Index: sm/workshop/2012/Talks/08_Inverse_methods/08_Inverse_methods.tex =================================================================== --- /issm/workshop/2012/Talks/08_Inverse_methods/08_Inverse_methods.tex (revision 14035) +++ (revision ) @@ -1,274 +1,0 @@ -%Preamble {{{ -\documentclass[9pt,notes=hide]{beamer} % Document class: beamer %8 pts is also a good option -\usetheme[jpl,headstyle=institute,fnolabel=,sidebar=.15\paperwidth,footstyle=low,footsep={ $\cdot$ },TPomitframeno]{Pasadena} -\useinnertheme{MLHacks} % ISSM bonuses (ghostframes for example) -\usepackage[T1]{fontenc} % T1 encoding is better for foreign langages -\usepackage{aeguill,aecompl} % Virtual fonts -\usepackage{amsmath,amssymb,amsfonts}% Mathematical symbols (useful for equations) -\usepackage{bm} % maths letters, real = IR -\usepackage{xcolor} % We can define new colors -\usepackage{graphicx} % Include pictures -\usepackage{times} % Use Times Font instead of default -\usepackage{appendixnumberbeamer} -\usepackage[framed,numbered,autolinebreaks,useliterate]{mcode} -\usepackage{booktabs} %nice tables (toprule, bottomrule,...) -\graphicspath{ % directories where the pictures are located -{Graphics/} -} -\hypersetup{pdfstartview={Fit}} % Acrobat reader: open fit to screen - -\setbeamercovered{transparent} %invisible, transparent, dynamic, highly dynamic -%}}} -%Newcommands {{{ -\newcommand*{\MAT}[1]{\ensuremath{\mathbf{#1}} } %Matrice name (Bold) -\newcommand*{\VEC}[1]{\ensuremath{\bm{#1}} } %Vector name (Bold Italic) -\newcommand*{\SET}[1]{\ensuremath{\mathcal{#1} }} %Set, Space (italic) -\newcommand*{\R}[0]{{\mathbb R}} %real set IR -\newcommand*{\N}[0]{{\mathbb N}} %natural set IN -\newcommand*{\Trace}[1]{\text{Tr}\left( #1\right)} %trace (Tr(A)) -\newcommand*{\abs}[1]{\left| #1\right|} %absolute value |x| -\newcommand*{\norm}[1]{\lVert#1\rVert} %norm ||x|| -\newcommand*{\grad}[1]{\nabla #1} -\newcommand*{\divergence}[1]{\nabla \cdot #1} -\newcommand*{\Divergence}[1]{\nabla \cdot #1} -\newcommand*{\eps}[0]{\dot{\bm \varepsilon}} %strain rate tensor -\newcommand*{\epseff}[0]{\ensuremath{\dot{\varepsilon}_{e} }} %effective strain rate -\newcommand*{\epscomp}[0]{\dot{\varepsilon}} %strain rate tensor component -\newcommand*{\stress}[0]{{\bm \sigma}} %stress tensor -\newcommand*{\devstress}[0]{\stress'} %deviatoric stress tensor -\newcommand*{\stresscomp}[0]{\ensuremath{\sigma}} %stress tensor component -\newcommand*{\vel}[0]{\VEC{v}} %velocity vector -\newcommand*{\velbar}[0]{\VEC{\bar{v} }} %velocity vector BAR -\newcommand*{\velobs}[0]{\ensuremath{\VEC{v}^{\text{obs}} }}%velocity obs -\newcommand*{\ubar}[0]{\ensuremath{\overline{v}_x}} %velocity vector BAR -\newcommand*{\vbar}[0]{\ensuremath{\overline{v}_y}} %velocity vector BAR -\newcommand*{\vx}[0]{\ensuremath{v_x}} %vx -\newcommand*{\vy}[0]{\ensuremath{v_y}} %vy -\newcommand*{\vz}[0]{\ensuremath{v_z}} %vz -\newcommand*{\vxobs}[0]{\ensuremath{v_x^{\text{obs}} }}%vx obs -\newcommand*{\vyobs}[0]{\ensuremath{v_y^{\text{obs}} }}%vy -\newcommand*{\pressure}[0]{p} %pressure -\newcommand*{\nx}[0]{\ensuremath{n_x}} %nx -\newcommand*{\ny}[0]{\ensuremath{n_y}} %ny -\newcommand*{\nz}[0]{\ensuremath{n_z}} %nz -\newcommand*{\Lagrangian}[0]{\ensuremath{\mathcal{L} }} -\newcommand*{\Cost}[0]{\ensuremath{\mathcal{J} }} -\newcommand*{\Ms}[0]{\ensuremath{\dot{M}_s}} -\newcommand*{\Mb}[0]{\ensuremath{\dot{M}_b}} -\newcommand*{\inner}[2]{\ensuremath{\langle #1 , #2 \rangle}} %inner product <.,.> -\newcommand*{\BigO}[1]{\mathcal{O}\left( #1\right) } %Big O landau notation - -%%Automatically insert outline slide at the begining of each new section (very nice!) -%\AtBeginSection[]{ -%\begin{frame} -% \ghostframe %do not count slide -% \frametitle{Outline} % title of the slide -% \tableofcontents[currentsection] -%\end{frame} -%} -%}}} -%Presentation properties{{{ -%between brackets []: shown on headers and footers of every slide (must be short) -%between parenthesis (): shown on title page only (long) -\title[Inverse Methods]{Ice Sheet System model} -\subtitle{Inverse Methods} -\author[M. Morlighem]{ -Eric \textsc{Larour}\inst{1}, -Eric \textsc{Rignot}\inst{1,3}, -\textbf{Mathieu \textsc{Morlighem}}\inst{1,2}, -H\'el\`ene \textsc{Seroussi}\inst{1,2} -Chris \textsc{Borstad}\inst{1}, -Feras \textsc{Habbal}\inst{1,3}, -Daria \textsc{Halkides}\inst{1,4}, -Behnaz \textsc{Khakbaz}\inst{1}, -John \textsc{Schiermeier}\inst{1}, -Nicole \textsc{Schlegel}\inst{1} -\vspace{1em} -} -\institute[Jet Propulsion Laboratory]{ -\inst{1}Jet Propulsion Laboratory - California Institute of Technology\\ -\inst{2}Laboratoire MSSMat, \'Ecole Centrale Paris, France\\ -\inst{3}University of California, Irvine\\ -\inst{4}Joint Institute for Regional Earth System Science \& Engineering, UCLA -} -\conference[ISSM Workshop 2012]{ISSM Workshop 2012} -\date[]{\hspace{-15em}December 2012\hspace{3em}\copyright Copyright 2012. All rights reserved} -\logo{\includegraphics[width=8em]{ISSMlogo}} -%}}} -\begin{document} - -%Title slide %{{{ -\begin{frame}[plain] %No headers or footers - %Do not change, this page is automatically generated - \ghostframe - \titlepage -\end{frame} %}}} -\begin{frame}{Outline}%{{{ - \tableofcontents -\end{frame}%}}} -\section{Introduction} -\subsection{Model equations} -\begin{frame} \frametitle{Ice flow model equations}%{{{ - \vspace{-10pt} - \uncover<+->{\begin{block}{Field equations} - \begin{enumerate} - \item Stokes flow: - \begin{equation} - \nabla \cdot \stress + \rho \VEC{g} = \VEC{0} - \end{equation} - \item Incompressibility: - \begin{equation} - \divergence{\vel}=0 - \end{equation} - \item Constitutive law: - \begin{equation} - \devstress - = \stress + \pressure\;\MAT{I} - = 2\mu\eps - \hspace{3em} - \mu = \dfrac{\only<3>{\color{red}}B}{2 \; \epseff^{1-1/n}} - \end{equation} - \end{enumerate} - \end{block}} - \uncover<+->{\begin{block}{Boundary conditions} - \renewcommand\arraystretch{1.7} - \begin{tabular}[b]{lll} - Ice/Air interface: Free surface & - $\stress \cdot\VEC{n}=P_{atm}\;\VEC{n}\simeq\VEC{0}$ & - on $\Gamma_s$ \\ - Ice/Ocean interface: water pressure & - $\stress \cdot\VEC{n}=P_w\;\VEC{n}$ & - on $\Gamma_w$ \\ - Ice/Bedrock interface (1): lateral friction & - $ \left(\stress \cdot\VEC{n} + {\only<3->{\color{red}}\alpha}^2\vel\right)_{\parallel} = \VEC{0}$ & - on $\Gamma_b$ \\ - Ice/Bedrock interface (2): impenetrability & - $\vel\cdot\VEC{n}=-\Mb\,\nz$ & - on $\Gamma_b$ \\ - + Dirichlet condition - \end{tabular} - \end{block}} - \note{Ice follows a Stokes flow: we neglect the acceleration. We also neglect the Coriolis - effects. - Ice is also an incompressible fluid so the divergence of the velocity is zero. - Ice being an incompressible fluid, it's consitutive law inly involves the deviatoric stress, and - ice is modeled as a viscous fluid so the deviatoric stress is proportional to the deviatoric - stress. The viscosity follows Glen's law, which is non linear because it depends on the effective - strain rate. - For the boundary conditions, the upper surface is considered stress free, air pressure being - negligible, water pressure is applied at the ice/ocean interface. Under grounded ice, it is a - combination of Dirichlet and Neuman conditions: there is lateral friction along the plane - tangential to the interface, and an impenetrability condition along the normal. (CLICK) - Now some parameters of these equations are not well understood and must be inferred from - observation. It is the case of the friction coefficient $\alpha$ in red here. To infer the - friction under the ice, we use an inversion method.} -\end{frame} %}}} -\subsection{Inverse problem} -\begin{frame} \frametitle{Inverse problems}%{{{ - \begin{itemize} - \item Basal friction and ice hardness are difficult to measure - \item Use extra datasets to infer unknows - \item[$\rightarrow$] ex: surface velocities derived from InSAR - \end{itemize} - \uncover<2->{\begin{block}{PDE-constrained optimization} - Minimize cost function - \begin{equation} - {\mathcal J}\left(\vel,\alpha\right) - = \frac{1}{2} \int_{\Gamma_s} \left(\vx-\vxobs\right)^2 + \left(\vy-\vyobs\right)^2 dS - + {\mathcal R}\left(\alpha\right) - \end{equation} - Subject to: - \renewcommand\arraystretch{1.5} - \begin{equation} - \begin{array}{rcll} - \nabla \cdot \mu\left(\grad{\vel}+\grad{\vel}^T\right)-\grad{\pressure}+ \rho \VEC{g} & = & \VEC{0} & \text{ in $\Omega$}\\ - \divergence{\vel} & = & 0 & \text{ in $\Omega$}\\ - \stress\cdot\VEC{n} & = & \VEC{f} & \text{ on $\Gamma_s\cup\Gamma_w$}\\ - \left(\stress \cdot\VEC{n} + \alpha^2\vel\right)_{\parallel} & = &\VEC{0} & \text{ on $\Gamma_b$} \\ - \vel\cdot\VEC{n} & = & -\Mb\,\nz & \text{ on $\Gamma_b$} - \end{array} - \end{equation} - \end{block}} - \note{We rely on surface velocities to infer those parameter using inverse methods. The - optimization problem is defined as follows:\\ - We seek for a minimum of a cost function J that is defined as the difference between the modeled - and the observed surface velocity, plus a regularizing term usualy Tikhonov, under the constraint - that the modeled velocity must obey the stress-balance equations.} -\end{frame} %}}} -\subsection{Algorithm} -\begin{frame} \frametitle{Algorithm of resolution}%{{{ - \begin{center} - \includegraphics[width=\textwidth]{algo_cm0.pdf} - \end{center} -\end{frame} %}}} - -\section{Hands on 1 (ice rigidity)} -\begin{frame} \frametitle{Hands on 1 (ice rigidity)}%{{{ - \lstinputlisting[firstnumber=3,firstline=3,lastline=10,basicstyle=\lstbasicfont\scriptsize]{Code/runme.m} - \vspace{-1.5em} - \lstinputlisting[firstnumber=25,firstline=25,lastline=29,basicstyle=\lstbasicfont\scriptsize]{Code/Square.par} - \vspace{-1em} - \begin{center} - \includegraphics[width=.4\textwidth]{TrueB} - \hspace{1em} - \includegraphics[width=.4\textwidth]{VelObs} - \end{center} -\end{frame} %}}} -\subsection{Initial state} -\begin{frame} \frametitle{Start from constant hardness}%{{{ - \lstinputlisting[firstnumber=16,firstline=16,lastline=25]{Code/runme.m} - \begin{center} - \includegraphics[width=.4\textwidth]{BInit} - \hspace{1em} - \includegraphics[width=.4\textwidth]{VelInit} - \end{center} -\end{frame} %}}} -\subsection{Inversion} -\begin{frame} \frametitle{Inverse method}%{{{ - \lstinputlisting[firstnumber=31,firstline=31,lastline=48,tabsize=1,basicstyle=\lstbasicfont\scriptsize]{Code/runme.m} -\end{frame} %}}} -\begin{frame} \frametitle{Inverse method}%{{{ - \begin{center} - \includegraphics[width=.45\textwidth]{BFinal} - \includegraphics[width=.45\textwidth]{VelFinal} - \end{center} -\end{frame} %}}} - -\section{Hands on 2 (friction)} -\begin{frame} \frametitle{Hands on 2 (friction)}%{{{ - \begin{itemize}\itemsep=+1em - \item Same example but now for a grounded glacier: - \item Changes step 1: - \begin{enumerate}\itemsep=+.5em - \item increase bed and surface elevation by 100 m - \item mask is now all grounded - \item $B=1.8\times10^8$ uniform - \item friction coefficient: 50, and 10 for $600000= becomes $\geq$, delta -%% becomes $\delta$ and so on. -%% -%% - framed if you want a frame around the source code blocks -%% -%% - final if you have ``gloablly'' set the draft option, the -%% listings package will not output the code at all. to -%% force it to do so anyway, load this package with the -%% final option (passes the ``final'' on to listings). -%% -%% For example, you may use \usepackage[numbered,framed]{mcode} -%% in your document preamble. -%% -%% Note: Inside code blocks you can escape to LaTeX text mode -%% using §...§. For ex. §text and some math: $x^2$§, which is -%% especially useful in comments for putting nicely typeset -%% equations etc. To get the same colour/style as in the rest -%% of the comment use \mcommentfont, i.e. §\mcommentfont $x^2$§ -%% -%% To change the font used, edit the first line in the "custo- -%% mise below" section. And feel free to edit other things as -%% well. Refer to the documentation of the listings package to -%% see what else you could do. If an extra small font is re- -%% quired, use {\fontfamily{pcr}\fontsize{3}{4.6}\selectfont} -%% in the definition of \lstbasicfont. -%% -%% Author: -%% Florian Knorn | florian@knorn.org | www.florian-knorn.com -%% -%% Version history: -%% 2.2 -- Bugfix (thanks Willi Gerbig!) -%% 2.1 -- Finally automatic detection between end and end -%% 2.0 -- New options for line breaking and literate prog. -%% 1.8 -- Fixed typo in documentation regarding §...§ -%% 1.7 -- Added MATLAB block comment syntax %{ ...... %} -%% 1.6 -- Added some infos, dealing with keyword ``end'' -%% 1.5 -- Tweaked check to see wether textcomp is loaded -%% 1.4 -- Fixed misconfig (mathescape now set to false) -%% 1.3 -- Purely cosmetic (tabs replaced by spaces) -%% 1.2 -- Added \lstset{showstringspaces=false} -%% 1.1 -- Added \mcode command and [final] option -%% 1.0 -- Release - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% D O N ' T T O U C H T H I S % -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\def\fileversion{2.2} -\def\filedate{2011/08/26} - -\typeout{-- Package: `mcode' \fileversion\space <\filedate> --} -\NeedsTeXFormat{LaTeX2e} -\ProvidesPackage{mcode}[\filedate\space\fileversion] - -% for bw-option -\newif\ifbw -\DeclareOption{bw}{\bwtrue} - -% numbered option -\newif\ifnumbered -\DeclareOption{numbered}{\numberedtrue} - -% final option -\newif\iffinal -\DeclareOption{final}{\finaltrue} - -% autolinebreaks option -\newif\ifautolinebreaks -\DeclareOption{autolinebreaks}{\autolinebreakstrue} - -% literate programming (replace certain characters/relations -\newif\ifuseliterate -\DeclareOption{useliterate}{\useliteratetrue} - -% framed option -\newif\ifframed -\DeclareOption{framed}{\framedtrue} - -\DeclareOption*{% default - \PackageWarning{mcode}{Unknown option `\CurrentOption' !}% -} -\ProcessOptions - -\ifbw\typeout{ - settings optimized for printing (bw formating)} -\else\typeout{ - settings optimized for display (colour formating)}\fi -\ifnumbered\typeout{ - line numbering enabled}\else\fi -\ifuseliterate\typeout{ - literate programming (character replacements) enabled}\else\fi -\ifautolinebreaks\typeout{ - automatic line breaking enabled (careful, buggy!)}\else\fi -\ifframed\typeout{ - framed listings}\else\fi - -% This command allows you to typeset syntax highlighted Matlab -% code ``inline''. 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