@@ -64,11 +39,11 @@ An algorithm is proposed combining several aspects of this theory and others to
\section{Introduction}
\label{mccoid_mini_17_sec:intro}
ASPIN \cite{Cai2002}, RASPEN \cite{Dolean2016}, and MSPIN \cite{liu2015field} rely on various Schwarz methods to precondition either Newton-Raphson or inexact Newton.
ASPIN \cite{mccoid_mini_17_Cai2002}, RASPEN \cite{mccoid_mini_17_Dolean2016}, and MSPIN \cite{mccoid_mini_17_liu2015field} rely on various Schwarz methods to precondition either Newton-Raphson or inexact Newton.
While \textit{a priori} convergence criteria have been found for the underlying Schwarz methods, so far none exist for their combination with Newton-Raphson.
Like in the linear case when combining a Krylov method and a Schwarz method,
there is an equivalence between preconditioning Newton-Raphson with a Schwarz method and accelerating that same Schwarz method with Newton-Raphson \cite{Gander2017}:
there is an equivalence between preconditioning Newton-Raphson with a Schwarz method and accelerating that same Schwarz method with Newton-Raphson \cite{mccoid_mini_17_Gander2017}:
A domain is first subdivided into subdomains, the problem solved on each subdomain, and the resulting formulation iterated through Krylov and Newton-Raphson, respectively.
We examine cycling behaviour in alternating Schwarz in one dimension that has been accelerated by applying Newton-Raphson.
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@@ -106,12 +81,12 @@ where $v_i(x)=\partial u_i(x) / \partial \gamma$ and $J(u_i)$ is the Jacobian of
\section{Convergence of generic fixed point iterations and Newton-Raphson}
A generic fixed point iteration $x_{n+1}=g(x_n)$ converges when $\abs{g(x_n)-x^*} < \abs{x_n-x^*}$, where $x^*$ is the fixed point.
A generic fixed point iteration $x_{n+1}=g(x_n)$ converges when $\left\vert g(x_n)-x^*\right\vert < \left\vert x_n-x^*\right\vert$, where $x^*$ is the fixed point.
This occurs when $g(x)$ lies between $x$ and $2x^*-x$.
The convergence or divergence of the fixed point iteration is monotonic if $\sign(g(x)-x^*)=\sign(x - x^*)$ and oscillatory otherwise.
This creates four lines, $y=x$, $y=2x^*-x$, $y=x^*$ and $x=x^*$, that divide the plane into octants.
The four pairs of opposite octants form four regions with distinct
behaviour of the fixed point iteration, see left of Figure \ref{mccoid_mini_17_fig:FP} or Figure 5.7 from \cite{gander2014scientific}:
behaviour of the fixed point iteration, see left of Figure \ref{mccoid_mini_17_fig:FP} or Figure 5.7 from \cite{mccoid_mini_17_gander2014scientific}:
\begin{description}
\item[\textcolor{red}{1}, $g(x) < x < x^*$ or $g(x) > x > x^*$:] monotonic divergence;
\item$\pdxdy{f(x,v,v')}{u}\leq0$ for all $x \in[a,b]$ and $v \in H_0^1([a,b])$ ,
\item$\abs{f(x,v,v')}\leq C(1+\abs{v'}^\eta)$ for all $x \in[a,b]$ and $v \in H_0^1([a,b])$ and some $C > 0$, $0 < \eta < 1$ .
\item$\frac{\partialf(x,v,v')}{\partialu}\leq0$ for all $x \in[a,b]$ and $v \in H_0^1([a,b])$ ,
\item$\left\vert f(x,v,v')\right\vert\leq C(1+\left\vert v' \right\vert^\eta)$ for all $x \in[a,b]$ and $v \in H_0^1([a,b])$ and some $C > 0$, $0 < \eta < 1$ .
\end{itemize}
The problem is solved using alternating Schwarz with two subdomains and Dirichlet transmission conditions.
Then $G(\gamma)$ for this problem lies within region 2.
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@@ -307,7 +280,7 @@ Then $G(\gamma)$ for this problem lies within region 2.
\begin{proof}
It suffices to prove that the problem is well posed and $0 < G'(\gamma) < 1$ for all $\gamma\in\mathbb{R}$.
The well-posedness of the problem is guaranteed by Proposition 2 from \cite{lui1999schwarz}.
The well-posedness of the problem is guaranteed by Proposition 2 from \cite{mccoid_mini_17_lui1999schwarz}.
As Lui points out, this also means the problem is well posed on any subdomain.
Using Theorem \ref{mccoid_mini_17_thm:mono} this gives monotonicity of $G(\gamma)$.
Moreover, if $u(x)=0$ for any $x \in(a,b)$ then the problem would be well posed on the domains $[a,x]$ and $[x,b]$.
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@@ -315,9 +288,9 @@ As such, $u(x)$ has the same sign as $\gamma$ and $G'(\gamma) > 0$.
\textbf{Left:} Results of Newton-Raphson accelerated alternating Schwarz as a function of initial condition in solving equation (\ref{mccoid_mini_17_eq:sin}).
The value of $a$ is 3.6 and the subdomains are $\Omega_1=(-1,0.2)$ and $\Omega_2=(-0.2,1)$.
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@@ -362,7 +335,7 @@ Where stable cycles exist so too must there be period doubling bifurcation.
Changing the value of the parameter $a$ we find that the 2-cycle found in Figure \ref{mccoid_mini_17_fig:ex1} (left) becomes two 2-cycles, then two 4-cycles, and so on until it devolves into chaos, see Figure \ref{mccoid_mini_17_fig:ex2}.
\textbf{Left:} value of $a$ at which bifurcation starts.
\textbf{Right:} width of basin of cycling in $\gamma$ and $a$.}
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@@ -389,10 +362,10 @@ Meanwhile, the length of the bifurcation diagram increases, meaning there are mo
\section{Accelerated alternating Schwarz with guaranteed convergence}
Given Theorem \ref{mccoid_mini_17_thm:lui} and the conditions of Table \ref{mccoid_mini_17_tab:conditions} one can construct a series of tests to see if the Newton-Raphson acceleration is suitable for a given iteration.
We present one further useful trick to strengthen convergence, a correction to Newton-Raphson due to Davidenko and Branin \cite{branin1972widely,brent1972davidenko,davidenko1953new}.
We present one further useful trick to strengthen convergence, a correction to Newton-Raphson due to Davidenko and Branin \cite{mccoid_mini_17_branin1972widely,mccoid_mini_17_brent1972davidenko,mccoid_mini_17_davidenko1953new}.
For $G(\gamma)$ within region 2 the Newton-Raphson acceleration will now always march in the direction of the fixed point.
It may still overshoot and cycle but the direction will always be correct.
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@@ -404,7 +377,7 @@ For a problem satisfying the conditions of Theorem \ref{mccoid_mini_17_thm:lui}
If $G'(\gamma_n)=1$ then set $\gamma_{n+1}= G(\gamma_n)$, increment $n$ and return to step 2.
If this is not true, proceed to step 3.
\item Perform step (3*), which is the Newton-Raphson acceleration using the Davidenko-Branin trick.
If $\abs{G'(\gamma_n)-1}\geq1/2$ then set $\gamma_{n+1}=\tilde{\gamma}_n$, increment $n$ and return to step 2.
If $\left\vertG'(\gamma_n)-1\right\vert\geq1/2$ then set $\gamma_{n+1}=\tilde{\gamma}_n$, increment $n$ and return to step 2.
If this is not true, calculate $\hat{\gamma}_n$, the average of $\gamma_n$ and $\tilde{\gamma}_n$, and proceed to step 4.
\item Calculate $G(\hat{\gamma}_n)$.
If $G(\hat{\gamma}_n)-\hat{\gamma}_n$ has the same sign as $G(\gamma_n)-\gamma_n$ then set $\gamma_{n+1}=\tilde{\gamma}_n$, increment $n$ and return to step 2.
