Skript

We finally mention that any suitable linear regularization method can be applied
to the Newton step, in particular also linear iterative methods such as
Landweber iteration or conjugate gradient methods, leading to methods called
Newton-Landweber or Newton-CG. The above decay of α k to zero corresponds
to an increase of the inner iteration number in such cases.
5.3 Iterative Methods als Zeit-Diskrete Flüße (time-discrete
flows)
Wir schließen diese Kapitel mit der Interpretation iterativer Regularisierungsmethoden
als Zeit-Diskretisierung von Gradientenflüßen. We starten wieder mit
der Landweber-Iteration die wir umschreiben als
x k+1 − x k
= −T ∗ (T x k − y)
τ
Wenn wir τ als Schrittweite und x k = x(kτ) als Zeitschritte eines Flußes interpretieren,
dann entspricht die Landweber-Iteration einer expliziten (forwärts
Euler) diskretisierung
dx
dt (t) = −T ∗ (T x(t) − y), (5.8)
d.h. einer asymptotischen Regularisierung. Analog erhalten wir für das Newton-
Verfahren
x k+1 − x k
= −F ′ (x k )(F (x k ) − y).
τ
underbrace
dx
dt (t) = −F ′ (x(t)) ∗ (F (x(t)) − y), (5.9)
From this correspondance to the flow (5.9) it seems natural to try other time discretizations.
The implicit time discretization (backward Euler) yields the nonlinear
equation
x k+1 − x k
= −F ′ (x k+1 )(F (x k+1 ) − y),
τ
which is the optimality condition of the optimization problem
‖F (x) − y‖ 2 + 1 τ ‖xk+1 − x k ‖ 2 → min
x∈X ,
well-known from Tikhonov regularization. The corresponding iterative procedure
is therefore called iterated Tikhonov regularization. If we perform a semiimplicit
time discretization, i.e., approximating F ′ (x) ∗ explicitely and F (x) by
a first-order Taylor expansion around the last time step x k we end up with the
Levenberg-Marquardt method. The iteratively regularized Gauss-Newton method
corresponds to a non-consistent semi-implicit time discretization.
From this motivation it is not surprising that general Runge-Kutta methods (even
non-consistent ones) applied to the flow (5.9) yield convergent iterative regularization
methods, as recently shown by Rieder [?].
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