# Output Least Squares Identifiability and Quadratically Wellposed NLS Problems

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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