-
Notifications
You must be signed in to change notification settings - Fork 12
Commit
This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository.
Merge pull request #94 from ubcecon/note-differential-operators-on-ir…
…regular-grids Add notes on differential operators on irregular grids
- Loading branch information
Showing
2 changed files
with
263 additions
and
0 deletions.
There are no files selected for viewing
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,3 @@ | ||
| # Notes | ||
|
|
||
| - [Notes on differential operators on irregular grids](https://github.com/ubcecon/computing_and_datascience/blob/master/continuous_time_methods/notes/differential-operator-on-irregular-grids.tex) |
260 changes: 260 additions & 0 deletions
260
continuous_time_methods/notes/differential-operator-on-irregular-grids.tex
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,260 @@ | ||
| % !TEX program = pdflatex | ||
|
|
||
| \documentclass[11pt]{article} | ||
| \usepackage{amsmath,amsfonts,amsthm,amssymb,geometry,dsfont} | ||
| \usepackage[usenames,dvipsnames,svgnamesable]{xcolor} | ||
| \usepackage[capitalise,noabbrev]{cleveref} % | ||
| \crefname{equation}{}{} % | ||
| \crefname{assumption}{Assumption}{Assumptions} | ||
| \crefname{property}{Property}{Properties} | ||
| \geometry{left=1in,right=1in,top=0.6in,bottom=1in} | ||
|
|
||
| \newcommand{\D}[1][]{\ensuremath{\boldsymbol{\partial}_{#1}}} | ||
| \newcommand{\R}{\ensuremath{\mathbb{R}}} | ||
| \newcommand{\diff}{\ensuremath{\mathrm{d}}} | ||
| \newcommand{\set}[1]{\ensuremath{\left\{{#1}\right\}}} | ||
| \newcommand{\indicator}[1]{\ensuremath{\mathds{1}\left\{{#1}\right\}}} | ||
| \newcommand{\condexpec}[3][]{\ensuremath{\mathbb{E}_{#1}\left[{#2} \; \middle| \; {#3} \right]}} | ||
| \newcommand{\expec}[2][]{\ensuremath{\mathbb{E}_{{#1}}\left[ {#2} \right]}} | ||
|
|
||
| \newenvironment{psmallmatrix} | ||
| {\left(\begin{smallmatrix}} | ||
| {\end{smallmatrix}\right)} | ||
|
|
||
| \begin{document} | ||
| \title{Notes on differential operators on irregular grids with reflecting barrier conditions} | ||
| \author{@chiyahn} | ||
| \maketitle | ||
|
|
||
|
|
||
| \section{Setup} | ||
|
|
||
| \begin{itemize} | ||
| \item Define an irregular grid $\set{z_i}_{i=1}^P$ with $z_1 = 0$ and $z_P = \bar{z}$ is a ``large'' number. Denote the grid with the variable name, i.e. $z \equiv \set{z_i}_{i=1}^P$. | ||
| \item Denote the distance between the grid points as the \textit{backwards} difference | ||
| \begin{align} | ||
| \Delta_{i,-} &\equiv z_i - z_{i-1},\, \text{for } i = 2,\ldots, P\\ | ||
| \Delta_{i,+} &\equiv z_{i+1} - z_i,\, \text{for } i = 1,\ldots, P-1 | ||
| \end{align} | ||
|
|
||
| \item Assume $\Delta_{1, -} = \Delta_{1, +}$ and $\Delta_{P, +} = \Delta_{P, -}$, due to ghost points, $z_0$ and $z_{P+1}$ on both boundaries. (i.e.he distance to the ghost nodes are the same as the distance to the closest nodes). Then define the vector of backwards and forwards first differences as | ||
| \begin{align} | ||
| \Delta_{-} &\equiv \begin{bmatrix} z_2 - z_1 \\ | ||
| \text{diff}(z) | ||
| \end{bmatrix}\\ | ||
| \Delta_{+} &\equiv \begin{bmatrix} \text{diff}(z)\\ | ||
| z_P - z_{P-1} | ||
| \end{bmatrix} | ||
| \end{align} | ||
| \item Reflecting barrier conditions: | ||
| \begin{align} | ||
| \xi v(0) + \D[z]v(0 ) &= 0\label{eq:new-BC1}\\ | ||
| \xi v(\bar{z}) + \D[z]v(\bar{z}) &= 0\label{eq:new-BC2} | ||
| \end{align} | ||
| \end{itemize} | ||
|
|
||
| Let $L_1^{-}$ be the discretized backwards first differences and $L_2$ be the discretized central differences subject to the Neumann boundary conditions in \cref{eq:new-BC1,eq:new-BC2} such that $L_1^{-} v(t)$ and $L_2 v(t)$ represent the first and second derivatives of $v(z)$ respectively at $t$. For second derivatives, we use the following numerical scheme from Achdou et al. (2017): | ||
|
|
||
| \begin{equation} | ||
| v''(z_i) \approx \dfrac{ \Delta_{i,-} v( z_i + \Delta_{i,+}) - (\Delta_{i,+} + \Delta_{i,-}) v( z_i ) + \Delta_{i,+} v( z_i - \Delta_{i,-})}{\frac{1}{2}(\Delta_{i,+} + \Delta_{i,-}) \Delta_{i,+} \Delta_{i,-} }, \text{for } i = 1, \ldots, P | ||
| \end{equation} | ||
|
|
||
|
|
||
|
|
||
|
|
||
|
|
||
| \subsection{Regular grids} | ||
| Suppose that the grids are regular, i.e., elements of $\text{diff}(z)$ are all identical with $\Delta$ for some $\Delta > 0$. | ||
|
|
||
| Using the backwards first-order difference, \eqref{eq:new-BC1} can be alternatively represented as | ||
| \begin{align} | ||
| \dfrac{v(0) - v(-\Delta)}{\Delta} &= - \xi v(0) | ||
| \end{align} | ||
| Similarly, using discretized central differences of second orders, \eqref{eq:new-BC1} can be shown as | ||
| \begin{align} | ||
| \dfrac{v (\Delta) - 2 v(0) + v(-\Delta)}{\Delta^2} &= \dfrac{v(\Delta) - v(0)}{\Delta^2} - \dfrac{1}{\Delta}\dfrac{v (0) - v(-\Delta) }{\Delta} \\ | ||
| &= \dfrac{v(\Delta) - v(0)}{\Delta^2} + \dfrac{1}{\Delta} \xi v(0) \\ | ||
| &= \dfrac{1}{\Delta^2} (- 1 + \Delta \xi) v(0) + \dfrac{1}{\Delta^2} v(\Delta) | ||
| \end{align} | ||
| Similarly, for \eqref{eq:new-BC2}, we have | ||
| \begin{align} | ||
| \dfrac{v (\bar{z} + \Delta) - 2 v(\bar{z} ) + v(\bar{z} -\Delta)}{\Delta^2} &= \dfrac{v(\bar{z} - \Delta) - v(\bar{z})}{\Delta^2} + \dfrac{1}{\Delta}\dfrac{ v(\bar{z}+\Delta) - v (\bar{z}) }{\Delta} \\ | ||
| &= \dfrac{v(\bar{z} - \Delta) - v(\bar{z})}{\Delta^2} - \dfrac{1}{\Delta} \xi v(\bar{z}) \\ | ||
| &= \dfrac{1}{\Delta^2} (- 1 - \Delta \xi) v(\bar{z}) + \dfrac{1}{\Delta^2} v(\bar{z} - \Delta) | ||
| \end{align} | ||
|
|
||
| Thushe corresponding $L_1^{-}$ and $L_2$ matrices are defined as | ||
|
|
||
| \begin{align} | ||
| L_1^{-} &\equiv \frac{1}{\Delta}\begin{pmatrix} | ||
| 1 - (1 + \xi \Delta) &0&0&\dots&0&0&0\\ | ||
| -1&1&0&\dots&0&0&0\\ | ||
| \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ | ||
| 0&0&0&\dots&-1&1&0\\ | ||
| 0&0&0&\cdots&0&-1&1 | ||
| \end{pmatrix}_{P\times P}\label{eq:L-1-regular} \\ | ||
| L_2 &\equiv \frac{1}{\Delta^2}\begin{pmatrix} | ||
| -2 + (1 + \xi\Delta) &1&0&\dots&0&0&0\\ | ||
| 1&-2&1&\dots&0&0&0\\ | ||
| \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ | ||
| 0&0&0&\dots&1&-2&1\\ | ||
| 0&0&0&\cdots&0&1&-2 + (1- \xi\Delta) | ||
| \end{pmatrix}_{P\times P}\label{eq:L-2-regular} | ||
| \end{align} | ||
|
|
||
| \subsection{Irregular grids} | ||
| Using the backwards first-order difference, \eqref{eq:new-BC1} can be alternatively represented as | ||
| \begin{align} | ||
| \dfrac{v(0) - v(-\Delta_{1, -})}{\Delta_{1, -}} &= - \xi v(0) | ||
| \end{align} | ||
|
|
||
| Note that we have assumed that $\Delta_{1,-} = \Delta_{1,+}$ and $\Delta_{P,+} = \Delta_{P,-}$ for the ghost notes. Using discretized central differences of second orders, \eqref{eq:new-BC1} can be shown as | ||
| \begin{align} | ||
| &\dfrac{\Delta_{1,-} v( \Delta_{1,+}) - (\Delta_{1,+} + \Delta_{1,-}) v( 0 ) + \Delta_{1,+} v( - \Delta_{1,-})}{\frac{1}{2}(\Delta_{1,+} + \Delta_{1,-}) \Delta_{1,+} \Delta_{1,-} } \\ | ||
| &= | ||
| \dfrac{v (\Delta_{1, +}) - 2 v(0) + v(-\Delta_{1, +})}{\Delta_{1, +}^2} \\ &= \dfrac{v(\Delta_{1, +}) - v(0)}{\Delta_{1, +}^2} - \dfrac{1}{\Delta_{1, +}}\dfrac{v (0) - v(-\Delta_{1, +}) }{\Delta_{1, +}} \\ | ||
| &= \dfrac{v(\Delta_{1, +}) - v(0)}{\Delta_{1, +}^2} + \dfrac{1}{\Delta_{i,+}} \xi v(0) \\ | ||
| &= \dfrac{1}{\Delta_{1, +}^2} (- 1 + \Delta_{1, +} \xi) v(0) + \dfrac{1}{\Delta_{1, +}^2} v(\Delta_{1, +}) | ||
| \end{align} | ||
| Similarly, for \eqref{eq:new-BC2}, we have | ||
| \begin{align} | ||
| &\dfrac{\Delta_{P,-} v( \bar{z} + \Delta_{P,+}) - (\Delta_{P,+} + \Delta_{P,-}) v(\bar{z} ) + \Delta_{P,+} v( \bar{z} - \Delta_{P,-})}{\frac{1}{2}(\Delta_{P,+} + \Delta_{P,-}) \Delta_{P,+} \Delta_{P,-} } \\ | ||
| &=\dfrac{v (\bar{z} + \Delta_{P,-}) - 2 v(\bar{z} ) + v(\bar{z} -\Delta_{P,-})}{\Delta_{P,-}^2} \\ | ||
| &= \dfrac{v(\bar{z} - \Delta_{P,-}) - v(\bar{z})}{\Delta_{P,-}^2} + \dfrac{1}{\Delta_{P,-}}\dfrac{ v(\bar{z}+\Delta_{P,-}) - v (\bar{z}) }{\Delta_{P,-}} \\ | ||
| &= \dfrac{v(\bar{z} - \Delta_{P,-}) - v(\bar{z})}{\Delta_{P,-}^2} - \dfrac{1}{\Delta_{P,-}} \xi v(\bar{z}) \\ | ||
| &= \dfrac{1}{\Delta_{P,-}^2} (- 1 - \Delta_{P,-} \xi) v(\bar{z}) + \dfrac{1}{\Delta_{P,-}^2} v(\bar{z} - \Delta_{P,-}) | ||
| \end{align} | ||
|
|
||
| Thushe corresponding $L_1^{-}$ and $L_2$ matrices are defined as | ||
|
|
||
| \begin{align} | ||
| L_1^{-} &\equiv \begin{pmatrix} | ||
| \Delta^{-1}_{1,-} [1 - (1 + \xi \Delta^{-1}_{1,-})] &0&0&\dots&0&0&0\\ | ||
| -\Delta_{2,-}^{-1}&\Delta_{2,-}^{-1}&0&\dots&0&0&0\\ | ||
| \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ | ||
| 0&0&0&\dots&-\Delta_{P-1,-}^{-1}&\Delta_{P-1,-}^{-1}&0\\ | ||
| 0&0&0&\cdots&0&-\Delta_{P,-}^{-1}&\Delta_{P,-}^{-1} | ||
| \end{pmatrix}_{P\times P}\label{eq:L-1} \\ | ||
| L_2 &\equiv \begin{psmallmatrix} | ||
| \Delta_{1,+}^{-2}[-2 + (1+\xi \Delta_{1,+})] &\Delta_{1,+}^{-2}&0&\cdots&0&0&0 \\ | ||
| \vdots&\ddots&\ddots&\ddots&\ddots&\vdots&\vdots\\ | ||
| 0&\cdots&2(\Delta_{i,+}+\Delta_{i,-})^{-1} \Delta_{i,-}^{-1} &-2\Delta_{i,-}^{-1} \Delta_{i,+}^{-1} & 2 (\Delta_{i,+}+\Delta_{i,-})^{-1} \Delta_{i,+}^{-1}&\cdots&0 \\ | ||
| \vdots&\vdots&\vdots&\ddots&\ddots&\ddots&\vdots\\ | ||
| 0&0&0&\cdots&0&\Delta_{P,-}^{-2}&\Delta_{P,-}^{-2} [-2 + (1- \xi\Delta_{P,-})] | ||
| \end{psmallmatrix}_{P\times P}\label{eq:L-2} | ||
| \end{align} | ||
|
|
||
| \subsection{Differential operators by basis} | ||
| Define the following basis matrices: | ||
|
|
||
| \begin{align} | ||
| U_1^{-} &\equiv \begin{pmatrix} | ||
| 1 &0&0&\dots&0&0&0\\ | ||
| -1&1&0&\dots&0&0&0\\ | ||
| \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ | ||
| 0&0&0&\dots&-1&1&0\\ | ||
| 0&0&0&\cdots&0&-1&1 | ||
| \end{pmatrix}_{P\times P}\label{eq:L-1-basis} \\ | ||
| U_1^{+} &\equiv \begin{pmatrix} | ||
| -1 &1&0&\dots&0&0&0\\ | ||
| 0&-1&1&\dots&0&0&0\\ | ||
| \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ | ||
| 0&0&0&\dots&0&-1&1\\ | ||
| 0&0&0&\cdots&0&0&-1 | ||
| \end{pmatrix}_{P\times P}\label{eq:L-1+-basis} \\ | ||
| U_2 &\equiv \begin{pmatrix} | ||
| -2 &1&0&\dots&0&0&0\\ | ||
| 1&-2&1&\dots&0&0&0\\ | ||
| \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ | ||
| 0&0&0&\dots&1&-2&1\\ | ||
| 0&0&0&\cdots&0&1&-2 | ||
| \end{pmatrix}_{P\times P}\label{eq:L-2-basis} | ||
| \end{align} | ||
|
|
||
| and the boundary conditions for the reflecting conditions: | ||
|
|
||
| \begin{align} | ||
| B_{1} &\equiv \begin{pmatrix} | ||
| (1 + \xi \Delta^{-1}_{1,-}) &0&\dots&0&0\\ | ||
| 0&0&\dots&0&0\\ | ||
| \vdots&\vdots&\ddots&\vdots&\vdots\\ | ||
| 0&0&\cdots&0&0\\ | ||
| 0&0&\cdots&0&0 | ||
| \end{pmatrix}_{P\times P} \\ | ||
| B_{P} &\equiv \begin{pmatrix} | ||
| 0 &0&\dots&0&0\\ | ||
| 0&0&\dots&0&0\\ | ||
| \vdots&\vdots&\ddots&\vdots&\vdots\\ | ||
| 0&0&\cdots&0&0\\ | ||
| 0&0&\cdots&0&(1 + \xi \Delta^{-1}_{P,+}) | ||
| \end{pmatrix}_{P\times P} | ||
| \end{align} | ||
|
|
||
| \subsubsection{Regular grids} | ||
| For regular grids with the uniform distance of $\Delta > 0$, \eqref{eq:L-1-regular} and \eqref{eq:L-2-regular} can be represented by | ||
|
|
||
| \begin{align} | ||
| L_1^{-} &= \dfrac{1}{\Delta} U_1^{-} - B_1 \\ | ||
| L_2 &= \dfrac{1}{\Delta^2} U_2 + B_1 + B_P | ||
| \end{align} | ||
|
|
||
| Note that $U_2 = U_1^+ - U_1^-$. Hence, $L_2$ can be also represented as | ||
| \begin{align} | ||
| L_2 &= \dfrac{1}{\Delta^2} (U_1^+ - U_1^-) + B_1 + B_P | ||
| \end{align} | ||
|
|
||
| \subsubsection{Irregular grids} | ||
| For irregular grids we need further decomposition of $L_2$. Define $U_2^-, U_2^0, U_2^+$ be the matrices that keep only the lower diagonal, diagonal, upper diagonal elements respectively and are zero on all the other elements, i.e., | ||
|
|
||
| \begin{align} | ||
| U_2^- &\equiv \begin{pmatrix} | ||
| 0&0&0&\dots&0&0&0\\ | ||
| 1&0&0&\dots&0&0&0\\ | ||
| \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ | ||
| 0&0&0&\dots&1&0&0\\ | ||
| 0&0&0&\cdots&0&1&0 | ||
| \end{pmatrix}_{P\times P} \\ | ||
| U_2^0 &\equiv \begin{pmatrix} | ||
| -2 &0&0&\dots&0&0&0\\ | ||
| 0&-2&0&\dots&0&0&0\\ | ||
| \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ | ||
| 0&0&0&\dots&0&-2&0\\ | ||
| 0&0&0&\cdots&0&0&-2 | ||
| \end{pmatrix}_{P\times P} \\ | ||
| U_2^+ &\equiv \begin{pmatrix} | ||
| 0&1&0&\dots&0&0&0\\ | ||
| 0&0&1&\dots&0&0&0\\ | ||
| \vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\ | ||
| 0&0&0&\dots&0&0&1\\ | ||
| 0&0&0&\cdots&0&0&0 | ||
| \end{pmatrix}_{P\times P} | ||
| \end{align} | ||
|
|
||
| For notational brevity, for vectors with the same size, $x_1, x_2$, define $x_1 x_2$ as the elementwise-multiplied vector. Then, we have | ||
| \begin{align} | ||
| L_1^{-} &= \text{diag}(\Delta_{-} )^{-1} U_1^{-} - B_1 \\ | ||
| L_2 &= \text{diag} \left[ \frac{1}{2} ( \Delta_+ + \Delta_- ) \Delta_+ \right]^{-1} U_2^{+} - | ||
| \text{diag} \left[ \frac{1}{2} ( \Delta_+ + \Delta_- ) \Delta_- \right]^{-1} U_2^{-} | ||
| + B_1 + B_P | ||
| \end{align} | ||
| We can simplify this expression further by introducing a new notation. Let $x^{-1}$ be defined as the elementwise inverse of a vector $x$ that contains no zero element. Then, $L_2$ can be represented as | ||
| \begin{align} | ||
| L_2 &= | ||
| 2\left[ \text{diag} \left( ( \Delta_+ + \Delta_- )^{-1} \Delta_+^{-1} \right) U_2^{+} - | ||
| \text{diag} \left( ( \Delta_+ + \Delta_- )^{-1} \Delta_-^{-1} \right) U_2^{-} \right] | ||
| + B_1 + B_P \\ \label{eq:L-2-by-basis} | ||
| &= 2 \text{diag} \left( ( \Delta_+ + \Delta_- )^{-1} \right) \left[ \text{diag} \left( \Delta_+^{-1} \right) U_2^{+} - | ||
| \text{diag} \left( \Delta_-^{-1} \right) U_2^{-} \right] | ||
| + B_1 + B_P | ||
| \end{align} | ||
|
|
||
|
|
||
| The diagonal elements of \eqref{eq:L-2-by-basis} are also identical with the one provided in \eqref{eq:L-2} -- to see this, note that the diagonal elements of \eqref{eq:L-2-by-basis}, modulo $B_1$ and $B_P$, are | ||
| \begin{align} | ||
| -2 \left[ (\Delta_+ + \Delta_-)^{-1} \Delta_+^{-1} + (\Delta_+ + \Delta_-)^{-1} \Delta_-^{-1} \right] &= -2 (\Delta_+ + \Delta_-)^{-1} ( \Delta_+^{-1} + \Delta_-^{-1} ) \\ | ||
| &= -2(\Delta_+ + \Delta_-)^{-1} (\Delta_+^{-1} \Delta_-^{-1}) (\Delta_+ + \Delta_- ) \\ | ||
| &= -2 (\Delta_+^{-1} \Delta_-^{-1}) | ||
| \end{align} | ||
| which is identical with $\text{diag} (L_2)$ with $L_2$ from \eqref{eq:L-2} except the first row and last row that are affected by $B_1$ and $B_P$. | ||
|
|
||
| \end{document} |