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Update README.md and add FILTER_PATTERN for Doxygen
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- Allows single README.md to be reused for MATHJAX markup on Github
  and with Doxygen
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kuberry committed Dec 9, 2024
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15 changes: 1 addition & 14 deletions README.md
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Expand Up @@ -7,19 +7,6 @@ The Compadre Toolkit provides a performance portable solution for the parallel e
This toolkit focuses on the 'on-node' aspects of meshless PDE solution and remap, namely the parallel construction of small dense matrices and their inversion. What it does **not** provide is the tools for managing fields, inverting globally sparse matrices, or neighbor search that requires orchestration over many MPI processes. This toolkit is designed to be easily dropped-in to an existing MPI (or serial) based framework for PDE solution or remap, with minimal dependencies ([Kokkos](https://github.com/kokkos/kokkos) and [KokkosKernels](https://github.com/kokkos/kokkos-kernels)).

### Generalized Moving Least Squares (GMLS)
<!---
A GMLS problem requires the specification of a target functional ![equation](https://latex.codecogs.com/gif.latex?\tau) (Compadre::TargetOperation), a reconstruction space ![equation](https://latex.codecogs.com/gif.latex?V) (Compadre::ReconstructionSpace), and a sampling functional ![equation](https://latex.codecogs.com/gif.latex?\lambda) (Compadre::SamplingFunctional).
The Compadre Toolkit is designed to efficiently assemble, factorize, and solve large batches of minimization problems having the form:
![equation](https://latex.codecogs.com/png.latex?%5Cbg_white%20%5Clarge%20%5C%5C%20%5Cbegin%7Balign*%7D%20p%5E%7B*%7D%26%20%3D%26%20%5Cunderset%7Bp%20%5Cin%20V%7D%7B%5Ctext%7Barg%20min%7D%7D%5C%3B%5Cfrac%7B1%7D%7B2%7D%5Csum_%7Bj%3D1%7D%5EN%20%28%5Clambda_j%28u%29-%5Clambda_j%28p%29%29%5E%7B2%7D%5Comega%28%5Ctau%3B%5Clambda_j%29%5C%5C%5C%5C%20%26%26%5Ctau%28u%29%20%5Capprox%20%5Ctau%28p%5E%7B*%7D%29%20%5Cend%7Balign*%7D)
https://www.codecogs.com/latex/eqneditor.php
\[\large \begin{align*}
p^{*}& =& \underset{p \in V}{\text{arg min}}\;\frac{1}{2}\sum_{j=1}^N (\lambda_j(u)-\lambda_j(p))^{2}\omega(\tau;\lambda_j)\\\\
&&\tau(u) \approx \tau(p^{*})
\end{align*} \]
--->

Here is a brief overview of the GMLS framework:

Expand All @@ -34,7 +21,7 @@ With an optimal reconstruction $p$ in hand, the target functional is approximate
As an unconstrained $\ell_2$-optimization problem, this process admits the explicit form:


$$\tau^h_{\tilde{x}}(\phi) = \tau_{\tilde{x}}(\mathbf{P})^\intercal \left(\Lambda(\mathbf{P})^\intercal \mathbf{W} \Lambda(\mathbf{P})\right)^{-1} \Lambda(\mathbf{P})^\intercal \mathbf{W} \Lambda(\phi),$$
$$\tau^h_{\tilde{x}}(\phi) = \tau_{\tilde{x}}(\mathbf{P})^\top \left(\Lambda(\mathbf{P})^\top \mathbf{W} \Lambda(\mathbf{P})\right)^{-1} \Lambda(\mathbf{P})^\top \mathbf{W} \Lambda(\phi),$$

where:
* $\tau_{\tilde{x}}(\mathbf{P}) \in \mathbb{R}^{dim(V_h)}$ is a vector with components consisting of the target functional applied to each basis function,
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4 changes: 2 additions & 2 deletions doc/Doxyfile
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Expand Up @@ -926,7 +926,7 @@ INPUT_FILTER =
# filters are used. If the FILTER_PATTERNS tag is empty or if none of the
# patterns match the file name, INPUT_FILTER is applied.

FILTER_PATTERNS =
FILTER_PATTERNS += "*.md=sed 's/\([^$]\)\$\([^$]*\)\$\([^$]\)/\1\\f(\2\\f)\3/g;s/\\{/\{/g;s/\\}/\}/g'"

# If the FILTER_SOURCE_FILES tag is set to YES, the input filter (if set using
# INPUT_FILTER ) will also be used to filter the input files that are used for
Expand Down Expand Up @@ -1500,7 +1500,7 @@ FORMULA_TRANSPARENT = YES
# The default value is: NO.
# This tag requires that the tag GENERATE_HTML is set to YES.

USE_MATHJAX = NO
USE_MATHJAX = YES

# When MathJax is enabled you can set the default output format to be used for
# the MathJax output. See the MathJax site (see:
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