Commit e89b60b4 by Conor McCoid

### Extrap: started explaining the structure of extrapolation methods

parent bf79d994
 ... ... @@ -34,11 +34,12 @@ \newtheorem{prop}{Proposition} \newtheorem{conj}{Conjecture} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \begin{document} Nonlinear preconditioning: Krylov as extrapolation methods. There exists proof of quadratic convergence (where? copy here) There exists proof of quadratic convergence (Sidi, \textit{Extrapolation vs. projection methods for linear systems of equations}, copy here) Method of Richardson: \begin{equation*} ... ... @@ -89,4 +90,85 @@ References to look into: \item Sidi, sequence of papers \end{itemize} Questions: \begin{itemize} \item what characterizes a quadratic acceleration method? \item what is the general framework of acceleration methods? \item do the sequences of Schwarz methods have distinct forms? Can they be predicted/exploited? \end{itemize} \section{Framework of acceleration methods} Take a sequence $(S_n) \to S$. Let $T$ be a transformation such that $T(S_n) = T_n$. The transformation $T$ is an acceleration if $(T_n) \to S$ faster than $(S_n)$, ie. \lim_{n \to \infty} \frac{T_n - S}{S_n - S} = 0. The kernel of a transformation $T$ is the set of sequences such that $T_n = S$ for all $n \in \bbn$. \begin{example} T_n = \frac{S_n S_{n+2} - S_{n+1}^2}{S_{n+2} - 2 S_{n+1} + S_n} then \begin{align*} \liminfty{n} \frac{T_n - S}{S_n - S} = & \liminfty{n} \frac{S_n S_{n+2} - S_{n+1}^2 - S (S_{n+2} - 2S_{n+1} + S_n)}{S_{n+2} - 2S_{n+1} + S_n} \frac{1}{S_n - S} \\ = & \liminfty{n} \frac{S_{n+2} (S_n - S) - S_{n+1} (S_{n+1} - S) - S (S_n - S_{n+1})}{(S_n-S) (S_{n+2} - 2 S_{n+1} + S_n)} \\ = & \liminfty{n} \frac{S_{n+2} - \frac{S_{n+1} - S}{S_n - S} S_{n+1} - \frac{S_n - S + S - S_{n+1}}{S_n - S} S}{S_{n+2} - 2 S_{n+1} + S_n} \\ = & \liminfty{n} \frac{S_{n+2} - S - \frac{S_{n+1} - S}{S_n - S} (S_{n+1} - S)}{S_{n+2} - 2 S_{n+1} + S_n} . \end{align*} If $\lim (S_{n+1} - S)/(S_n - S) \in [-1,1)$ then the sequence is accelerated (missing a step?). The kernel of this transformation is $S_n = S + a \lambda^n$ for $\lambda \neq 1$ and $a \neq 0$: \begin{align*} T_n = & \frac{(S + a \lambda^n)(S + a \lambda^{n+2}) - (S + a \lambda^{n+1})^2}{S + a \lambda^{n+2} - 2S - 2 a \lambda^{n+1} + S + a \lambda^n} \\ = & \frac{S^2 + a \lambda^n S + a \lambda^{n+2} S + a^2 \lambda^{2n + 2} - S^2 - 2 a \lambda^{n+1} S - a^2 \lambda^{2n + 2}}{a \lambda^n (\lambda^2 - 2 \lambda + 1)} \\ = & \frac{a \lambda^n S ( 1 + \lambda^2 - 2 \lambda)}{a \lambda^n (\lambda - 1)^2} = S . \end{align*} \end{example} \newcommand{\kernel}{\mathcal{K}_T} A transformation may be expressed as a relation on $q$ elements of a sequence, $R(S_n,S_{n+1}, \dots, S_{n+q}, S)$. In this form, if $(S_n) \in \kernel$, the kernel of the transformation, then $R(S_n, \dots, S_{n+q}, S) = 0$ for all $n \in \bbn$. \begin{defn}[Extrapolation method] A transformation $T$ is an extrapolationmethod if T_n = S \ \forall n \in \bbn \iff (S_n) \in \kernel. \end{defn} (Somehow this already defines the kernel of the transformation; is this definition then vacuous? It is stated in Brezinski that because of this definition any sequence transformation can be viewed as an extrapolation method.) This relation may be expressed as R(S_n, \dots, S_{n+q}, S) =0 \iff S = T_n = \sum_{i=0}^q a_i S_{n+i}. \begin{example} Suppose $R(S_n,S_{n+1},S) = a_1 (S_n - S) + a_2 (S_{n+1} - S) = 0$ for all $n$. Then \begin{align*} a_1 & (S_n-S) && + & a_2 (S_{n+1} - S) = & 0 \\ a_1 & (S_{n+1} - S) && + & a_2 (S_{n+2} - S) = & 0. \end{align*} WLOG (can you prove it?) $a_1 + a_2 = 1$ and \begin{align*} S = & a_1 S_n + (1-a_1) S_{n+1} \\ S = & a_1 S_{n+1} + (1 - a_1) S_{n+2} \\ \implies 0 = & a_1 \Delta S_n + (1 - a_1) \Delta S_{n+1} \end{align*} where $\Delta S_n = S_{n+1} - S_n$. And so $a_1 = \Delta S_{n+1} / \Delta^2 S_n$. This leads to the transformation described in the previous example, T_n = \frac{S_n S_{n+2} - S_{n+1}^2}{S_{n+2} - 2 S_{n+1} + S_n}. Note that $dR/dS = -(a_1 + a_2)$. \end{example} \end{document} \ No newline at end of file
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