Commit 86a25c0f authored by conmccoid's avatar conmccoid
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Extrapolation: initial commit

parent cdd78439
%--Introduction
This research project is an extension of the ASPN investigations looking for cycling behaviour in Schwarz methods preconditioned by Newton-Raphson.
Acceleration can be done by extrapolating from previous results, often improving speed up to quadratic convergence without significant increase in cost.
Some of this extrapolation can be posed as CG or other Krylov subspace methods.
Keywords:
--Method of Richardson
--Modified polynomial extrapolation (MPE)
--Conjugate gradient and Krylov subspace
Goals: --Arrive at a different method from Newton's (?)
--When is convergence quadratic?
--Apply such a method to the altS cyclic example
--Can all extrapolation methods be posed under a single unifying framework?
%--Images
Images are expected to be generated primarily through MATLAB experiments and TIKZ.
They will follow the naming convention:
FORMAT_extrap_SpecificTopic_v1_YYYYMMDD.svg
FORMAT: 3 to 4 letter indicator of how the file was created, ie. MTLB (MATLAB) or TIKZ.
extrap: indicates image is related to extrapolation research project.
SpecificTopic: identifies topic within research project the image relates to.
v1: version number.
YYYYMMDD: date of initial creation.
.svg: file format; use .eps for vector graphics when possible.
%--Code
Code is expected to be almost entirely MATLAB.
TYPE_extrap_Function_v1_YYYYMMDD.m
TYPE: 3 to 4 letters denoting intended result of the script:
PLOT: produces a plot, MTLB_extrap_Function_v1_YYYYMMDD.eps;
SUB: subfunction, used in another function;
EXP: computational experiment/test/example problem;
ALGO: implementation of algorithm.
extrap: indicates code relates to extrapolation research project.
Function: name to indicate related topic or result.
v1: version number.
YYYYMMDD: date of initial creation.
.m: file format; .m is for MATLAB files.
%--LaTeX
Notes and results will be typed in LaTeX.
Both types of documents will use the custom MathArticle template.
This provides a list of commands I seem to use often as well as the packages I tend to need.
Since this does not come from a style file any changes can be made to suit the individual document's needs.
A references.bib file will be created especially for this project, generated by Mendeley most likely.
Mendeley uses as its preferred citation key AuthorYYYY.
My preferred citation key is authorYYYYfirst, where first is the first word of the paper cited.
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\begin{document}
Nonlinear preconditioning: Krylov as extrapolation methods.
There exists proof of quadratic convergence (where? copy here)
Method of Richardson:
\begin{equation*}
u^{k+1} = u^k + \frac{1}{\alpha_k} (f-Au^k)
\end{equation*}
where $\alpha_k$ is a parameter of the acceleration.
This leads to a formula for the errors (?)
\begin{equation*}
e_k^{n+1} = \dots = \Pi_{j=0}^n \left ( 1 - \frac{\lambda_k}{\alpha_j} \right ) e_k^0.
\end{equation*}
Choose $\alpha_j$ to minimize the polynomial.
This leads to Chebyshev polynomials.
Need to know spectrum.
Alternative: take linear combinations of previous iterates.
\begin{align*}
e^n = & u - u^n \\
\sum \gamma_j u^j = & u \sum \gamma_j + \sum \gamma_j e^j \\
v^n = & u + \sum \gamma_j e^j \\
u - v^n = \sum \gamma_j e^j
\end{align*}
and the goal is to minimize the last line.
Set $d^n = u^{n+1} - u^n$ and
\begin{align*}
u = & M^{-1} N u + M^{-1} f \\
u^{n+1} = & M^{-1} N u^{n-1} + M^{-1} f \\
e^{n+1} = & u - u^{n+1} = M^{-1} N e^n \\
\implies \sum \gamma_j e^j = & \sum \gamma_j G^j e^0 = p_n(G) e^0 \\
d^n = & G d^{n-1} = G^n d^0 \\
\sum \gamma_j d^j = & \sum \gamma_j G^j d^0 = p_n(G) d^0.
\end{align*}
Minimizing this last equation is called Modified Polynomial Extrapolation.
Overdetermine the system:
\begin{equation*}
\sum \tilde{\gamma}_j d^j = -d^n
\end{equation*}
which leads to Krylov (thm, where? copy).
Try to get to different method from Newton.
Which methods have quadratic convergence?
Apply methods to altS example.
References to look into:
\begin{itemize}
\item book with Walter and Felix
\item Brezinski (book)
\item Sidi, sequence of papers
\end{itemize}
\end{document}
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