Update README.md

Author: kuberry

README.md

@@ -7,20 +7,48 @@ 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)
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+<!---
 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)
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+
 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:
+
+Consider $\phi$ of function class $\mathbf{V}$ as well as a collection of samples $\Lambda = \\{\lambda_ i(\phi)\\}_ {i=1}^{N}$ (Compadre::SamplingFunctional) corresponding to a quasiuniform collection of data sites $\mathbf{X}_ h = \\{ \mathbf{x}_ i \\} \subset \mathbb{R}^d$ characterized by fill distance $h$. To approximate a given linear target functional $\tau_{\tilde{x}}$ (Compadre::TargetOperation) associated with a target site $\tilde{x}$, we seek a reconstruction $p \in \mathbf{V}_ h$, where $\mathbf{V}_ h \subset \mathbf{V}$ is a finite dimensional space (Compadre::ReconstructionSpace) chosen to provide good approximation properties, with basis $\mathbf{P} = \\{P\\}_{i=1}^{dim(V_h)}$. We perform this reconstruction in the following weighted $\ell_2$ sense:
+
+$$p = \underset{{q \in \mathbf{V}_ h}}{\mathrm{argmin}} \sum_{i=1}^N ( \lambda_i(\phi) -\lambda_i(q) )^2 \omega(\lambda_i,\tau_{\tilde{x}}),$$
+
+where $\omega$ is a locally supported positive function, $\omega = \Phi(|\tilde{x}-\mathbf{x}_i|)$ and $|\cdot|$ denotes the Euclidean norm. $\Phi(r,\epsilon)$ is selected by the user, having a parameter controlling the support of $\omega$.
+
+With an optimal reconstruction $p$ in hand, the target functional is approximated via $\tau_{\tilde{x}} (\phi) \approx \tau^h_{\tilde{x}} (\phi) := \tau_{\tilde{x}} (p)$.
+
+As an unconstrained $\ell_2$-optimization problem, this process admits the explicit form:
+
+
+$$\tau^h_{\tilde{x}}(\phi) = \tau_{\tilde{x}}(\mathbf{P})^\intercal \left(\Lambda(\mathbf{P})^\intercal \mathbf{W} \Lambda(\mathbf{P})\right)^{-1} \Lambda(\mathbf{P})^\intercal \mathbf{W} \Lambda(\phi),$$
+
+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,
+* $\mathbf{W} \in \mathbb{R}^{N \times N}$ is a diagonal matrix with diagonal entries consisting of $\\{\omega(\lambda_i,\tau_{\tilde{x}})\\}_{i=1,...,N}$,
+* $\Lambda(\mathbf{P}) \in \mathbb{R}^{N \times dim(V_h)}$ is a rectangular matrix whose $(i,j)$ entry corresponds to the application of the $i^{th}$ sampling functional applied to the $j^{th}$ basis function,
+* and $\Lambda(\phi) \in \mathbb{R}^N$ is a vector consisting of the $N$ samples of the function $\phi$.
+
+Compadre forms and solves the GMLS problem for $\\{\alpha_i\\}$ used in the approximation $\tau^h_{\tilde{x}}(\phi) = \sum_{\mathbf{x}_i \in B^\epsilon(\tilde{x})} \alpha_i \lambda_i(\phi)$,
+where $B^\epsilon(\tilde{x})$ denotes the $\epsilon$-ball neighborhood of the target site $\tilde{x}$. 
+
+As such, GMLS admits an interpretation as an automated process for generating generalized finite difference methods on unstructured point clouds. Note that the computational cost of solving the GMLS problem amounts to inverting a small linear system which may be assembled using only information from neighbors within the support of $\omega$, and construction of such stencils across the entire domain is embarrassingly parallel.
+
+The Compadre Toolkit is designed to efficiently assemble, factorize, and solve large batches of GMLS problems.
+
 ## Wiki Information
 Details about building and using the Compadre toolkit can be found on the [Wiki](https://github.com/sandialabs/compadre/wiki).