Skip to content

Commit

Permalink
docs: notation fixes
Browse files Browse the repository at this point in the history
  • Loading branch information
Theresa Pollinger committed Dec 19, 2024
1 parent ede7dd8 commit 9d03c7d
Showing 1 changed file with 7 additions and 4 deletions.
11 changes: 7 additions & 4 deletions docs/combination_technique.md
Original file line number Diff line number Diff line change
Expand Up @@ -36,23 +36,26 @@ above figure.
Here, we use the truncated combination technique, which sets a minimum and maximum
level, $\vec{\ell^\text{min}}$ and $\vec{\ell^\text{max}}$, to select the
component grids $\vec{\ell} \in \mathcal{I}$.
The set of all selected component grids $\mathcal{I}$ is referred to as the
"combination scheme".
For regularity, we assume a constant difference between $\vec{\ell^\text{min}}$
and $\vec{\ell^\text{max}}$ in each dimension, but [other schemes can be useful](./advanced_topics).
<!-- The resulting component grids will be $d$ simplex "layers" in the space of
level vectors $\vec{\ell}$. -->

One can then compute suitable combination coefficients $c^c$ that are used to
obtain a sparse grid function $f_\text{SG}$ by way of a linear superposition of
One can then compute suitable combination coefficients $c_{\vec{\ell}}^c$
that are used to obtain a sparse grid function
$f_\text{SG}$ by way of a linear superposition of
functions $f_{\vec{\ell}}$ defined on the component grid:

$f_\text{SG} = \sum_{\vec{\ell}} c_{\vec{ell}}^c\cdot f_{\vec{\ell}}$
$f_\text{SG} = \sum_{\vec{\ell} \in \mathcal{I}} c_{\vec{\ell}}^c\cdot f_{\vec{\ell}}$

In the two-dimensional combination scheme in the above figure, all combination
coefficients are 1 and -1, respectively.
The sparse grid is illustrated on the upper right.

This sparse grid function can be expected to be accurate in
$\mathcal{O}(h^2 \cdot log(h^{d-1}))$, given some assumptions on the bounds of
$\mathcal{O}(h^2 \cdot \log(h^{d-1}))$, given some assumptions on the bounds of
mixed-dimension partial derivatives, where $N$ is the number of points corresponding
to the finest resolution occurring in the scheme.
At the same time, the number of points is only $\mathcal{O}(d\cdot N(\log N)^{d-1})$,
Expand Down

0 comments on commit 9d03c7d

Please sign in to comment.