| 311 |
\label{fig:3d-probe-code-opencl} |
\label{fig:3d-probe-code-opencl} |
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\end{figure} |
\end{figure} |
| 313 |
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|
| 314 |
|
\section{Generalizing to derivative fields} |
| 315 |
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To generalize the probe operation to work on derivative fields (\eg{}, $(\nabla^k F)\mkw{@}\vecp$), |
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we represent the $\nabla^k$ operation as a $k$-order tensor of partial derivative operators. |
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To simplify the presentation, we will restrict the discussion to $3$-dimensional space, |
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but it easily generalizes. |
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In 3D, a partial-derivative operator has the form |
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\begin{displaymath} |
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\frac{\partial}{\partial{}x^i \partial{}y^j \partial{}z^k} |
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\end{displaymath}% |
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where $i$, $j$, and $k$ are the number of levels of differentiation in the $x$, $y$, |
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and $z$-axes (respectively). |
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The normal convention is to omit explicit mention of axes with zero levels of differentiation, |
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but we will be explicit here. |
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We use $\mathbf{Id}^n$ to represent the identity operator in $n$-dimensions (i.e., the |
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one with zero=-levels of differentiation in all axes). |
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|
| 330 |
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The product of two partial-derivative operators is formed by summing the exponents; |
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|
for example in 3D we have |
| 332 |
|
\begin{displaymath} |
| 333 |
|
\frac{\partial}{\partial{}x^i \partial{}y^j \partial{}z^k} |
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\frac{\partial}{\partial{}x^{i'} \partial{}y^{j'} \partial{}z^{k'}} = |
| 335 |
|
\frac{\partial}{\partial{}x^{i+i'} \partial{}y^{j+j'} \partial{}z^{k+k'}} |
| 336 |
|
\end{displaymath}% |
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The last bit of notation that we need is the $n$-vector of partial-derivative |
| 338 |
|
operators $\delta^n$, in which the $i$th element has one level of differentiation in the $i$th |
| 339 |
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axis (and zero in all other axes). |
| 340 |
|
For example, |
| 341 |
|
\begin{displaymath} |
| 342 |
|
\delta^3 = \left[ |
| 343 |
|
\frac{\partial}{\partial{}x^1 \partial{}y^0 \partial{}z^0} \quad |
| 344 |
|
\frac{\partial}{\partial{}x^0 \partial{}y^1 \partial{}z^0} \quad |
| 345 |
|
\frac{\partial}{\partial{}x^0 \partial{}y^0 \partial{}z^1} |
| 346 |
|
\right] |
| 347 |
|
\end{displaymath}% |
| 348 |
|
For $k > 0$ levels of differentiation in $n$-dimensional space, we have |
| 349 |
|
\begin{displaymath} |
| 350 |
|
\nabla^k = \left\{\begin{array}{ll} |
| 351 |
|
\mathbf{Id}^n & \text{when $k = 0$} \\ |
| 352 |
|
\delta^n \otimes \nabla^{k-1} & \text{when $k > 0$} \\ |
| 353 |
|
\end{array}\right. |
| 354 |
|
\end{displaymath}% |
| 355 |
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Consider, for example, the case where $k = 2$. |
| 356 |
|
\begin{displaymath} |
| 357 |
|
\nabla^2 = \delta^3 \otimes \delta^3 = \left[\begin{array}{ccc} |
| 358 |
|
\frac{\partial}{\partial{}x^2 \partial{}y^0 \partial{}z^0} |
| 359 |
|
& \frac{\partial}{\partial{}x^1 \partial{}y^1 \partial{}z^0} |
| 360 |
|
& \frac{\partial}{\partial{}x^1 \partial{}y^0 \partial{}z^1} \\ |
| 361 |
|
\frac{\partial}{\partial{}x^1 \partial{}y^1 \partial{}z^0} |
| 362 |
|
& \frac{\partial}{\partial{}x^0 \partial{}y^2 \partial{}z^0} |
| 363 |
|
& \frac{\partial}{\partial{}x^0 \partial{}y^1 \partial{}z^1} \\ |
| 364 |
|
\frac{\partial}{\partial{}x^1 \partial{}y^0 \partial{}z^1} |
| 365 |
|
& \frac{\partial}{\partial{}x^0 \partial{}y^1 \partial{}z^1} |
| 366 |
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& \frac{\partial}{\partial{}x^0 \partial{}y^0 \partial{}z^2} \\ |
| 367 |
|
\end{array}\right] |
| 368 |
|
\end{displaymath}% |
| 369 |
|
|
| 370 |
|
When we have a probe of a derivative field $(\nabla^k F)\mkw{@}\vecp$, we will generate code for |
| 371 |
|
each partial derivative operator $\frac{\partial}{\partial{}x^i\partial{}y^j\partial{}z^k}$ |
| 372 |
|
separately. |
| 373 |
|
If $F = V\circledast{}h$, then we will use $h^i(x) h^j(y) h^k(z)$ as the interpolation |
| 374 |
|
coefficients, where $h^i$ is the $i$th derivative of $h$. |
| 375 |
|
|
| 376 |
\section{Probing a 3D derivative field} |
\section{Probing a 3D derivative field} |
| 377 |
We next consider the case of probing the derivative of a scalar field $F = V\circledast{}h$, where $s$ is the support |
We next consider the case of probing the derivative of a scalar field $F = V\circledast{}h$, where $s$ is the support |
| 378 |
of $h$. |
of $h$. |
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