diff --git a/Lectures_my/NumMet/2016/Lecture12/Makefile b/Lectures_my/NumMet/2016/Lecture12/Makefile
new file mode 100644
index 0000000..cb8768a
--- /dev/null
+++ b/Lectures_my/NumMet/2016/Lecture12/Makefile
@@ -0,0 +1,7 @@
+all: lecture12.pdf
+
+lecture12.pdf: lecture12.tex
+	xelatex lecture12.tex
+	xelatex lecture12.tex
+clean:
+	rm *.aux *.bbl *.blg *.log *.out *.pdf *.sh *.xml *.snm *.toc *.nav 
diff --git a/Lectures_my/NumMet/2016/Lecture12/beamercolorthemeFlip.sty b/Lectures_my/NumMet/2016/Lecture12/beamercolorthemeFlip.sty
new file mode 100755
index 0000000..d6b9c50
--- /dev/null
+++ b/Lectures_my/NumMet/2016/Lecture12/beamercolorthemeFlip.sty
@@ -0,0 +1,179 @@
+% Color style file for the BEAMER FLIP THEME
+% Copyright 2012 by Flip Tanedo
+% This file may be distributed and/or modified
+% 	1. under the LaTeX Project Public License and/or
+% 	2. under the GNU Public License.
+
+
+
+\mode<presentation>
+\newif\ifbeamer@Flip@dark
+\beamer@Flip@darkfalse
+\DeclareOptionBeamer{dark}{\beamer@Flip@darktrue}
+\ProcessOptionsBeamer
+
+
+% These are a few colors that I like.
+
+\definecolor{crimsonred}{RGB}{153,0,0}		% Neurtal red, good for dark or light bg
+\definecolor{darkcharcoal}{RGB}{25,25,25}		% Darker gray
+\definecolor{charcoal}{RGB}{51,51,51}		% Darker gray
+\definecolor{ash}{RGB}{100,100,100}			% medium gray
+\definecolor{paleblue}{RGB}{0,102,102}		% More of an `ocean' color
+\definecolor{turtlegreen}{RGB}{51,153,0}	% A more neutral green
+\definecolor{paleale}{RGB}{204,204,102}		% Only for dark BG
+\definecolor{lager}{RGB}{140,110,10}		% Use instead of pale ale for white BG
+\definecolor{regal}{RGB}{90,0,120}			% A more neutral purple
+\definecolor{jeans}{RGB}{20,30,150}			% A more neutral blue
+
+
+% Define some "invariant colors" so that a color that doen't work with a
+% light/dark background will automatically be substituted with one that does.
+% These will be rewritten in beamercolorthemeFlipDark
+
+\ifbeamer@Flip@dark
+	\definecolor{FlipSand}{RGB}{204,204,102}		% paleale, Only for dark BG
+	\definecolor{FlipGreen}{RGB}{0,255,0}			% Neon-like green, only for dark BG
+\else
+	\definecolor{FlipSand}{RGB}{140,110,10}			% lager, Only for light BG
+	\definecolor{FlipGreen}{RGB}{51,153,0}			% turtlegreen, works for both
+\fi
+
+%% Future revision: It would be nice to define these in terms of the palette.
+
+
+\ifbeamer@Flip@dark
+	\setbeamercolor{normal text}{fg=white, bg=black}		
+	\setbeamercolor{structure}{fg=gray, bg=black}		% fg controls bullets
+	\setbeamercolor*{footerlike}{fg=white,bg=black}		% Controls the footerline
+\else
+	% Default to black on white
+	\setbeamercolor*{footerlike}{fg=white,bg=charcoal}	% for charcoal footer
+	\setbeamercolor{structure}{fg=charcoal, bg=white}	% fg controls bullets
+	% \setbeamercolor{normal text}{fg=white, bg=black}	
+\fi
+
+
+
+
+
+% These defined colors can be used in \textcolor{colorname}{text in that color}.
+
+% Colors for Keynote background: DARK
+\definecolor{keynotebottom}{rgb}{0.32,0.3,0.38}
+\definecolor{keynotemiddle}{rgb}{0.08,0.08,0.16}
+\definecolor{keynotetop}{rgb}{0,0,0}
+%%% Example of usage:
+%% \setbeamertemplate{background canvas}[vertical shading]
+%% [bottom=keynotebottom, middle=keynotemiddle, top=keynotetop]
+
+
+
+
+% ALERT AND COMMENT COLORS (just copies of the above colors)
+% (I don't know how ot do this more elegantly)
+\ifbeamer@Flip@dark
+	\definecolor{ALERT}{RGB}{153,0,0}			% crimsonred
+	\definecolor{Alert}{RGB}{51,153,0}			% turtlegreen
+	\definecolor{alert}{RGB}{204,204,102}		% paleale		
+	\definecolor{charcoal}{RGB}{51,51,51}		% charcoal
+	\definecolor{comment}{RGB}{80,80,80}		% light charcoal
+	\definecolor{Comment}{RGB}{100,100,100}		% ash
+	\definecolor{COMMENT}{RGB}{80,20,120}		% regal
+\else
+%	\definecolor{ALERT}{RGB}{153,0,0}			% crimsonred
+	\definecolor{ALERT}{RGB}{28,134,238}		% DodgerBlue2
+%	\definecolor{Alert}{RGB}{51,153,0}			% turtlegreen
+	\definecolor{Alert}{RGB}{255,127,36}		% chocolate1
+%	\definecolor{alert}{RGB}{140,110,10}		% lager
+	\definecolor{alert}{RGB}{0,205,102} 		% SpringGreen3
+	\definecolor{charcoal}{RGB}{80,80,80}		% light charcoal
+	\definecolor{comment}{RGB}{51,51,51}		% charcoal
+	\definecolor{Comment}{RGB}{100,100,100}		% ash
+	\definecolor{COMMENT}{RGB}{80,20,120}		% regal
+\fi
+
+
+% \definecolor{crimsonred}{RGB}{153,0,0}		% Neurtal red, good for dark or light bg
+% \definecolor{charcoal}{RGB}{51,51,51}		% Darker gray
+% \definecolor{ash}{RGB}{100,100,100}			% medium gray
+% \definecolor{paleblue}{RGB}{0,102,102}		% More of an `ocean' color
+% \definecolor{turtlegreen}{RGB}{51,153,0}	% A more neutral green
+% \definecolor{paleale}{RGB}{204,204,102}		% Only for dark BG
+% \definecolor{lager}{RGB}{140,110,10}		% Use instead of pale ale for white BG
+% \definecolor{regal}{RGB}{90,0,120}			% A more neutral purple
+% \definecolor{jeans}{RGB}{20,30,150}			% A more neutral blue
+
+
+% A few useful colors
+\setbeamercolor{alerted text}{fg=alert}
+
+%% I would like ot use these colors, but there seems to be a bug in Beamer
+%% where the footnote color infects the normal text color
+%% See:  http://tex.stackexchange.com/questions/8264/beamer-changing-the-footnote-color-also-changes-the-normal-text-color/8268#8268
+
+% \setbeamercolor{footnote}{parent=normal text, fg=FlipSand}
+% \setbeamercolor{footnote mark}{parent=normal text, fg=FlipSand}
+
+% Blocks
+\setbeamercolor{block title}{fg=gray!30!white,bg=charcoal}
+\setbeamercolor{block body}{parent=normal text,bg=gray, fg=white}
+
+\setbeamercolor{block title example}{fg=turtlegreen!30!white,bg=turtlegreen!40!black}
+\setbeamercolor{block body example}{parent=normal text,bg=turtlegreen, fg=white}
+% 
+\setbeamercolor{block title alerted}{fg=crimsonred!30!white,bg=crimsonred!40!black}
+\setbeamercolor{block body alerted}{parent=normal text,bg=crimsonred, fg=white}
+
+
+
+% These "setbeamercolor" commands are more standardized. 
+
+
+%% From the Beamer User Guide:
+% Outer themes base the color of navigational elements and, possibly, also of other elements, on the 
+% four palette colors. The “primary” palette should be used for the most important navigational elements, 
+% which are usually the ones that change most often and hence require the most attention by the audience. 
+% The “secondary” and “tertiary” are less important, the “quaternary” one is least important. 
+% By default, the palette colors do not have a background and the foreground ranges from structure.fg to black. 
+% For the sidebar, there is an extra set of palette colors, see palette sidebar primary.
+
+\setbeamercolor*{palette primary}{fg=crimsonred,bg=charcoal}
+\setbeamercolor*{palette secondary}{fg=paleale,bg=lager}
+\setbeamercolor*{palette tertiary}{fg=turtlegreen,bg=jeans}
+\setbeamercolor*{palette quaternary}{fg=paleblue,bg=regal}
+
+% \setbeamercolor*{palette primary}{fg=crimsonred,bg=white}
+% \setbeamercolor*{palette secondary}{fg=charcoal,bg=white}
+% \setbeamercolor*{palette tertiary}{fg=paleblue,bg=white}
+% \setbeamercolor*{palette quaternary}{fg=turtlegreen,bg=white}
+
+
+
+%% These are some structures that I define for use in the outer theme.
+
+\setbeamercolor*{crimsontown}{fg=paleblue,bg=crimsonred}
+\setbeamercolor*{bluetown}{fg=white,bg=DodgerBlue2}
+%\setbeamercolor*{titlelike}{fg=crimsonred}
+\setbeamercolor*{titlelike}{fg=DodgerBlue2}
+\setbeamercolor*{topbarlike}{fg=white, bg=charcoal}
+
+
+
+
+
+%% Some old definitions that I might want to get back to
+%
+% \setbeamercolor*{frametitle}{bg=white,fg=charcoal}
+% \setbeamercolor*{part title}{bg=white,fg=charcoal}
+% \setbeamercolor*{item}{fg=charcoal, bg=white}
+% 
+% \setbeamercolor*{separation line}{}
+% \setbeamercolor*{fine separation line}{}
+
+
+
+
+
+\mode
+<all>
diff --git a/Lectures_my/NumMet/2016/Lecture12/beamerinnerthemeFlip.sty b/Lectures_my/NumMet/2016/Lecture12/beamerinnerthemeFlip.sty
new file mode 100755
index 0000000..8dc12a0
--- /dev/null
+++ b/Lectures_my/NumMet/2016/Lecture12/beamerinnerthemeFlip.sty
@@ -0,0 +1,95 @@
+% Inner style file for the BEAMER FLIP THEME
+% Copyright 2010 by Flip Tanedo
+% This file may be distributed and/or modified
+% 	1. under the LaTeX Project Public License and/or
+% 	2. under the GNU Public License.
+%
+% Based on: beamerinnerthemeFlip by Marco Barisione
+
+\mode<presentation>
+
+% % Use alternative title page style.
+% \DeclareOptionBeamer{alternativetitlepage}[true]{\def\beamer@Flip@alternativetitlepage{#1}}
+
+% Logo to use in the alternative title page.
+% \def\beamer@Flip@titlepagelogo{}
+% \DeclareOptionBeamer{titlepagelogo}{\def\beamer@Flip@titlepagelogo{#1}}
+
+% Bullet shape.
+\DeclareOptionBeamer{bullet}{\def\beamer@Flip@bullet{#1}}
+\DeclareOptionBeamer{shadow}[true]{\def\beamer@themerounded@shadow{#1}}
+\ExecuteOptionsBeamer{shadow=false}
+
+
+% \ExecuteOptionsBeamer{alternativetitlepage=false,bullet=square}
+\ProcessOptionsBeamer
+
+
+
+% Margins.
+\newlength{\beamer@Flip@normalmargin}
+\setlength{\beamer@Flip@normalmargin}{.06\paperwidth}
+\setbeamersize{text margin left=\beamer@Flip@normalmargin}
+\setbeamersize{text margin right=\beamer@Flip@normalmargin}
+\setlength\leftmargini{.6\beamer@Flip@normalmargin}
+\setlength\leftmarginii{.6\beamer@Flip@normalmargin}
+\setlength\leftmarginiii{.6\beamer@Flip@normalmargin}
+
+% Normal title page.
+\defbeamertemplate*{title page normal}{Flip theme}[1][]
+{
+  \vbox{}
+  \vfill
+  \begin{centering}
+    \begin{beamercolorbox}[wd=\paperwidth,sep=8pt,center,#1]{title page header}
+      \usebeamerfont{title}\inserttitle\par%
+      \ifx\insertsubtitle\@empty%
+      \else%
+        \vskip0.25em%
+        {\usebeamerfont{subtitle}\usebeamercolor[fg]{subtitle}\insertsubtitle\par}%
+      \fi%     
+    \end{beamercolorbox}%
+    %\vskip0.5em\par
+    \begin{beamercolorbox}[sep=8pt,center,#1]{author}
+      \usebeamerfont{author}\insertauthor
+    \end{beamercolorbox}
+    \begin{beamercolorbox}[sep=8pt,center,#1]{institute}
+      \usebeamerfont{institute}\insertinstitute
+    \end{beamercolorbox}
+    \begin{beamercolorbox}[sep=8pt,center,#1]{date}
+      \usebeamerfont{date}\insertdate
+    \end{beamercolorbox}\vskip0.5em
+    {\usebeamercolor[fg]{titlegraphic}\inserttitlegraphic\par}
+  \end{centering}
+  \vfill
+}
+
+
+% Items.
+\defbeamertemplate{itemize item}{squarealt}%
+{\tiny\raise.5ex\hbox{\donotcoloroutermaths$\blacksquare$}}
+\defbeamertemplate{itemize subitem}{squarealt}%
+{\tiny\raise.4ex\hbox{\donotcoloroutermaths$\square$}}
+\defbeamertemplate{itemize subsubitem}{squarealt}%
+{\tiny\raise.3ex\hbox{\donotcoloroutermaths$\blacksquare$}}
+% 
+\defbeamertemplate{itemize item}{circlealt}%
+{\small\raise.2ex\hbox{\donotcoloroutermaths$\bullet$}}
+\defbeamertemplate{itemize subitem}{circlealt}%
+{\small\raise.1ex\hbox{\donotcoloroutermaths$\circ$}}
+\defbeamertemplate{itemize subsubitem}{circlealt}%
+{\scriptsize\raise.1ex\hbox{\donotcoloroutermaths$\bullet$}}
+\setbeamertemplate{items}[circlealt]
+% 
+\def\circletext{circle}
+\ifx\beamer@Flip@bullet\circletext
+  \setbeamertemplate{items}[circlealt]
+\else
+  \setbeamertemplate{items}[squarealt]
+\fi
+
+\setbeamertemplate{blocks}[rounded][shadow=\beamer@themerounded@shadow]
+
+
+\mode<all>
+
diff --git a/Lectures_my/NumMet/2016/Lecture12/beamerouterthemeFlip.sty b/Lectures_my/NumMet/2016/Lecture12/beamerouterthemeFlip.sty
new file mode 100755
index 0000000..d12f6d6
--- /dev/null
+++ b/Lectures_my/NumMet/2016/Lecture12/beamerouterthemeFlip.sty
@@ -0,0 +1,189 @@
+% Outer style file for the BEAMER FLIP THEME 212
+% Copyright 2012 by Flip Tanedo
+% This file may be distributed and/or modified
+% 	1. under the LaTeX Project Public License and/or
+% 	2. under the GNU Public License.
+%
+% Based on: beamerouterthemesplit by Till Tantau
+
+
+\mode<presentation>
+
+% Watermark
+\DeclareOptionBeamer{watermark}{\def\beamer@Flip@watermark{#1}}
+\ExecuteOptionsBeamer{watermark=} % Default value is empty
+
+
+\newif\ifbeamer@Flip@bigpagenumber
+\beamer@Flip@bigpagenumberfalse
+\DeclareOptionBeamer{bigpagenumber}{\beamer@Flip@bigpagenumbertrue}
+
+
+\def\beamer@Flip@truetext{true} % Kludge for true/false so I can use ifx
+
+\DeclareOptionBeamer{topline}[true]{\def\beamer@Flip@topline{#1}}
+\ExecuteOptionsBeamer{topline=false} % Default
+
+
+
+% \ExecuteOptionsBeamer{topline=} % Default value is empty
+
+
+\ProcessOptionsBeamer
+
+
+\usepackage{tikz}	% For fancy decorations
+
+
+% I should fix these up to be more robust against color changes
+\setbeamercolor{section in head/foot}{parent=footerlike}
+\setbeamercolor{subsection in head/foot}{parent=footerlike}
+\setbeamercolor{author in head/foot}{parent=footerlike}
+\setbeamercolor{title in head/foot}{parent=footerlike}
+
+
+% \setbeamercolor{author in head/foot}{parent=section in head/foot}
+% \setbeamercolor{title in head/foot}{parent=subsection in head/foot}
+
+%\setbeamercolor{topbar}{parent=crimsontown}
+\setbeamercolor{topbar}{parent=bluetown}
+
+
+
+
+\usesectionheadtemplate
+  {\hfill\insertsectionhead}
+  {\hfill\color{fg!50!bg}\insertsectionhead}
+
+
+%%%%%%%%%%%%%%%%%%
+% The footer bar %
+%%%%%%%%%%%%%%%%%%
+
+\defbeamertemplate*{footline}{Flip theme}
+{%
+  \leavevmode% 
+% OBSERVATION: the "%" symbols inside hbox are all very important here.
+% The \hbox enviroment will insert spaces whenever there's whitespace
+% Adding a % at the END of each line ensures that any hard returns aren't
+% interpreted as white space. This allows the color boxes to be flush against
+% one another. If a faint white line appears one could probably do something like
+% hspace{-1px} and add an additional pixel-or-two wide beamercolorbox.	
+
+\hbox{%
+\begin{beamercolorbox}[wd=.4\paperwidth,ht=2.5ex,dp=1.125ex,leftskip=.3cm, rightskip=.3cm plus1fil]{author in head/foot}%
+	\usebeamerfont{author in head/foot}\insertshortauthor  \end{beamercolorbox}%
+%
+\begin{beamercolorbox}[wd=.6\paperwidth,ht=2.5ex,dp=1.125ex,leftskip=.3cm,rightskip=.3cm plus1fil]{title in head/foot}%
+    \usebeamerfont{title in head/foot}{\textit \insertshorttitle}%
+    \hskip2ex plus1fill%
+		% \insertpagenumber\,/\,\insertpresentationendpage %% Want FRAME, not SLIDE number
+		\insertframenumber/\inserttotalframenumber
+ \end{beamercolorbox}%
+}%
+
+% This is a complete kludge for the faint white line which appears
+% between the beamer color boxes above. (It doesnt appear in presentation mode
+% but sometimes looks funny when the pdfs are viewed in a window)
+\begin{tikzpicture}[overlay]
+	\usebeamercolor[bg]{author in head/foot}
+	\draw[thick] (0,3.625ex) -- (\paperwidth,3.625ex);
+	\draw (.4\paperwidth,0) -- (.4\paperwidth,3.625ex);
+	\draw[thick] (0,0) -- (\paperwidth,0);
+\end{tikzpicture}
+
+
+
+\ifbeamer@Flip@bigpagenumber
+	\begin{tikzpicture}[overlay]
+	\usebeamercolor[bg]{author in head/foot}
+	\draw[fill] (\paperwidth-5ex,3.625ex) circle (6ex);
+	\usebeamercolor[fg]{author in head/foot}
+	\draw[fill] (\paperwidth-5ex,3.625ex) circle (5ex);
+	\usebeamercolor[bg]{author in head/foot}
+	\node at (\paperwidth-5ex,4ex) {\small$^\text{\insertframenumber}/_\text{\inserttotalframenumber}$};
+	\end{tikzpicture}
+\fi
+}
+
+
+
+\defbeamertemplate*{frametitle}{Flip theme}[1][left]
+{
+% \ifbeamercolorempty[bg]{frametitle}{}{\nointerlineskip}%
+\@tempdima=\textwidth%
+\advance\@tempdima by\beamer@leftmargin%
+\advance\@tempdima by\beamer@rightmargin%
+
+\vbox{}\vskip-3.5ex%
+%
+\begin{beamercolorbox}[sep=0.3cm,#1,wd=\the\@tempdima]{titlelike}
+	\usebeamerfont{frametitle}%
+	\vbox{}\vskip-1ex%
+	\if@tempswa\else\csname beamer@fte#1\endcsname\fi%
+	% \strut{\bf\insertframetitle}\strut\par% 					%% Include the \bf
+	\strut{\insertframetitle}\strut\par%
+	{%
+  		\ifx\insertframesubtitle\@empty%
+  		\else
+			{\usebeamerfont{framesubtitle}
+			 \usebeamercolor[fg]{framesubtitle}
+			\insertframesubtitle\strut\par}%
+  		\fi
+ 	}%
+	%
+  	\vskip-1ex%
+  	\if@tempswa\else\vskip-.3cm\fi% set inside beamercolorbox... evil here...
+\end{beamercolorbox}%
+%%
+
+% \def\beamer@fteright{\vskip0.35cm\advance\leftskip by 1.7cm\advance\rightskip by1.7cm}
+}
+
+
+
+
+% Sidebar right is important: anything placed here will be
+% visible *behind* the main text. This is where to put any
+% interesting watermarks.
+%
+% 
+\defbeamertemplate*{sidebar right}{Flip theme}
+{
+% \llap{\includegraphics[width=50px]{footdecoration}}
+% \llap{\includegraphics[width=\paperwidth,height=\paperheight]{upperleft_watermark_transp}}
+%
+% THIS WOULD BE A NICE WAY TO PUT THE PARTICLE BUBBLE CHAMBER BG
+% ... make a png of the background with TRANSPARENT BG
+% ... then put the image here (use \hfill to get it on the left)
+% Now one can FURTHER apply a slight gradient in the BG
+%
+% 
+% % Top Line 
+\ifx\beamer@Flip@topline\beamer@Flip@truetext
+	\begin{tikzpicture}[overlay]
+	\usebeamercolor[fg]{titlelike}	% alternately, topbarlike would be a good beamercolor
+%	\usebeamercolor[fg]{topbarlike}	% alternately, topbarlike would be a good beamercolor
+	\draw[fill] (-\paperwidth,0) rectangle (0,-3pt);
+	\end{tikzpicture}
+\fi
+%
+% 
+  \vskip.2\beamer@leftmargin%
+   %\llap{\insertlogo\hskip.5\beamer@leftmargin}%
+   \llap{\insertlogo\hskip.08\beamer@leftmargin}%
+    \vfill%
+  \if \beamer@Flip@watermark\@empty
+	% This is kind of a kludge since if I put anything in here, I automatically get errors
+	% Somehow LaTeX wants to read the \else part!
+	\else%
+	\llap{\includegraphics[width=\paperwidth,height=\paperheight]{\beamer@Flip@watermark}}
+  \fi
+}
+
+
+
+\mode<all>
+\endinput
+
+
diff --git a/Lectures_my/NumMet/2016/Lecture12/beamerouterthemedecolines.sty b/Lectures_my/NumMet/2016/Lecture12/beamerouterthemedecolines.sty
new file mode 100755
index 0000000..669e493
--- /dev/null
+++ b/Lectures_my/NumMet/2016/Lecture12/beamerouterthemedecolines.sty
@@ -0,0 +1,168 @@
+% Copyright 2007 by Marco Barisione
+%
+% This file may be distributed and/or modified
+%
+% 1. under the LaTeX Project Public License and/or
+% 2. under the GNU Public License.
+
+\mode<presentation>
+
+% String used between the current page and the total page count.
+\def\beamer@decolines@pageofpages{/}
+\DeclareOptionBeamer{pageofpages}{\def\beamer@decolines@pageofpages{#1}}
+
+% Show a line below the frame title.
+\DeclareOptionBeamer{titleline}[true]{\def\beamer@decolines@titleline{#1}}
+
+% Image used for the watermark.
+\def\beamer@decolines@watermarkorig{}
+\DeclareOptionBeamer{watermark}{\def\beamer@decolines@watermarkorig{#1}}
+
+% Height of the watermark.
+\def\beamer@decolines@watermarkheight{100px}
+\DeclareOptionBeamer{watermarkheight}{\def\beamer@decolines@watermarkheight{#1}}
+
+% The original image height is watermarkheightmult * watermarkheight.
+\def\beamer@decolines@watermarkheightmult{1}
+\DeclareOptionBeamer{watermarkheightmult}{\def\beamer@decolines@watermarkheightmult{#1}}
+
+\ExecuteOptionsBeamer{titleline=false}
+\ProcessOptionsBeamer
+
+% Enable/disable the watermark.
+\def\watermarkon{%
+  \def\beamer@decolines@watermark{\beamer@decolines@watermarkorig}%
+}
+\def\watermarkoff{\def\beamer@decolines@watermark{}}
+
+% Initially enable the watermark.
+\watermarkon
+
+% Colors.
+\setbeamercolor*{lineup}{parent=palette primary}
+\setbeamercolor*{linemid}{parent=palette secondary}
+\setbeamercolor*{linebottom}{parent=palette tertiary}
+\setbeamercolor*{page header}{parent=titlelike}
+
+% Lengths
+\newlength{\headerheight}
+\setlength{\headerheight}{.045\paperheight}
+\newlength{\beamer@decolines@lineup}
+\setlength{\beamer@decolines@lineup}{.025\paperheight}
+\newlength{\beamer@decolines@linemid}
+\setlength{\beamer@decolines@linemid}{.015\paperheight}
+\newlength{\beamer@decolines@linebottom}
+\setlength{\beamer@decolines@linebottom}{.01\paperheight}
+
+% The height of the watermark part below the 3 bottom lines.
+\newlength{\beamer@decolines@watermarkheightbottom}
+\addtolength{\beamer@decolines@watermarkheightbottom}{\beamer@decolines@lineup}
+\addtolength{\beamer@decolines@watermarkheightbottom}{\beamer@decolines@linemid}
+\addtolength{\beamer@decolines@watermarkheightbottom}{\beamer@decolines@linebottom}
+
+% The height of the watermark part over the 3 bottom lines before shrinking.
+\newlength{\beamer@decolines@watermarkheightupperorig}
+\setlength{\beamer@decolines@watermarkheightupperorig}{\beamer@decolines@watermarkheight}
+\addtolength{\beamer@decolines@watermarkheightupperorig}{-\beamer@decolines@watermarkheightbottom}
+\multiply\beamer@decolines@watermarkheightupperorig by \beamer@decolines@watermarkheightmult
+
+% % Footer.
+% \defbeamertemplate*{footline}{decolines theme}
+% {
+%   \leavevmode%
+%   % Page number.
+%   \hbox{%
+%   \begin{beamercolorbox}[wd=.2\paperwidth,ht=0ex,dp=0ex,center]{}%
+%     \usebeamerfont{palette primary}\insertframenumber{} \beamer@decolines@pageofpages{} \inserttotalframenumber%
+%   \end{beamercolorbox}%
+%   \begin{beamercolorbox}[wd=.8\paperwidth,ht=0ex,dp=0ex]{}%
+%   \end{beamercolorbox}%
+%   } %
+%   % First line.
+%   \hbox{%
+%   \begin{beamercolorbox}[wd=.2\paperwidth,ht=\beamer@decolines@lineup,dp=0pt]{}%
+%   \end{beamercolorbox}%
+%   \begin{beamercolorbox}[wd=.8\paperwidth,ht=\beamer@decolines@lineup,dp=0pt]{lineup}%
+%   \end{beamercolorbox}%
+%   } %
+%   % Second line.
+%   \hbox{%
+%   \begin{beamercolorbox}[wd=\paperwidth,ht=\beamer@decolines@linemid,dp=0pt]{linemid}%
+%   \end{beamercolorbox}%
+%   } %
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+%   \hbox{%
+%   \begin{beamercolorbox}[wd=.1\paperwidth,ht=\beamer@decolines@linebottom,dp=0pt]{}%
+%   \end{beamercolorbox}%
+%   \begin{beamercolorbox}[wd=.9\paperwidth,ht=\beamer@decolines@linebottom,dp=0pt]{linebottom}%
+%   \end{beamercolorbox}%
+%   }%
+%   % This seems to fix some alignment problems with the watermark. It has to be
+%   % always applied if you do not want to see the footer moving up and down when
+%   % moving from a page with watermark to a page without or vice versa.
+%   \vskip-.5px%
+%   % Watermark.
+%   \if\beamer@decolines@watermark\@empty\else%
+%     \vskip-\beamer@decolines@watermarkheightbottom%
+%     \llap{\includegraphics[height=\beamer@decolines@watermarkheightbottom,clip=true,%
+%       trim=0pt 0pt 0pt \beamer@decolines@watermarkheightupperorig]{\beamer@decolines@watermark}\hskip-\paperwidth}%
+%   \fi%
+% }
+
+\defbeamertemplate*{headline}{decolines theme}
+{
+  \leavevmode%
+  \hbox{%
+  \begin{beamercolorbox}[wd=\paperwidth,ht=\headerheight,dp=0pt]{page header}%
+  \end{beamercolorbox}%
+  } %
+  \vskip0pt%
+}
+
+\defbeamertemplate*{frametitle}{decolines theme}[1][left]
+{
+  \ifbeamercolorempty[bg]{frametitle}{}{\nointerlineskip}%
+  \@tempdima=\textwidth%
+  \advance\@tempdima by\beamer@leftmargin%
+  \advance\@tempdima by\beamer@rightmargin%
+  \vbox{}\vskip-.5\beamer@leftmargin%
+  \begin{beamercolorbox}[sep=\beamer@leftmargin,#1,wd=\the\@tempdima]{}
+    \usebeamerfont{frametitle}\usebeamercolor[bg]{framesubtitle}%
+    \vbox{}\vskip0ex%
+    \if@tempswa\else\csname beamer@fte#1\endcsname\fi%
+    \strut\insertframetitle\strut\par%
+    {%
+      \ifx\insertframesubtitle\@empty%
+      \else%
+      {\usebeamerfont{framesubtitle}\usebeamercolor[bg]{framesubtitle}\insertframesubtitle\strut\par}%
+      \fi
+    }%
+    \vskip-1ex%
+    \if@tempswa\else\vskip-\beamer@leftmargin\fi
+  \end{beamercolorbox}%
+  \def\beamer@decolines@truetext{true}%
+  \ifx\beamer@decolines@titleline\beamer@decolines@truetext%
+    \vskip-.5\beamer@leftmargin%
+    \begin{beamercolorbox}[wd=\textwidth,ht=.1ex,dp=0ex]{linemid}%
+    \end{beamercolorbox}%
+  \fi
+}
+
+% Frame title continuations, default
+\defbeamertemplate*{frametitle continuation}{decolines theme}{(\insertcontinuationcount)}
+
+
+
+\defbeamertemplate*{sidebar right}{decolines theme}
+{
+  \vskip.1\beamer@leftmargin%
+  \llap{\insertlogo\hskip.5\beamer@leftmargin}%
+  \vfill%
+  \if\beamer@decolines@watermark\@empty\else%
+  \llap{\includegraphics[height=\beamer@decolines@watermarkheight]{\beamer@decolines@watermark}}%
+  \vskip-\beamer@decolines@watermarkheightbottom%
+  \fi
+}
+
+\mode
+<all>
diff --git a/Lectures_my/NumMet/2016/Lecture12/beamerthemeFlip.sty b/Lectures_my/NumMet/2016/Lecture12/beamerthemeFlip.sty
new file mode 100755
index 0000000..a615ff5
--- /dev/null
+++ b/Lectures_my/NumMet/2016/Lecture12/beamerthemeFlip.sty
@@ -0,0 +1,59 @@
+% Main style file for the BEAMER FLIP THEME 2012
+% Copyright 2012 by Flip Tanedo
+% This file may be distributed and/or modified
+% 	1. under the LaTeX Project Public License and/or
+% 	2. under the GNU Public License.
+%
+% Comments: This is still a work in progress.
+% A good way to present: http://www.cs.hmc.edu/~oneill/freesoftware/pdftokeynote.html
+
+% \usepackage{beamerthemesplit}
+\mode<presentation>
+
+
+%% These are the options that are fed in through the driver file
+%% Some of them get passed on to the other theme files
+\DeclareOptionBeamer{bullet}{\PassOptionsToPackage{bullet=#1}{beamerinnerthemeFlip}}
+\DeclareOptionBeamer{bigpagenumber}{\PassOptionsToPackage{bigpagenumber}{beamerouterthemeFlip}}
+\DeclareOptionBeamer{topline}[true]{\PassOptionsToPackage{topline=#1}{beamerouterthemeFlip}}
+\DeclareOptionBeamer{shadow}[false]{\PassOptionsToPackage{shadow=#1}{beamerinnerthemeFlip}}
+
+\DeclareOptionBeamer{watermark}{\PassOptionsToPackage{watermark=#1}{beamerouterthemeFlip}}
+
+\ProcessOptionsBeamer
+
+\useinnertheme{Flip}	% Calls beamerinnerthemeFlip.sty
+\useoutertheme{Flip}	% Calls beamerouterthemeFlip.sty
+\usecolortheme{Flip}	% Calls beamercolorthemeFlip.sty
+
+
+
+%%%%%%%%%%%%%%%%%%%%%%%%
+% Background  Gradient %
+%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+%% Usually I would leave this on
+%% However, there seems to be problems with XeLaTeX
+%% http://tex.stackexchange.com/questions/29497/xelatex-preventing-beamer-from-using-different-backgrounds
+\setbeamertemplate{background canvas}[vertical shading][bottom=blue!.1, middle=white, top=white]
+
+%% HOWEVER: it seems to make it impossible to change this later on in the document
+
+
+%% For a dark background: 
+% \setbeamertemplate{background canvas}[vertical shading][bottom=keynotebottom, middle=keynotemiddle, top=keynotetop]
+
+
+%% An alternate way: \beamertemplateshadingbackground{blue!.1}{red!2}
+%% In general: you want the gradient to be subtle!
+
+
+%%%%%%%%%%%%%%%%%%%%%%%
+% Navigation  symbols %
+%%%%%%%%%%%%%%%%%%%%%%%
+
+\setbeamertemplate{navigation symbols}{} 	% Turns off PDF navigation symbols
+
+\mode<all>
+\endinput
\ No newline at end of file
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diff --git a/Lectures_my/NumMet/2016/Lecture12/lecture12.tex b/Lectures_my/NumMet/2016/Lecture12/lecture12.tex
new file mode 100644
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+++ b/Lectures_my/NumMet/2016/Lecture12/lecture12.tex
@@ -0,0 +1,331 @@
+% vim: set sts=4 et :
+\documentclass[11 pt,xcolor={dvipsnames,svgnames,x11names,table}]{beamer}
+
+\usetheme[
+	bullet=circle,		% Other option: square
+	bigpagenumber,		% circled page number on lower right
+	topline=true,			% colored bar at the top of the frame 
+	shadow=false,			% Shading for beamer blocks
+	watermark=BG_lower,	% png file for the watermark
+	]{Flip}
+
+\usepackage{bm}
+\usepackage{tikz,pgfplots,pgfplotstable}
+\usepgfplotslibrary{fillbetween}
+\usepackage{graphicx}
+
+% SOME COMMANDS THAT I FIND HANDY
+% \renewcommand{\tilde}{\widetilde} % dinky tildes look silly, dosn't work with fontspec
+\newcommand{\comment}[1]{\textcolor{comment}{\footnotesize{#1}\normalsize}} % comment mild
+\newcommand{\Comment}[1]{\textcolor{Comment}{\footnotesize{#1}\normalsize}} % comment bold
+\newcommand{\COMMENT}[1]{\textcolor{COMMENT}{\footnotesize{#1}\normalsize}} % comment crazy bold
+\newcommand{\Alert}[1]{\textcolor{Alert}{#1}} % louder alert
+\newcommand{\ALERT}[1]{\textcolor{ALERT}{#1}} % loudest alert
+%% "\alert" is already a beamer pre-defined
+\newcommand*{\Scale}[2][4]{\scalebox{#1}{$#2$}}%
+
+\graphicspath{{images/}}	% Put all images in this directory. Avoids clutter.
+
+% suppress frame numbering for backup slides
+% you always need the appendix for this!
+\newcommand{\backupbegin}{
+	\newcounter{framenumberappendix}
+	\setcounter{framenumberappendix}{\value{framenumber}}
+}
+\newcommand{\backupend}{
+	\addtocounter{framenumberappendix}{-\value{framenumber}}
+	\addtocounter{framenumber}{\value{framenumberappendix}} 
+}
+
+\begin{document}
+\tikzstyle{every picture}+=[remember picture]
+
+{
+\setbeamertemplate{sidebar right}{\llap{\includegraphics[width=\paperwidth,height=\paperheight]{bubble2}}}
+\begin{frame}[c]%{\phantom{title page}} 
+\begin{center}
+\begin{center}
+	\begin{columns}
+		\begin{column}{0.9\textwidth}
+			\flushright%\fontspec{Trebuchet MS}
+			\bfseries \Huge {Function Minimization (I)}
+		\end{column}
+		\begin{column}{0.2\textwidth}
+		  %\includegraphics[width=\textwidth]{SHiP-2}
+		\end{column}
+	\end{columns}
+\end{center}
+	\quad
+	\vspace{3em}
+\begin{columns}
+\begin{column}{0.6\textwidth}
+\flushright \vspace{-1.8em} {%\fontspec{Trebuchet MS}
+    \large Marcin Chrzaszcz, Danny van Dyk\\\vspace{-0.1em}\small \href{mailto:mchrzasz@cern.ch}{mchrzasz@cern.ch}, \href{mailto:danny.van.dyk@gmail.com}{danny.van.dyk@gmail.com}}
+
+\end{column}
+\begin{column}{0.4\textwidth}
+\includegraphics[height=1.3cm]{uzh-transp}
+\end{column}
+\end{columns}
+
+\vspace{1em}
+\vspace{0.5em}
+	\textcolor{normal text.fg!50!Comment}{Numerical Methods, \\ 07. November, 2016}
+\end{center}
+\end{frame}
+}
+
+\begin{frame}{Plan for today}
+    \begin{itemize}
+        \item \alert{What types of function do we usually need to minimize?}\\
+            \begin{itemize}
+                \item least-squares
+                \item likelihood/posterior
+                \item general non-linear problem
+            \end{itemize}
+        \vfill
+        \item \alert{What information can we use?}\\
+            \begin{itemize}
+                \item the target function
+                \item its first derivatives
+                \item its second derivatives
+            \end{itemize}
+    \end{itemize}
+\end{frame}
+
+\begin{frame}{General Minimization Problem}
+    The general problem can be posited as follows: There exists
+    a function
+    \begin{equation*}
+        f(a_1, \dots, a_N)
+    \end{equation*}
+    with real-valued parameters $\lbrace a_n \rbrace$.\\
+    \vfill
+    We intend to find
+    \begin{itemize}
+        \item \alert{one},
+        \item \alert{any}, or
+        \item \alert{all}
+    \end{itemize}
+    of its minima on a support $A$,
+    \begin{equation*}
+        A = \lbrace (a_1, \dots, a_N)\,|\,g(a_1, \dots, a_N) > 0\rbrace\,.
+    \end{equation*}
+\end{frame}
+
+\begin{frame}{Example}
+    Posterior with non-gaussian likelihood. For example, exponential likelihood.
+\end{frame}
+
+\begin{frame}{Limitations and Monte-Carlo Methods}
+    In general we have no further information on $f$, i.e., we do not
+    \emph{analytically} know any of its derivatives.\\
+    \vfill
+    A very popular way to explore $f$ is by using Monte Carlo methods,
+    e.g. plain random walks, Markov chain methods, or genetic algorithms.
+    Most of these methds are very good at delineating \emph{local environments}
+    around some/most/all(?) of the modes. As always, analytic knowledge
+    of the problem will help. For example, symmetry relations among the parameters
+    or (a)periodic boundary conditions should be exploited if possible.
+
+    The specific Monte-Carlo methods are beyond the scope of these lectures.
+\end{frame}
+
+\begin{frame}{Simplex Method \hfill (aka Nelder-Mead Method)}
+    We will now discuss the work-horse of minimization algorithms,
+    the \alert{Simplex} method. While it has very good convergence
+    properties, but a rather slow convergence rate.\\
+    \vfill
+    The basic idea is as follows:
+    \begin{itemize}
+        \item[0] A set of parameter points must be provided as an initial
+        value. If all else fails, random points can be used.
+        \item[1] Order the points by their respetive size of $f$
+        \item[2] Compute the midpoint $\vec{a}_\text{mid}$ of all points except the worst, 
+        and reflect the worst point $\vec{a}_{N+1}$ at the midpoint yielding $\vec{a}_\text{ref}$.
+        \begin{itemize}
+            \item[a] If $f(\vec{a}_\text{ref})$ fulfills some minimization criteria, replace
+            one of the existing simplex points with $\vec{a}_\text{ref}$.
+            Continue at step 1.
+            \item[b] Otherwise compute $\vec{a}_\text{contr}$ as a linear combination of the worst
+            point of the simplex $\vec{a}_N$, and $\vec{a}_\text{ref}$. If $\vec{a}_\text{contr}$
+            is better than the worst point, replace the worst point.
+            Continue at step 1.
+            \item[c] Otherwise compress the the simplex by moving the points
+            $\vec{x}_1$ to $\vec{x}_N$ closer to $\vec{x}_0$ on their
+            respective connecting lines. Continue at step 1.
+        \end{itemize}
+        \item[3] If at any point the volume of the simplex falls below a given
+        treshold, then stop.
+    \end{itemize}
+\end{frame}
+
+\begin{frame}{Simplex Method (continued)}
+    More details on the previous steps:\\
+    \vspace{-\medskipamount}
+    \begin{overlayarea}{\textwidth}{.95\textwidth}
+    \begin{itemize}
+        \only<1-2>{
+        \item[2] Compute the midpoint as
+        \begin{equation*}
+            \vec{a}_\text{mid} = \frac{1}{N} \sum_{n=0}^{N - 1} \vec{a}_n
+        \end{equation*}
+        The reflection is computed as\hfill~[default: $\alpha = 1$]
+        \begin{equation*}
+            \vec{a}_\text{ref} = (1 + \alpha) \vec{a}_\text{mid}  - \alpha \vec{a}_N
+        \end{equation*}
+        }
+        \only<2>{
+        \item[2a.1] If $\vec{a}_\text{ref}$ is better than $\vec{a}_0$, then replace
+        $\vec{a}_N$ with $\vec{a}_\text{ref}$. Continue with step 1.
+        \item[2a.1] \alert{alternative}: Compute\hfill~\alert{[default: $\epsilon = 2$]}
+        \begin{equation*}
+            \vec{a}_\text{exp} = (1 + \epsilon) \vec{a}_\text{mid} - \epsilon \vec{a}_N
+        \end{equation*}
+        and use in step \alert{2a} the better of the two ($\vec{a}_\text{ref}$ or $\vec{a}_\text{exp}$).
+        Continue with step 1.
+        }
+        \only<3-6>{
+        \item[2a.2] If $\vec{a}_\text{ref}$ is better than the second worst point
+        $\vec{a}_{N-1}$, then replace the worst point by $\vec{a}_\text{ref}$.
+        }
+        \only<4-6>{
+        \item[2b] Let $\vec{a}_\text{tmp}$ be the better of $\vec{a}_\text{ref}$ and $\vec{a}_\text{N}$.
+        Compute\hfill~\alert{[default: $\gamma = 1/2$]}
+        \begin{equation*}
+            \vec{a}_\text{contr} = \gamma \vec{a}_\text{mid} + (1 - \gamma) \vec{a}_\text{tmp}\,.
+        \end{equation*}
+        If $\vec{a}_\text{contr}$ is better than the worst point, replace the latter. Continue
+        at step 1.
+        }
+        \only<5-6>{
+        \item[2c] Compress the entire simplex\hfill~\alert{[default: $\kappa = 1/2$]}
+        \begin{equation*}
+            \vec{a}_n = \sigma \vec{a}_0 + (1 - \sigma) \vec{a}_n\qquad \forall n=1,\dots,N
+        \end{equation*}
+        }
+        \only<6>{
+        \item[0] The $N+1$ initial points must span a simplex. Therefore, care must be
+        taken that they do not lie in a linear subspace of the simplex. This is similar
+        to picking points in a plane when constructing a 3D volume.
+        }
+    \end{itemize}
+    \end{overlayarea}
+\end{frame}
+
+\begin{frame}{Simplex Method: Properties}
+    \begin{itemize}
+        \item The method is very robust against problems such as overshooting.
+        \item It will usually converge toward a close-by minimum.
+        \item However, the convergence rate is smaller than for many other methods,
+        which e.g. work on more specialised problems.
+        \item No guarantee is given that the simplex method converges toward
+        the global minimum. Possible relief comes in the form of several starting
+        simplex obtained using Monte-Carlo methods.
+    \end{itemize}
+\end{frame}
+
+\begin{frame}{Example}
+\end{frame}
+
+\begin{frame}{Least-squares Problems}
+    The least-squares problem arises from the case of a Gaussian
+    likelihood function if all uncertainties are equal. It is a
+    very good example for understanding a non-linear problem
+    through linearization.\\
+    \vfill
+    The target function is called the residue $r(a_1, \dots, a_N)$
+    with $N$ parameters.
+    \begin{equation*}
+        r_k \equiv f(\vec{a}, \vec{x}_k) - y_k
+    \end{equation*}
+    \vfill
+    Here $k$ denotes one of the $K$ possible coordinate on a curve,
+    with (external, e.g. experimental) inputs $(\vec{x}_k, y_k)$.\\
+    \vfill
+    The problem now aims to minimize
+    \begin{equation*}
+        \sum_k^K |r_k|^2
+    \end{equation*}
+\end{frame}
+
+\begin{frame}{Gauss-Newton Method}
+    This iterative method requires partial derivatives of the resiudes
+    $r_k$ with respect to the parameters $a_n$:
+    \begin{equation*}
+        r'_{k,n} \equiv \frac{\partial r(\vec{a}, \vec{x})}{\partial a_n}\Big|_{\vec{x} = \vec{x}_k}
+    \end{equation*}
+    \vfill
+    The algorithm now involves the following quantities:
+    \begin{align*}
+        D   & = \left(\begin{matrix}
+            r'_{1,1} & r'_{1,2} & \dots  & r'_{1,n}\\
+            r'_{2,1} & r'_{2,2} & \dots  & r'_{2,n}\\
+            \vdots   & \vdots   & \ddots & \vdots  \\
+            r'_{k,1} & r'_{k,2} & \dots  & r'_{k,n}
+        \end{matrix}\right) &
+        \vec{r}
+            & = (r_1, \dots, r_k)^T
+    \end{align*}
+\end{frame}
+
+\begin{frame}{Gauss-Newton Method (cont'd)}
+    \begin{itemize}
+        \item[0] As always, we will be requiring a starting point $\vec{a}_0$ in parameter space.\\
+        \item[1] Update the current point:
+        \begin{equation*}
+            \vec{a}_{i + 1} = \vec{a}_i - (D^T\cdot D)^{-1} \cdot D \cdot \vec{r}
+        \end{equation*}
+        \item[2] If $||\vec{a}_{i + 1} - \vec{a}_i|| < T$, where $T$ is some a-prior threshold,
+        we stop. Otherwise, continue with step 1.
+    \end{itemize}
+    Some comments
+    \begin{itemize}
+        \item The literature usually recommends to compute the auxiliary variable $\vec{s}$ via:
+        \begin{equation*}
+            (D^T \cdot D) \cdot \vec{s} = D^T \cdot \vec{r}
+        \end{equation*}
+        The above linear system of equations can be solved with known methods.
+        \item Since $(D^T \cdot D)$ is symmetric, it is a good idea to use Cholesky decomposition.
+    \end{itemize}
+\end{frame}
+
+\begin{frame}{Derivation}
+    \url{https://en.wikipedia.org/wiki/Gauss-Newton_algorithm}
+\end{frame}
+
+\begin{frame}{Example}
+    \url{https://en.wikipedia.org/wiki/Gauss-Newton_algorithm}
+\end{frame}
+
+\begin{frame}{Levenberg-Marquardt Method}
+    The Levenberg-Marquardt method arises from a modification to the Gauss-Newton method.
+
+    Trust region.
+
+    Dampen:
+    \begin{itemize}
+        \item[0] As always, we will be requiring a starting point $\vec{a}_0$ in parameter space.\\
+        \item[1] Update the current point:
+        \begin{equation*}
+            \vec{a}_{i + 1} = \vec{a}_i - (D^T\cdot D {\color{red}\,+ \lambda I})^{-1} \cdot D \cdot \vec{r}
+        \end{equation*}
+        where $I$ is the unit matrix in $N \times N$ {\color{red} and $\lambda$ a real-valued parameter}.
+        \item[2] If $||\vec{a}_{i + 1} - \vec{a}_i|| < T$, where $T$ is some a-prior threshold,
+        we stop. Otherwise, continue with step 1.
+    \end{itemize}
+
+    The optimal choice of {\color{red}$\lambda$} is specific to the problem.
+\end{frame}
+
+\begin{frame}
+\end{frame}
+
+\backupbegin
+
+\begin{frame}\frametitle{Backup}
+
+\end{frame}
+
+\backupend
+\end{document}