minor changes to tdem writeup and code (still buggy!)

notes/tem/tem.tex

@@ -1,33 +1,33 @@
 \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)}
+\newcommand{\grad}{\vec \nabla}
+\newcommand{\curl}{{\vec \nabla}\times\,}
+\newcommand {\J}{{\vec J}}
+\renewcommand{\H}{{\vec H}}
+\newcommand {\E}{{\vec E}}
+\newcommand{\dcurl}{{\mathbf C}}
+\newcommand{\dgrad}{{\mathbf G}}
+\newcommand{\Acf}{{\mathbf A_c^f}}
+\newcommand{\Ace}{{\mathbf A_c^e}}
+\renewcommand{\S}{{\mathbf \Sigma}}
+\newcommand{\St}{{\mathbf \Sigma_\tau}}
+\newcommand{\T}{{\mathbf T}}
+\newcommand{\Tt}{{\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{\M}{{\mathbf M}}
+\newcommand{\MfMui}{{\M^f_{\mu^{-1}}}}
+\newcommand{\MeSig}{{\M^e_\sigma}}
+\newcommand{\MeSigInf}{{\M^e_{\sigma_\infty}}}
+\newcommand{\MeSigO}{{\M^e_{\sigma_0}}}
+\newcommand{\Me}{{\M^e}}
+\newcommand{\Mes}[1]{{\M^e_{#1}}}
+\newcommand{\Mee}{{\M^e_e}}
+\newcommand{\Mej}{{\M^e_j}}
 
-\newcommand{\BigO}[1]{\ensuremath{\mathcal{O}\bigl(#1\bigr)}}
+\newcommand{\BigO}[1]{\mathcal{O}\bigl(#1\bigr)}
 
 % ********** TDIP paper
 
@@ -75,27 +75,27 @@ Using Gauss-Newton to solve the inverse problem requires the ability to calculat
 where
 \begin{subequations}
     \begin{align}
-        \mathbf{A} = 
+        \mathbf{A} =
         \left[
             \begin{array}{cc}
                 \frac{1}{\delta t} \mathbf{I} & \dcurl \\
-                \dcurl^\top & -\MeSig
+                \dcurl^\top \MfMui & -\MeSig
             \end{array}
         \right] \\
-        \mathbf{B} = 
+        \mathbf{B} =
         \left[
             \begin{array}{cc}
                 -\frac{1}{\delta t} \mathbf{I} & 0 \\
                 0 & 0
             \end{array}
         \right] \\
-        \u^{(k)} = \left[ 
+        \u^{(k)} = \left[
         \begin{array}{c}
             \b^{(k)}\\
             \e^{(k)}
         \end{array}
         \right] \\
-        \s^{(k)} = \left[ 
+        \s^{(k)} = \left[
         \begin{array}{c}
             0\\
             \Me \j^{(k)}_s
@@ -153,7 +153,7 @@ Defining the function $\vec{c}(m,\vec{u})$ to be
 \end{align}
 then
 \begin{align}
-    \frac{\partial \vec{c}}{\partial m} \partial m 
+    \frac{\partial \vec{c}}{\partial m} \partial m
     + \frac{\partial \vec{c}}{\partial \u} \partial \vec{u} = 0
 \end{align}
 or
@@ -168,8 +168,8 @@ Differentiating, we find that
 \end{align}
 and
 \begin{align}
-    \frac{\partial \vec{c}}{\partial \sigma} = \mathbf{G}_\sigma = 
-    \left[ 
+    \frac{\partial \vec{c}}{\partial \sigma} = \mathbf{G}_\sigma =
+    \left[
         \begin{array}{c}
             g_\sigma^{(1)}\\
             g_\sigma^{(2)}\\
@@ -181,8 +181,8 @@ and
 with
 \begin{subequations}
     \begin{align}
-        g_\sigma^{(n)} = 
-        \left[ 
+        g_\sigma^{(n)} =
+        \left[
             \begin{array}{c}
                 \mathbf{0} \\
                 - \diag{\e^{(n)}} \Ace \diag{\vec{V}}
@@ -215,7 +215,7 @@ Multiplying $\mathbf{J}$ onto a vector can be broken into three steps
 
 \begin{subequations}
     \begin{align}
-        \frac{1}{\delta t} \vec{y}_{b}^{(1)} + \dcurl \vec{y}_{e}^{(1)} = 0 \\
+        \dcurl \vec{y}_{e}^{(1)} + \frac{1}{\delta t} \vec{y}_{b}^{(1)} = 0 \\
         \dcurl^\top \MfMui \vec{y}_b^{(1)} - \MeSig \vec{y}_e^{(1)} = \vec{p}_e^{(1)}
     \end{align}
 \end{subequations}
@@ -231,14 +231,18 @@ Multiplying $\mathbf{J}$ onto a vector can be broken into three 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)}
+        \dcurl \vec{y}_{e}^{(t+1)} + \frac{1}{\delta t} \vec{y}_{b}^{(t+1)}
+        {\color{red}- \frac{1}{\delta t} \vec{y}_{b}^{(t)} }
+        = 0 \\
+        \dcurl^\top \MfMui \vec{y}_b^{(t+1)} - \MeSig \vec{y}_e^{(t+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)} \\
+            \left( \MfMui \dcurl \MeSig^{-1} \dcurl^\top \MfMui + \frac{1}{\delta t} \MfMui \right) \vec{y}_{b}^{(t+1)} =
+            {\color{red} \frac{1}{\delta t} \MfMui \vec{y}_b^{(t)} }
+            + \MfMui \dcurl \MeSig^{-1} \vec{p}_e^{(t+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}
@@ -247,4 +251,4 @@ Multiplying $\mathbf{J}$ onto a vector can be broken into three steps
 
 
 
-\end{document}
+\end{document}