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47 lines
1.9 KiB
TeX
47 lines
1.9 KiB
TeX
\newsection
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\subsection{Gedämpftes Newton-Verfahren}
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Wir wenden einen einen Dämpfungsfaktor $\lambda^{(k)}$ an, welcher heuristisch gewählt wird:
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\rmvspace
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\begin{align*}
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x^{(k + 1)} := x^{(k)} - \lambda^{(k)}DF(x^{(k)})^{-1} F(x^{(k)})
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\end{align*}
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\drmvspace
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Wir wählen $\lambda^{(k)}$ so, dass für $\Delta x^{(k)} = DF(x^{(k)})^{-1} F(x^{(k)})$ und $\Delta(\lambda^{(k)}) = DF(x^{(k)})^{-1} F(x^{(k)} - \lambda^{(k)} \Delta x^{(k)})$
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\rmvspace
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\begin{align*}
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||\Delta x(\lambda^{(k)})||_2 \leq \left( 1 - \frac{\lambda^{(k)}}{2} \right) ||\Delta x^{(k)}||_2
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\end{align*}
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\innumpy Das gedämpfte Newton-Verfahren lässt sich mit Funktionen aus \verb|scipy.linalg| implementieren:
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\begin{code}{python}
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def dampened_newton(x: np.ndarray, F, DF, q=0.5, rtol=1e-10, atol=1e-12):
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""" Dampened Newton with dampening factor q """
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lup = lu_factor(DF(x)) # LU factorization for efficiency, direct works too
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s = lu_solve(lup, F(x)) # 1st proper Newton correction
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damp = 1 # Start with no dampening
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x_damp = x - damp*s
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s_damp = lu_solve(lup, F(x_damp)) # 1st simplified Newton correction (Reuse Jacobian)
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while norm(s_damp) > rtol * norm(x_damp) and norm(s_damp) > atol:
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while norm(s_damp) > (1-damp*q) * norm(s): # Reduce dampening if step aggresive
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damp *= q
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if damp < 1e-4: return x # Conclude dampening doesn't work anymore
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x_damp = x - damp*s # Try weaker dampening instead
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s_damp = lu_solve(lup, F(x_damp))
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x = x_damp # Accept this dampened iteration, continue with next proper step
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lup = lu_factor(DF(x))
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s = lu_solve(lup, F(x)) # Next proper Newton correction
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damp = np.min( damp/q, 1 )
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x_damp = x - damp*s
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s_damp = lu_solve(lup, F(x_damp)) # Next simplified Newton correction
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return x_damp
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\end{code}
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\newpage
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