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+1,3 @@ \newsection \subsection{Nichtlineare Ausgleichsrechnung} +Es ist natürlich auch möglich, dass das Modell für das Ausgleichsproblem nicht linear ist. diff --git a/semester3/numcs/parts/05_curve-fitting/02_non-linear/01_newton-method-in-rn.tex b/semester3/numcs/parts/05_curve-fitting/02_non-linear/01_newton-method-in-rn.tex index e69de29..b952726 100644 --- a/semester3/numcs/parts/05_curve-fitting/02_non-linear/01_newton-method-in-rn.tex +++ b/semester3/numcs/parts/05_curve-fitting/02_non-linear/01_newton-method-in-rn.tex @@ -0,0 +1,15 @@ +\subsubsection{Newton-Verfahren} +Aus der Analysis ist bekannt, dass für gesuchtes $x \in \R^n$, so dass $\Phi(x)$ minimal ist, eine notwendige Bedingung durch $\text{grad}(\Phi(x)) = 0$ gegeben ist. + +Da wir also eine Nullstellensuche in $\R^n$ haben, können wir dies mit dem Newton-Verfahren lösen: +\rmvspace +\begin{align*} + x^{(k + 1)} = x^{(k)} - (D \text{grad}(\Phi(x)))^{-1}\text{grad}(\Phi(x)) +\end{align*} + +\drmvspace +Da $H_\Phi(x) := D(\text(grad)(\Phi(x)))$ ist, haben wir also: +\rmvspace +\begin{align*} + x^{(k + 1)} = x^{(k)} - (H_\Phi(x))^{-1}\text{grad}(\Phi(x)) +\end{align*} diff --git a/semester3/numcs/parts/05_curve-fitting/02_non-linear/02_gauss-newton.tex b/semester3/numcs/parts/05_curve-fitting/02_non-linear/02_gauss-newton.tex index e69de29..2311c6c 100644 --- a/semester3/numcs/parts/05_curve-fitting/02_non-linear/02_gauss-newton.tex +++ b/semester3/numcs/parts/05_curve-fitting/02_non-linear/02_gauss-newton.tex @@ -0,0 +1,20 @@ +\subsubsection{Gauss-Newton Verfahren} +Direkt das Newton-Verfahren auf ein Problem anzuwenden kann unmöglich oder schwer praktikabel sein. + +Die Idee des Gauss-Newton Verfahrens ist es, die komplizierte Funktion $F(x)$ lokal durch eine lineare Funktion approximiert, also: +\begin{align*} + F(x) \approx F(y) + DF(y) (x - y) = F(y) + DF(y)x - DF(y)y +\end{align*} +Falls man $A := DF(y)$ und $b = DF(y)y - F(y)$ definiert, so erhält man ein lineares Ausgleichsproblem: +\rmvspace +\begin{align*} + \argmin{x \in \R^n} \frac{1}{2} ||F(x)||^2_2 \approx \argmin{x \in \R^n} \frac{1}{2} ||F(y) + DF(y) x||^2_2 = \argmin{x \in \R^n} \frac{1}{2} ||Ax - b||^2_2 +\end{align*} + +\drmvspace +wobei $y$ eine Näherung der Lösung $x$ ist. +Die Iterationsvorschrift ist gegeben durch: +\rmvspace +\begin{align*} + x^{(k + 1)} = x^{(k)} - s \smallhspace \text{ mit } s := \argmin{z \in \R^n} ||F(x^{(k)}) - DF(x^{(k)})z||^2_2 +\end{align*}