From cc50202cce35b9faa003ab7efcdf1339d4e6b64b Mon Sep 17 00:00:00 2001 From: Stefan Dresselhaus Date: Tue, 10 Oct 2017 12:24:15 +0200 Subject: [PATCH] added evo-section --- arbeit/bibma.bib | 11 +++++++ arbeit/ma.md | 51 ++++++++++++++++++++++++++++++++- arbeit/ma.pdf | Bin 182523 -> 189418 bytes arbeit/ma.tex | 73 +++++++++++++++++++++++++++++++++++++++++++---- 4 files changed, 128 insertions(+), 7 deletions(-) diff --git a/arbeit/bibma.bib b/arbeit/bibma.bib index 3f3b250..ed165e3 100644 --- a/arbeit/bibma.bib +++ b/arbeit/bibma.bib @@ -115,3 +115,14 @@ eprint = {https://doi.org/10.1137/0111030} organization={IEEE}, url={http://www.graphics.uni-bielefeld.de/publications/cec15.pdf} } +@article{back1993overview, + title={An overview of evolutionary algorithms for parameter optimization}, + author={B{\"a}ck, Thomas and Schwefel, Hans-Paul}, + journal={Evolutionary computation}, + volume={1}, + number={1}, + pages={1--23}, + year={1993}, + publisher={MIT Press}, + url={https://www.researchgate.net/profile/Hans-Paul_Schwefel/publication/220375001_An_Overview_of_Evolutionary_Algorithms_for_Parameter_Optimization/links/543663d00cf2dc341db30452.pdf} +} diff --git a/arbeit/ma.md b/arbeit/ma.md index 7f6c4ed..3890936 100644 --- a/arbeit/ma.md +++ b/arbeit/ma.md @@ -174,7 +174,56 @@ mesh albeit the downsides. ## What is evolutional optimization? \label{sec:back:evo} -\change[inline]{Write this section} +In this thesis we are using an evolutional optimization strategy to solve the +problem of finding the best parameters for our deformation. This approach, +however, is very generic and we introduce it here in a broader sense. + +\begin{algorithm} +\caption{An outline of evolutional algorithms} +\label{alg:evo} +\begin{algorithmic} +\STATE t := 0; +\STATE initialize $P(0) := \{\vec{a}_1(0),\dots,\vec{a}_\mu(0)\} \in I^\mu$; +\STATE evaluate $F(0) : \{\Phi(x) | x \in P(0)\}$; +\WHILE{$c(F(t)) \neq$ \TRUE} + \STATE recombine: $P’(t) := r(P(t))$; + \STATE mutate: $P''(t) := m(P’(t))$; + \STATE evaluate $F''(t) : \{\Phi(x) | x \in P''(t)\}$ + \STATE select: $P(t + 1) := s(P''(t) \cup Q,\Phi)$; + \STATE t := t + 1; +\ENDWHILE +\end{algorithmic} +\end{algorithm} + +The general shape of an evolutional algorithm (adapted from +\cite{back1993overview}} is outlined in Algorithm \ref{alg:evo}. Here, $P(t)$ +denotes the population of parameters in step $t$ of the algorithm. The +population contains $\mu$ individuals $a_i$ that fit the shape of the parameters +we are looking for. Typically these are initialized by a random guess or just +zero. Further on we need a so-called *fitness-function* $\Phi : I \mapsto M$ that can take +each parameter to a measurable space along a convergence-function $c : I \mapsto +\mathbb{B}$ that terminates the optimization. + +The main algorithm just repeats the following steps: + +- **Recombine** with a recombination-function $r : I^{\mu} \mapsto I^{\lambda}$ to + generate new individuals based on the parents characteristics. + This makes sure that the next guess is close to the old guess. +- **Mutate** with a mutation-function $m : I^{\lambda} \mapsto I^{\lambda}$ to + introduce new effects that cannot be produced by mere recombination of the + parents. + Typically this just adds minor defects to individual members of the population + like adding a random gaussian noise or amplifying/dampening random parts. +- **Selection** takes a selection-function $s : (I^\lambda \cup I^{\mu + \lambda},\Phi) \mapsto I^\mu$ that + selects from the previously generated $I^\lambda$ children and optionally also + the parents (denoted by the set $Q$ in the algorithm) using the + fitness-function $\Phi$. The result of this operation is the next Population + of $\mu$ individuals. + +All these functions can (and mostly do) have a lot of hidden parameters that +can be changed over time. One can for example start off with a high +mutation--rate that cools off over time (i.e. by lowering the variance of a +gaussian noise). As the recombination and selection-steps are usually pure ## Advantages of evolutional algorithms \label{sec:back:evogood} diff --git a/arbeit/ma.pdf b/arbeit/ma.pdf index c50672934b8ef2c07809f4cae23fcc9d485d06e3..a4a7040c3a09b24ae725a9430526d3242aa6527b 100644 GIT binary patch delta 86646 zcmZU*Q*LVWZd;}JeOhHtfj)|TPhV1b6>J=t&kAMuo#>T|Y5993OWNK&&qHBfA!cu5yw3<@4dUhPHgrC`E^@+&Hq6wS!T;ei_2=(#T zLc-aipqWOSDIr|rEN9n2MU)yg<=DOqnLk!JI9BuHjr;STh7NFM6)5g_+MQaJVnz}G zDY;U>m`3;a)JesM7{S+APeW_tR?Wj3kU+G=4>kcFTn1_g?OXFHIpkE)AxhwZl*6O3Nf zRxV4o#^k`>oSQLW{$4Zu(L}cGpSiUeVp9DYat!j0E==!^ZILX zG9>x?8mz+sW>LpDF#XS#ZD{^QvLmENSHAtV` zRd{du<1T5!530(`^eD@kpeJxgaU=h-6?)Ol+))dY}Y=y43#6 zm?$);`@eGp?R=EYHn4SM5(zD`{Jx>YPvhdp1yb$X5D>J`NMTt!4rbp;FVmn5gM=Ol zUd&3QYTM_#T@T@KWUD`GrvYL_&}|%rDk#*G8$^oR3{yYL9a>%j^u3@6qRi)X969}% z$sDM>w8hGK&RD%8Lm{z<}Rj+ilOqUu>gO@zEGs*h^GQoSsJu+OI zhBx@fu~^CwqG?EqsD~9MPj1%zKI1FXNQiJy>d|F+!I^ZD9aukg(pIR`Ecr=Hdrl?> ziX~?yl!zQj;ww|S^EP9{^D#XdIy+zj^6Q0g50nJhqV0HFJ*#;LDctZlTam05V~ySz 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