From 847070f1a2f5cef2fe8b8a87c32a016987fcdd3d Mon Sep 17 00:00:00 2001 From: rowanc1 Date: Thu, 13 Feb 2014 18:16:57 -0800 Subject: [PATCH] updated and fixed em derivation --- docs/api_TDEM_derivation.rst | 79 ++++++++++++++++++++++++++++++++---- 1 file changed, 70 insertions(+), 9 deletions(-) diff --git a/docs/api_TDEM_derivation.rst b/docs/api_TDEM_derivation.rst index e38be920..8c927d14 100644 --- a/docs/api_TDEM_derivation.rst +++ b/docs/api_TDEM_derivation.rst @@ -83,14 +83,14 @@ where \mathbf{A} = \left[ \begin{array}{cc} - \frac{1}{\delta t} \mathbf{I} & \dcurl \\ + \frac{1}{\delta t} \MfMui & \MfMui\dcurl \\ \dcurl^\top \MfMui & -\MeSig \end{array} \right] \\ \mathbf{B} = \left[ \begin{array}{cc} - -\frac{1}{\delta t} \mathbf{I} & 0 \\ + -\frac{1}{\delta t} \MfMui & 0 \\ 0 & 0 \end{array} \right] \\ @@ -108,6 +108,10 @@ where \right] \end{align} +.. note:: + + Here we have multiplied through by \\(\\MfMui\\) to make A and B symmetric! + The entire time dependent system can be written in a single matrix expression .. math:: @@ -256,7 +260,7 @@ First time step .. math:: \begin{align} - \dcurl \vec{y}_{e}^{(1)} + \frac{1}{\delta t} \vec{y}_{b}^{(1)} = \vec{p}_b^{(1)} \\ + \frac{1}{\delta t} \MfMui \vec{y}_{b}^{(1)} + \MfMui \dcurl \vec{y}_{e}^{(1)} = \vec{p}_b^{(1)} \\ \dcurl^\top \MfMui \vec{y}_b^{(1)} - \MeSig \vec{y}_e^{(1)} = \vec{p}_e^{(1)} \end{align} @@ -264,7 +268,7 @@ First time step .. math:: \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)} + \MfMui \vec{p}_b^{(1)} \\ + \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{p}_b^{(1)} \\ \vec{y}_e^{(1)} = \MeSig^{-1} \dcurl^\top \MfMui \vec{y}_b^{(1)} - \MeSig^{-1} \vec{p}_e^{(1)} \end{align} @@ -274,8 +278,8 @@ Remaining time steps: .. math:: \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)} + \frac{1}{\delta t} \MfMui\vec{y}_{b}^{(t+1)} + \MfMui\dcurl \vec{y}_{e}^{(t+1)} + - \frac{1}{\delta t} \MfMui \vec{y}_{b}^{(t)} = \vec{p}_b^{(t+1)} \\ \dcurl^\top \MfMui \vec{y}_b^{(t+1)} - \MeSig \vec{y}_e^{(t+1)} = \vec{p}_e^{(t+1)} \end{align} @@ -287,7 +291,7 @@ and \begin{align} \left( \MfMui \dcurl \MeSig^{-1} \dcurl^\top \MfMui + \frac{1}{\delta t} \MfMui \right) \vec{y}_{b}^{(t+1)} = \frac{1}{\delta t} \MfMui \vec{y}_b^{(t)} - + \MfMui \dcurl \MeSig^{-1} \vec{p}_e^{(t+1)} + \MfMui \vec{p}_b^{(t+1)} \\ + + \MfMui \dcurl \MeSig^{-1} \vec{p}_e^{(t+1)} + \vec{p}_b^{(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} @@ -299,6 +303,63 @@ Implementing \\(\\mathbf{J}^\\top\\) times a vector Multiplying \\(\\mathbf{J}^\\top\\) onto a vector can be broken into three steps -* Compute \\(\\vec{u} = \\mathbf{Q}^\\top \\vec{v}\\) -* Solve \\(\\hat{\\mathbf{A}}^\\top \\vec{y} = \\vec{u}\\) +* Compute \\(\\vec{p} = \\mathbf{Q}^\\top \\vec{v}\\) +* Solve \\(\\hat{\\mathbf{A}}^\\top \\vec{y} = \\vec{p}\\) * Compute \\(\\vec{w} = -\\mathbf{G}^\\top y\\) + + +.. math:: + + \mathbf{\hat{A}}^\top = \left[ + \begin{array}{cccc} + A & B & & \\ + & \ddots & \ddots & \\ + & & A & B \\ + & & 0 & A + \end{array} + \right] + +For the last time-step \\(t=N\\): + +.. math:: + + \begin{align} + \frac{1}{\delta t} \MfMui \vec{y}_{b}^{(N)} + \MfMui \dcurl \vec{y}_{e}^{(N)} = \vec{p}_b^{(N)} \\ + \dcurl^\top \MfMui \vec{y}_b^{(N)} - \MeSig \vec{y}_e^{(N)} = \vec{p}_e^{(N)} + \end{align} + + +.. math:: + + \begin{align} + \left( \MfMui \dcurl \MeSig^{-1} \dcurl^\top \MfMui + \frac{1}{\delta t} \MfMui \right) \vec{y}_{b}^{(N)} = \MfMui \dcurl \MeSig^{-1} \vec{p}_e^{(N)} + \vec{p}_b^{(N)} \\ + \vec{y}_e^{(N)} = \MeSig^{-1} \dcurl^\top \MfMui \vec{y}_b^{(N)} - \MeSig^{-1} \vec{p}_e^{(N)} + \end{align} + +For the rest of the time-steps (going backwards in time) + + +.. math:: + + A \vec{y}^{(t-1)} + B \vec{y}^{(t)} = \vec{p}^{(t-1)} + + +.. math:: + + \begin{align} + \frac{1}{\delta t} \MfMui\vec{y}_{b}^{(t-1)} + \MfMui\dcurl \vec{y}_{e}^{(t-1)} + - \frac{1}{\delta t} \MfMui \vec{y}_{b}^{(t)} + = \vec{p}_b^{(t-1)} \\ + \dcurl^\top \MfMui \vec{y}_b^{(t-1)} - \MeSig \vec{y}_e^{(t-1)} = \vec{p}_e^{(t-1)} + \end{align} + +and + +.. math:: + + \begin{align} + \left( \MfMui \dcurl \MeSig^{-1} \dcurl^\top \MfMui + \frac{1}{\delta t} \MfMui \right) \vec{y}_{b}^{(t-1)} = + \frac{1}{\delta t} \MfMui \vec{y}_b^{(t)} + + \MfMui \dcurl \MeSig^{-1} \vec{p}_e^{(t-1)} + \vec{p}_b^{(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}