D}_{t}\,\varphi {((1 - t){x}_{0} + t{x}_{1})\|}_{F} \leq \ {\alpha }_{M}\|{x}_{1} - {x{}_{0}\|}_{E},\end{array} \right.$$
and we recall the definition of stationary points


Inverse Problem Singular Value Decomposition Linear Stability Nonlinear Little Square Deflection Condition 
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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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