From 80fff7722edfa6279459ee486c505775c130a17f Mon Sep 17 00:00:00 2001 From: Dave Marchant Date: Wed, 12 Feb 2014 14:07:07 -0800 Subject: [PATCH] Added some notes. --- notes/tem/tem.tex | 250 ++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 250 insertions(+) create mode 100644 notes/tem/tem.tex diff --git a/notes/tem/tem.tex b/notes/tem/tem.tex new file mode 100644 index 00000000..4d7058c8 --- /dev/null +++ b/notes/tem/tem.tex @@ -0,0 +1,250 @@ +\documentclass[]{article} + +\renewcommand{\div}{\nabla\cdot\,} +\newcommand{\grad}{\ensuremath {\vec \nabla}} +\newcommand{\curl}{\ensuremath{{\vec \nabla}\times\,}} +\newcommand {\J} { {\vec J} } +\renewcommand {\H} { {\vec H} } +\newcommand {\E} { {\vec E} } +\newcommand{\dcurl}{\ensuremath{{\mathbf C}}} +\newcommand{\dgrad}{\ensuremath{{\mathbf G}}} +\newcommand{\Acf}{\ensuremath{{\mathbf A_c^f}}} +\newcommand{\Ace}{\ensuremath{{\mathbf A_c^e}}} +\renewcommand{\S}{\ensuremath{{\mathbf \Sigma}}} +\newcommand{\St}{\ensuremath{{\mathbf \Sigma_\tau}}} +\newcommand{\T}{\ensuremath{{\mathbf T}}} +\newcommand{\Tt}{\ensuremath{{\mathbf T_\tau}}} +\newcommand{\diag}[1]{\, {\sf diag}\left( #1 \right)} + +%Common mass matricies +\newcommand{\M}{\ensuremath{{\mathbf M}}} +\newcommand{\MfMui}{\ensuremath{{\M^f_{\mu^{-1}}}}} +\newcommand{\MeSig}{\ensuremath{{\M^e_\sigma}}} +\newcommand{\MeSigInf}{\ensuremath{{\M^e_{\sigma_\infty}}}} +\newcommand{\MeSigO}{\ensuremath{{\M^e_{\sigma_0}}}} +\newcommand{\Me}{\ensuremath{{\M^e}}} +\newcommand{\Mes}[1]{\ensuremath{{\M^e_{#1}}}} +\newcommand{\Mee}{\ensuremath{{\M^e_e}}} +\newcommand{\Mej}{\ensuremath{{\M^e_j}}} + +\newcommand{\BigO}[1]{\ensuremath{\mathcal{O}\bigl(#1\bigr)}} + +% ********** TDIP paper + +\newcommand{\bE}{\mathbf{E}} +\newcommand{\bH}{\mathbf{H}} +\newcommand{\B}{\vec{B}} +\newcommand{\D}{\vec{D}} +\renewcommand{\H}{\vec{H}} +\newcommand{\s}{\vec{s}} +\newcommand{\bfJ}{\bf{J}} +\newcommand{\vecm}{\vec m} +\renewcommand{\Re}{\mathsf{Re}} +\renewcommand{\Im}{\mathsf{Im}} + +\renewcommand {\j} { {\vec j} } +\newcommand {\h} { {\vec h} } +\renewcommand {\b} { {\vec b} } +\newcommand {\e} { {\vec e} } +\renewcommand {\d} { {\vec d} } +\renewcommand {\u} { {\vec u} } + +\newcommand{\I}{\vec{I}} + + +\usepackage{pslatex,palatino,avant,graphicx,color,amsmath} +% \usepackage[margin=2cm]{geometry} + +\begin{document} +\title{TEM} + +\section{Sensitivity Calculation} + +\begin{subequations} + \begin{align} + \dcurl \e^{(t+1)} + \frac{\b^{(t+1)} - \b^{(t)}}{\delta t} = 0 \\ + \dcurl^\top \MfMui \b^{(t+1)} - \MeSig \e^{(t+1)} = \Me \j_s^{(t+1)} + \end{align} +\end{subequations} + +Using Gauss-Newton to solve the inverse problem requires the ability to calculate the product of the Jacobian and a vector, as well as the transpose of the Jacobian times a vector. The above system can be rewritten as + +\begin{align} + \mathbf{A} \u^{(t+1)} + \mathbf{B} \u^{(t)}= \s^{(t+1)} +\end{align} +where +\begin{subequations} + \begin{align} + \mathbf{A} = + \left[ + \begin{array}{cc} + \frac{1}{\delta t} \mathbf{I} & \dcurl \\ + \dcurl^\top & -\MeSig + \end{array} + \right] \\ + \mathbf{B} = + \left[ + \begin{array}{cc} + -\frac{1}{\delta t} \mathbf{I} & 0 \\ + 0 & 0 + \end{array} + \right] \\ + \u^{(k)} = \left[ + \begin{array}{c} + \b^{(k)}\\ + \e^{(k)} + \end{array} + \right] \\ + \s^{(k)} = \left[ + \begin{array}{c} + 0\\ + \Me \j^{(k)}_s + \end{array} + \right] + \end{align} +\end{subequations} + +The entire time dependent system can be written in a single matrix expression +\begin{align} + \hat{\mathbf{A}} \hat{u} = \hat{s} +\end{align} +where +\begin{subequations} + \begin{align} + \mathbf{\hat{A}} = \left[ + \begin{array}{cccc} + A & 0 & & \\ + B & A & & \\ + & \ddots & \ddots & \\ + & & B & A + \end{array} + \right] \\ + \hat{u} = \left[ + \begin{array}{c} + \u^{(1)} \\ + \u^{(2)} \\ + \vdots \\ + \u^{(N)} + \end{array} \right]\\ + \hat{s} = \left[ + \begin{array}{c} + \s^{(1)} - \mathbf{B} \u^{(0)} \\ + \s^{(2)} \\ + \vdots \\ + \s^{(N)} + \end{array} + \right] + \end{align} +\end{subequations} + +For the fields $\u$, the measured data is given by +\begin{align} + \vec{d} = \mathbf{Q} \u +\end{align} +The sensitivity matrix $\mathbf{J}$ is then defined as +\begin{align} + \mathbf{J} = \mathbf{Q} \frac{\partial \u}{\partial \sigma} +\end{align} + + +Defining the function $\vec{c}(m,\vec{u})$ to be +\begin{align} + \vec{c}(m,\u) = \hat{\mathbf{A}} \vec{u} - \vec{q} = \vec{0} +\end{align} +then +\begin{align} + \frac{\partial \vec{c}}{\partial m} \partial m + + \frac{\partial \vec{c}}{\partial \u} \partial \vec{u} = 0 +\end{align} +or +\begin{align} + \frac{\partial \vec{u}}{\partial m} = -\left(\frac{\partial \vec{c}}{\partial \u} \right)^{-1} \frac{\partial \vec{c}}{\partial m} +\end{align} + + +Differentiating, we find that +\begin{align} + \frac{\partial \vec{c}}{\partial \hat{u}} = \hat{\mathbf{A}} +\end{align} +and +\begin{align} + \frac{\partial \vec{c}}{\partial \sigma} = \mathbf{G}_\sigma = + \left[ + \begin{array}{c} + g_\sigma^{(1)}\\ + g_\sigma^{(2)}\\ + \vdots \\ + g_\sigma^{(N)} + \end{array} + \right] +\end{align} +with +\begin{subequations} + \begin{align} + g_\sigma^{(n)} = + \left[ + \begin{array}{c} + \mathbf{0} \\ + - \diag{\e^{(n)}} \Ace \diag{\vec{V}} + \end{array} + \right] + \end{align} +\end{subequations} + +\subsection{Implementing $\mathbf{J}$ times a vector} +Multiplying $\mathbf{J}$ onto a vector can be broken into three steps +\begin{enumerate} +\item Compute $\vec{p} = \mathbf{G}m$ +\item Solve $\hat{\mathbf{A}} \vec{y} = \vec{p}$ +\item Compute $\vec{w} = -\mathbf{Q} \vec{y}$ +\end{enumerate} + +\begin{subequations} + \begin{align} + \vec{p}^{(n)} = \left[ + \begin{array}{c} + 0 \\ + \vec{p}_e^{(n)} + \end{array} + \right] \\ + \vec{p}_e^{(n)} = - \diag{\e^{(n)}} \Ace \diag{V} m + \end{align} +\end{subequations} + +\paragraph{First time step} + +\begin{subequations} + \begin{align} + \frac{1}{\delta t} \vec{y}_{b}^{(1)} + \dcurl \vec{y}_{e}^{(1)} = 0 \\ + \dcurl^\top \MfMui \vec{y}_b^{(1)} - \MeSig \vec{y}_e^{(1)} = \vec{p}_e^{(1)} + \end{align} +\end{subequations} + +\begin{subequations} + \begin{align} + \left( \MfMui \dcurl \MeSig^{-1} \dcurl^\top \MfMui + \frac{1}{\delta t} \MfMui \right) \vec{y}_{b}^{(1)} = \MfMui \dcurl \MeSig^{-1} \vec{p}_e^{(1)} \\ + \vec{y}_e^{(1)} = \MeSig^{-1} \dcurl^\top \MfMui \vec{y}_b^{(1)} - \MeSig^{-1} \vec{p}_e^{(1)} + \end{align} +\end{subequations} + +\paragraph{Remaining time steps} + +\begin{subequations} + \begin{align} + \dcurl \vec{y}_{e}^{(t+1)} + \frac{1}{\delta t} \vec{y}_{b}^{(t+1)} - \frac{1}{\delta t} \vec{y}_{b}^{(t)} = 0 \\ + \vec{y}_e^{(t+1)} = \MeSig^{-1} \dcurl^\top \MfMui \vec{y}_b^{(t+1)} - \MeSig^{-1} \vec{p}_e^{(t+1)} + \end{align} +\end{subequations} + +\begin{subequations} + \begin{align} + \left( \MfMui \dcurl \MeSig^{-1} \dcurl^\top \MfMui + \frac{1}{\delta t} \MfMui \right) \vec{y}_{b}^{(1)} = \frac{1}{\delta t} \MfMui \vec{y}_b^{(t)} + \MfMui \dcurl \MeSig^{-1} \vec{p}_e^{(1)} \\ + \vec{y}_e^{(t+1)} = \MeSig^{-1} \dcurl^\top \MfMui \vec{y}_b^{(t+1)} - \MeSig^{-1} \vec{p}_e^{(t+1)} + \end{align} + \end{subequations} + +\subsection{Implementing $\mathbf{J}^\top$ onto a vector} + + + +\end{document} \ No newline at end of file