software/Nstrophy
1
Fork 0
Code Issues Pull Requests Releases Activity

Rewrite cost function for adaptive step

Browse Source
This commit is contained in:
Ian Jauslin
2024-11-18 17:09:17 -05:00
parent 9fa10c8db4
commit d81ad18618
6 changed files with 211 additions and 264 deletions
Download Patch File Download Diff File Expand all files Collapse all files

85
docs/nstrophy_doc.tex
View File

@ -115,18 +115,18 @@ In addition, several variable step methods are implemented:
\end{itemize}
In these adaptive step methods, two steps are computed at different orders: $\hat u_k^{(n)}$ and $\hat U_k^{(n)}$, the step size is adjusted at every step in such a way that the error is small enough:
\begin{equation}
\|\hat u^{(n)}-\hat U^{(n)}\|
D(\hat u^{(n)},\hat U^{(n)})
<\epsilon_{\mathrm{target}}
\end{equation}
for some given $\epsilon_{\mathrm{target}}$, set using the {\tt adaptive\_tolerance} parameter.
The choice of the norm matters, and will be discussed below.
for some given {\it cost function} $D$, and $\epsilon_{\mathrm{target}}$, set using the {\tt adaptive\_tolerance} parameter.
The choice of the cost function matters, and will be discussed below.
If the error is larger than the target, then the step size is decreased.
How this is done depends on the order of algorithm.
If the order is $q$ (here we mean the smaller of the two orders, so 4 for {\tt RKDP54} and {\tt RKF45} and 2 for {\tt RKBS32}), then we expect
If the order is $q$ (here we mean the smaller of the two orders, so 4 for {\tt RKDP54} and {\tt RKF45} and 2 for {\tt RKBS32}), then we expect (as long as the cost function is such that $D(\hat u,\hat u+\varphi)\sim\|\varphi\|$ in some norm)
\begin{equation}
\|\hat u^{(n)}-\hat U^{(n)}\|=\delta_n^qC_n
.
D(\hat u^{(n)},\hat U^{(n)})=\delta_n^qC_n
\end{equation}
for some number $C_n$.
We wish to set $\delta_{n+1}$ so that
\begin{equation}
\delta_{n+1}^qC_n=\epsilon_{\mathrm{target}}
@ -135,7 +135,7 @@ so
\begin{equation}
\delta_{n+1}
=\left(\frac{\epsilon_{\mathrm{target}}}{C_n}\right)^{\frac1q}
=\delta_n\left(\frac{\epsilon_{\mathrm{target}}}{\|\hat u^{(n)}-\hat U^{(n)}\|}\right)^{\frac1q}
=\delta_n\left(\frac{\epsilon_{\mathrm{target}}}{D(\hat u^{(n)},\hat U^{(n)})}\right)^{\frac1q}
.
\label{adaptive_delta}
\end{equation}
@ -145,59 +145,49 @@ To be safe, we also set a maximal value for $\delta$ via the {\tt max\_delta} pa
\bigskip
\indent
The choice of the norm $\|\cdot\|$ matters.
It can be made by specifying the parameter {\tt adaptive\_norm}.
The choice of the cost function $D$ matters.
It can be made by specifying the parameter {\tt adaptive\_cost}.
\begin{itemize}
\item A naive choice is to take $\|\cdot\|$ to be the normalized $L_1$ norm:
\item
For computations where the main focus is the enstrophy\-~(\ref{enstrophy}), one may want to set the cost function to the relative difference of the enstrophies:
\begin{equation}
\|f\|:=
\frac1{\mathcal N}\sum_k|f_k|
,\quad
\mathcal N:=\sum_k|\hat u_k^{(n)}-\hat u_k^{(n-1)}|
D(\hat u,\hat U):=\frac{|\mathcal En(\hat u)-\mathcal En(\hat U)|}{\mathcal En(\hat u)}
.
\end{equation}
This norm is selected by choosing {\tt adaptive\_norm=L1}.
This cost function is selected by choosing {\tt adaptive\_cost=enstrophy}.
\item Empirically, we have found that $|\hat u-\hat U|$ behaves like $k^{-3}$ for {\tt RKDP54} and {\tt RKF45}, and like $k^{-\frac32}$ for {\tt RKBS32}, so a norm of the form
\item
For computations where the main focus is the value of $\alpha$\-~(\ref{alpha}), one may want to set the cost function to the relative difference of $\alpha$:
\begin{equation}
\|f\|:=\frac1{\mathcal N}\sum_k|f_k|k^{-3}
D(\hat u,\hat U):=\frac{|\alpha(\hat u)-\alpha(\hat U)|}{|\alpha(\hat u)|}
.
\end{equation}
This cost function is selected by choosing {\tt adaptive\_cost=alpha}.
\item Alternatively, one my take $D$ to be the normalized $L_1$ norm:
\begin{equation}
D(\hat u,\hat U):=
\frac1{\mathcal N}\sum_k|\hat u_k-\hat U_k|
,\quad
\mathcal N:=\sum_k|\hat u_k^{(n)}-\hat u_k^{(n-1)}|k^{-3}
\mathcal N:=\sum_k|\hat u_k|
.
\end{equation}
This function is selected by choosing {\tt adaptive\_cost=L1}.
\item Empirically, we have found that $|\hat u-\hat U|$ behaves like $k^{-3}$ for {\tt RKDP54} and {\tt RKF45}, and like $k^{-\frac32}$ for {\tt RKBS32}, so a cost function of the form
\begin{equation}
D(\hat u,\hat U):=\frac1{\mathcal N}\sum_k|\hat u_k-\hat U_k|k^{-3}
,\quad
\mathcal N:=\sum_k|\hat u_k|k^{-3}
\end{equation}
or
\begin{equation}
\|f\|:=\frac1{\mathcal N}\sum_k|f_k|k^{-\frac32}
D(\hat u,\hat U):=\frac1{\mathcal N}\sum_k|\hat u_k-\hat U_k|k^{-\frac32}
,\quad
\mathcal N:=\sum_k|\hat u_k^{(n)}-\hat u_k^{(n-1)}|k^{-\frac32}
\mathcal N:=\sum_k|\hat u_k|k^{-\frac32}
\end{equation}
are sensible choices.
These norms are selected by choosing {\tt adaptive\_norm=k3} and {\tt adaptive\_norm=k32} respectively.
\item
Another option is to define a norm based on the expression of the enstrophy\-~(\ref{enstrophy}):
\begin{equation}
\|f\|:=\frac1{\mathcal N}\sqrt{\sum_k k^2|f_k|^2}
,\quad
\mathcal N:=\frac{\sqrt{\sum_k k^2|\hat u_k^{(n)}|^2}+\sqrt{\sum_k k^2|\hat U_k^{(n)}|^2}}{\sum_k k^2|\hat u_k^{(n)}|^2}
.
\end{equation}
Doing so controls the error of the enstrophy through
\begin{equation}
\frac1{\mathcal N^2}|\mathcal En(\hat u)-\mathcal En(\hat U)|\equiv|\|\hat u\|^2-\|\hat U\|^2|\leqslant \|\hat u-\hat U\|(\|\hat u\|+\|\hat U\|)
\end{equation}
so
\begin{equation}
\frac1{\mathcal N^2}
|\mathcal En(\hat u)-\mathcal En(\hat U)|\leqslant
\|\hat u-\hat U\|\frac1{\mathcal N}\left(\sqrt{\sum_k k^2|\hat u_k|^2}+\sqrt{\sum_k k^2|\hat U_k|^2}\right)
\end{equation}
and thus
\begin{equation}
\frac{|\mathcal En(\hat u)-\mathcal En(\hat U)|}{\mathcal En(\hat u)}\leqslant
\|\hat u-\hat U\|
.
\end{equation}
This norm is selected by choosing {\tt adaptive\_norm=enstrophy}.
These cost functions are selected by choosing {\tt adaptive\_cost=k3} and {\tt adaptive\_cost=k32} respectively.
\end{itemize}
\subsubsection{Reality}.
@ -511,6 +501,7 @@ and so
\alpha(\hat u)
=\frac{\frac{L^2}{4\pi^2}\sum_k k^2\hat u_k^*\hat g_k+\sum_k|k|\hat u_k^*T(\hat u,k)}{\sum_kk^4|\hat u_k|^2}
.
\label{alpha}
\end{equation}
Note that, by\-~(\ref{realu})-(\ref{realT}),
\begin{equation}