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fix equation labels authored by snuverink_j's avatar snuverink_j
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...@@ -83,27 +83,27 @@ an observation location latexmath:[(x,y,z)]. ...@@ -83,27 +83,27 @@ an observation location latexmath:[(x,y,z)].
For an isolated distribution of charge this reduces to For an isolated distribution of charge this reduces to
[[eq:convolutionsolution]]
[latexmath] [latexmath]
++++ ++++
\phi(x,y,z)=\int\int\int{\mathrm{d}x' \,\mathrm{d}y' \,\mathrm{d}z'}\rho(x',y',z') G(x-x',y-y',z-z'), \phi(x,y,z)=\int\int\int{\mathrm{d}x' \,\mathrm{d}y' \,\mathrm{d}z'}\rho(x',y',z') G(x-x',y-y',z-z'),
\label{eq:convolutionsolution}
++++ ++++
where where
[[eq:isolatedgreenfunction]]
[latexmath] [latexmath]
++++ ++++
G(u,v,w)={\frac{1}{\sqrt{u^2+v^2+w^2}}}. G(u,v,w)={\frac{1}{\sqrt{u^2+v^2+w^2}}}.
\label{eq:isolatedgreenfunction}
++++ ++++
A simple discretization of Equation [convolutionsolution] on a Cartesian A simple discretization of Equation [convolutionsolution] on a Cartesian
grid with cell size latexmath:[(h_x,h_y,h_z)] leads to, grid with cell size latexmath:[(h_x,h_y,h_z)] leads to,
[[eq:openbruteforceconvolution]]
[latexmath] [latexmath]
++++ ++++
\phi_{i,j,k}=h_x h_y h_z \sum_{i'=1}^{M_x}\sum_{j'=1}^{M_y}\sum_{k'=1}^{M_t} \rho_{i',j',k'}G_{i-i',j-j',k-k'}, \phi_{i,j,k}=h_x h_y h_z \sum_{i'=1}^{M_x}\sum_{j'=1}^{M_y}\sum_{k'=1}^{M_t} \rho_{i',j',k'}G_{i-i',j-j',k-k'},
\label{eq:openbruteforceconvolution}
++++ ++++
where latexmath:[\rho_{i,j,k}] and latexmath:[G_{i-i',j-j',k-k'}] where latexmath:[\rho_{i,j,k}] and latexmath:[G_{i-i',j-j',k-k'}]
...@@ -118,6 +118,7 @@ FFTs can be used to compute convolutions by appropriate zero-padding of ...@@ -118,6 +118,7 @@ FFTs can be used to compute convolutions by appropriate zero-padding of
the sequences. Discrete convolutions arise in solving the Poisson the sequences. Discrete convolutions arise in solving the Poisson
equation, and one is typically interested in the following, equation, and one is typically interested in the following,
[[eq:bruteforceconvolution]]
[latexmath] [latexmath]
++++ ++++
\bar{\phi}_j=\sum_{k=0}^{K-1}\bar{\rho}_k \bar{G}_{j-k}\quad, \bar{\phi}_j=\sum_{k=0}^{K-1}\bar{\rho}_k \bar{G}_{j-k}\quad,
...@@ -126,7 +127,6 @@ j=0,\ldots,J-1 \\ ...@@ -126,7 +127,6 @@ j=0,\ldots,J-1 \\
k=0,\ldots,K-1 \\ k=0,\ldots,K-1 \\
j-k=-(K-1),\ldots,J-1 \\ j-k=-(K-1),\ldots,J-1 \\
\end{array} \end{array}
\label{eq:bruteforceconvolution}
++++ ++++
where latexmath:[\bar{G}] corresponds to the free space Green where latexmath:[\bar{G}] corresponds to the free space Green
...@@ -153,6 +153,7 @@ latexmath:[\rho], ...@@ -153,6 +153,7 @@ latexmath:[\rho],
Define a periodic Green function, Define a periodic Green function,
latexmath:[G_m], as follows, latexmath:[G_m], as follows,
[[eq:periodicgreenfunction]]
[latexmath] [latexmath]
++++ ++++
G_m=\left\{ G_m=\left\{
...@@ -161,18 +162,17 @@ G_m=\left\{ ...@@ -161,18 +162,17 @@ G_m=\left\{
0 & \quad \text{if }m=J,\ldots,N-K, \\ 0 & \quad \text{if }m=J,\ldots,N-K, \\
G_{m+iN}=G_{m} & \quad \text{for } i \text{ integer }. G_{m+iN}=G_{m} & \quad \text{for } i \text{ integer }.
\end{array}\right. \end{array}\right.
\label{eq:periodicgreenfunction}G
++++ ++++
Now consider the sum Now consider the sum
[[eq:fftconvolution]]
[latexmath] [latexmath]
++++ ++++
{\phi}_j=\frac{1}{N}\sum_{k=0}^{N-1} W^{-jk} {\phi}_j=\frac{1}{N}\sum_{k=0}^{N-1} W^{-jk}
\left(\sum_{n=0}^{N-1} \rho_n W^{nk}\right) \left(\sum_{n=0}^{N-1} \rho_n W^{nk}\right)
\left(\sum_{m=0}^{N-1} G_m W^{mk}\right), \left(\sum_{m=0}^{N-1} G_m W^{mk}\right),
~~~~~~0 \le j \le N-1, ~~~~~~0 \le j \le N-1,
\label{eq:fftconvolution}
++++ ++++
where latexmath:[W=e^{-2\pi i/N}]. This where latexmath:[W=e^{-2\pi i/N}]. This
...@@ -206,11 +206,11 @@ It follows that ...@@ -206,11 +206,11 @@ It follows that
But latexmath:[G] is periodic with period But latexmath:[G] is periodic with period
latexmath:[N]. Hence, latexmath:[N]. Hence,
[[eq:finaleqn]]
[latexmath] [latexmath]
++++ ++++
{\phi}_j=\sum_{n=0}^{K-1}~\bar{\rho}_n G_{j-n} {\phi}_j=\sum_{n=0}^{K-1}~\bar{\rho}_n G_{j-n}
~~~~~~0 \le j \le N-1. ~~~~~~0 \le j \le N-1.
\label{eq:finaleqn}
++++ ++++
In the physical (unpadded) region, In the physical (unpadded) region,
...@@ -246,6 +246,7 @@ latexmath:[W^{j k}] and latexmath:[W^{-m k}], respectively, in ...@@ -246,6 +246,7 @@ latexmath:[W^{j k}] and latexmath:[W^{-m k}], respectively, in
Equation [fftconvolution]. In other words, the FFT-based approach can be Equation [fftconvolution]. In other words, the FFT-based approach can be
used to compute used to compute
[[eq:bruteforcecorrelation]]
[latexmath] [latexmath]
++++ ++++
\bar{\phi}_j=\sum_{k=0}^{K-1}\bar{\rho}_k \bar{G}_{j+k}\quad, \bar{\phi}_j=\sum_{k=0}^{K-1}\bar{\rho}_k \bar{G}_{j+k}\quad,
...@@ -254,7 +255,6 @@ j=0,\ldots,J-1 \\ ...@@ -254,7 +255,6 @@ j=0,\ldots,J-1 \\
k=0,\ldots,K-1 \\ k=0,\ldots,K-1 \\
j-k=-(K-1),\ldots,J-1 \\ j-k=-(K-1),\ldots,J-1 \\
\end{array} \end{array}
\label{eq:bruteforcecorrelation}
++++ ++++
simply by changing the direction of simply by changing the direction of
...@@ -268,10 +268,10 @@ Algorithm used in _OPAL_ ...@@ -268,10 +268,10 @@ Algorithm used in _OPAL_
As a result, the solution of Equation [openbruteforceconvolution] is As a result, the solution of Equation [openbruteforceconvolution] is
then given by then given by
[[eq:oneterm]]
[latexmath] [latexmath]
++++ ++++
\phi_{i,j,k}=h_x h_y h_z \text{FFT}^{-1} \{ ( \text{FFT}\{\rho_{i,j,k}\}) ( \text{FFT}\{G_{i,j,k}\}) \} \phi_{i,j,k}=h_x h_y h_z \text{FFT}^{-1} \{ ( \text{FFT}\{\rho_{i,j,k}\}) ( \text{FFT}\{G_{i,j,k}\}) \}
\label{eq:oneterm}
++++ ++++
where the notation has been introduced that where the notation has been introduced that
......