# Line-integral & Interpolation Experiment

Fundamental Theorem of Line Integrals
\begin{align}
F(\mathbf{s}_2) - F(\mathbf{s}_1) = \int_{\mathbf{s}_1}^{\mathbf{s}_2}F'(\mathbf{s}) d\mathbf{s}
\end{align}

According to our estimation of LRE
\begin{align}
F(\mathbf{s}_2) - F(\mathbf{s}_1) &\approx \mathcal{W} (\mathbf{s}_2 - \mathbf{s}_1).
\end{align}

If $u \propto F(\mathbf{s}_2) - F(\mathbf{s}_1)$ and $v \propto \mathbf{s}_2 - \mathbf{s}_1$ in the row and column space respectively, then

\begin{align}
||F(\mathbf{s}_2) - F(\mathbf{s}_1)|| &\approx u^T \mathcal{W} v ||\mathbf{s}_2 - \mathbf{s}_1 || \\
|| F(\mathbf{s}_2) - F(\mathbf{s}_1) || &= \int_{\mathbf{s}_1}^{\mathbf{s}_2} u^T F'(\mathbf{s}) d\mathbf{s}
\approx \sum_{\mathbf{s}_1}^{\mathbf{s}_2} u^T F'(\mathbf{s}) \Delta\mathbf{s}
\end{align}

\begin{align}
r = \frac{\sum_{\mathbf{s}_1}^{\mathbf{s}_2} u^T F'(\mathbf{s}) \Delta\mathbf{s}}{u^T \mathcal{W} v ||\mathbf{s}_2 - \mathbf{s}_1 ||}
\end{align}

The *hope* is that we will get $r$ close to $2.25$ allowing us the justify our $\beta$

Tried:
* Calculating $\mathcal{W}$, $u$, and $v$ on each $\mathbf{s}_i$. $\Delta\mathbf{s} = \mathbf{s}_{i+1} - \mathbf{s}_{i}$

* Calculating $\mathcal{W}$ on $\mathbf{s}_1$
* $\mathcal{W}$ from LRE (our flagship approach in the paper)
* Also tried walking along a path from $\overline{\mathbf{s}_1}$ to $\overline{\mathbf{s}_2}$. Walk along an averaged path with the hope that averaged $\overline{\mathcal{W}}$ will be less noisy.

:::info
# notebook
https://github.com/evandez/relations/blob/tables_and_figures/notebooks/interpolation_experiment.ipynb
:::