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playing with shapes called superquadrics

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Kohlbrecher committed Nov 15, 2019
1 parent b9e5522 commit d7529f08af301aa220abda1ed2b99432af2d8226
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@@ -64,3 +64,31 @@ saskit/saskit_*
/src/fftw/fftw-3.3.8.tar.gz
/sasfit.vfs/lib/app-sasfit/tcl/CET_CLB2.rgb
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@@ -183,10 +183,11 @@ \subsection{ASCII data without error bar} \hspace{1pt}\\
This procedure gives reasonable error bars as long as the data are oversampled and do not show sharp features within the data points $i-n$ and $i+n$. A diffraction peak or a minimum in a form factor like for monodisperse particle might not reasonable well described by the polynomial and consequently the resulting error bar will get large. Furthermore systematic errors in the absolute intensity like an overall scaling factor will not be recognized. The procedure to guess the error bar is basing on the assumption that the scattering curve behaves locally approximately like a polynomial function of degree $p=2$.

\begin{figure}[htb]
\begin{center}
\subfigure[simulated model data set with supplied error bars]{\label{fig:ModelDataWithError}\includegraphics[width=0.47\textwidth]{../images/ErrorBar/ModelDataWithError.png}}
\subfigure[Ratio between supplied error bar and the error bar guessed by \SASfit]{\label{fig:ErrorGUI2}\includegraphics[width=0.47\textwidth]{../images/ErrorBar/RatioErrSuppGuess.png}}
\end{center}
\captionsetup[subfigure]{position=b}
\centering
\subcaptionbox{simulated model data set with supplied error bars \label{fig:ModelDataWithError} }{\includegraphics[width=0.47\textwidth]{../images/ErrorBar/ModelDataWithError.png}}
\hfill
\subcaptionbox{Ratio between supplied error bar and the error bar guessed by \SASfit \label{fig:ErrorGUI2} }{\includegraphics[width=0.47\textwidth]{../images/ErrorBar/RatioErrSuppGuess.png}}
\caption{Comparison of a simulated model data set with supplied error bar and an error bar guessed by \SASfit.}
\label{fig:ErrBar}
\end{figure}
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@@ -51,7 +51,7 @@ \subsection{Size distribution $N(R)$}
&= \int_0^\infty N(s) \int_0^\infty R_{av}\left(Q,\langle
Q\rangle,\sigma\right) \abs{F(Q,s)}^2 \, \mathrm{d}Q \, \mathrm{d}s \label{eq:IQ}
\end{align}
As already mentioned in eq.\ \ref{eq:KernelScaling} also here the squared form factor is scaling with the squared sample volume, which is the case of spheres proportional to $R^6$. Therefore the distribution function to be determined often is choosen to be the intensity distribution $N(R)R^\alpha$ with $\alpha=6$ so that eq.\ \ref{eq:IQ} is slightly modified to
As already mentioned in eq.\ \ref{eq:KernelScaling} also here the squared form factor is scaling with the squared sample volume, which is in the case of spheres proportional to $R^6$. Therefore the distribution function to be determined often is choosen to be the intensity distribution $N(R)R^\alpha$ with $\alpha=6$ so that eq.\ \ref{eq:IQ} is slightly modified to
\begin{align}
I_m(\langle Q\rangle) &= \int_0^\infty N(s)s^\alpha \int_0^\infty R_{av}\left(Q,\langle
Q\rangle,\sigma\right) s^{-\alpha}\abs{F(Q,s)}^2 \, \mathrm{d}Q \, \mathrm{d}s
@@ -305,13 +305,13 @@ \subsection{EM with linear and non-linear smoothing operator}~\\

\begin{figure}[htb]
\centering
\includegraphics[width=0.63\textwidth]{../images/form_factor/EM/EMsmoothLcurves.png}
\includegraphics[width=0.65\textwidth]{../images/form_factor/EM/EMsmoothLcurves.png}
\caption{Optimum smoothing parameter $h$ given by the corner of the L-curve.\label{fig:EMhLcurve}}
\end{figure}

\begin{figure}[htb]
\centering
\includegraphics[width=0.56\textwidth]{../images/form_factor/EM/EMsmoothNRR3.png}
\includegraphics[width=0.6\textwidth]{../images/form_factor/EM/EMsmoothNRR3.png}
\caption{The smoothing parameters has been obtained by the corners of the L-curves in fig.\ \ref{fig:EMhLcurve} and the solution of the EM iteration scheme with double smoothing is compared with the exact solution.\label{fig:EMsmoothNRR3}}
\end{figure}

@@ -339,24 +339,25 @@ \subsection{EM with maximum entropy regularization}~\\
\begin{align}
\label{eq:LucyRichardsonInversionMethodMEadaptive}
x_j^{(k+1)} &= x_j^{(k)} + \Delta x_j^{(k)} + \Delta s_j^{(k)}= \mathcal{O}_\mathrm{EM,ME,adaptive}\left[x_j^{(k)}\right]\\
\Delta x_j^{(k)} &= x_j^{(k)}\left[\sum_{i=0}^{M-1}\frac{A_{ij}}{\left(\displaystyle \sum_{m=0}^{M-1}A_{mj}\right)}\frac{b_i}{\displaystyle \sum_{n=0}^{N-1}A_{in}x_n^{(k)}}-1\right] \\
\Delta x_j^{(k)} &= x_j^{(k)}\left[\sum_{i=0}^{M-1}\frac{A_{ij}}{\left( \sum_{m=0}^{M-1}A_{mj}\right)}\frac{b_i}{ \sum_{n=0}^{N-1}A_{in}x_n^{(k)}}-1\right] \\
\Delta s_j^{(k)} &= \lambda x_j^{(k)}\left(-\ln\frac{x_j}{m_j^{(k)}}
+\frac{1}{\sum_{j=0}^{N-1}x_j^{(k)}}\sum_{j=0}^{N-1}x_j^{(k)}\ln\frac{x_j^{(k)}}{m_j^{(k)}}
-1+\sum_{j=0}^{N-1}\Pi_{ij}\frac{x_i^{(k)}}{m_i^{(k)}}\right)
-1+\sum_{i=0}^{N-1}\Pi_{ij}\frac{x_i^{(k)}}{m_i^{(k)}}\right) \\
m_i^{(k)} &= \sum_{j=0}^{N-1}\Pi_{ij}x_j^{(k)}
\end{align}
As a smoothing operator for the prior
\begin{align}\label{eq:smoothingOp4prior}
\Pi_{ij} &= \frac{1}{c}\exp\left(-\frac{(i-j)^2}{2\sigma^2}\right)
\end{align}
has been assumed with the normalization constant $c=\sum_j \Pi_{ij}$ and a $\sigma^2$ typically between $\frac12$ and 2. For large values of $\sigma^2$ the results converge to that one of a constant prior. For too small values the regularization is suppressed as the prior converges to the $k^\mathrm{th}$ iteration step.

The optimum regularization parameter $\lambda$ can be determined similar to the previous section via the corner in the L-curve. The L-curves for both an constant and adaptive prior are shown in fig.\ \ref{fig:EMlambdaLcurve} using the same data than shown in fig.\ \ref{fig:EMIQevolution}. The optimum Lagrange parameter in the corner are marked with a symbol. The corresponding size distribution are shown in fig.\ \ref{fig:EMconstadaptiveNRR3} together wit the exact solution and for an Lagrange parameter leading to a $\chi_r^2=1$. From the L-curves in fig.\ \ref{fig:EMlambdaLcurve} one can see, that the $\mathcal{G}$-test parameter for the constant prior varies over orders of magnitudes whereas for the adaptive prior also for a Lagrange parameter far away from its optimum value leads to a reasonable good result. Therefore the L-curve shows for the adaptive prior a much smaller range on the $x$-axis.
The optimum regularization parameter $\lambda$ can be determined similar to the previous section via the corner in the L-curve. The L-curves for both a constant and adaptive prior are shown in fig.\ \ref{fig:EMlambdaLcurve} using the same data than shown in fig.\ \ref{fig:EMIQevolution}. The optimum Lagrange parameter in the corner are marked with a symbol. The corresponding size distribution are shown in fig.\ \ref{fig:EMconstadaptiveNRR3} together wit the exact solution and for a Lagrange parameter leading to $\chi_r^2=1$. From the L-curves in fig.\ \ref{fig:EMlambdaLcurve} one can see, that the $\mathcal{G}$-test parameter for the constant prior varies over orders of magnitudes whereas for the adaptive prior also for a Lagrange parameter far away from its optimum value leads to a reasonable good result. Therefore the L-curve shows for the adaptive prior a much smaller range on the $x$-axis.

\begin{figure}[htb]
\centering
\includegraphics[width=0.42\textwidth]{../images/form_factor/EM/EMconstLcurve.png} \hspace{0.06\textwidth}
\includegraphics[width=0.42\textwidth]{../images/form_factor/EM/EMadaptiveLcurve.png}
\caption{Determine optimum smoothing parameter $\lambda$ by the corner of the L-curve, which is determined by a minimum radius of curvature.\label{fig:EMlambdaLcurve}}
\caption{Determine optimum smoothing parameter $\lambda$ by the corner of the L-curve, which is determined by a minimum radius of curvature. (left) constant prior, (right) adaptive prior\label{fig:EMlambdaLcurve}}
\end{figure}

\begin{figure}[htb]
@@ -395,7 +396,7 @@ \subsection{Acceleration of EM iteration scheme} ~\\
with $\lambda^{\max} \simeq 4$ so that $1\leq \lambda^{(k)}\leq \lambda^{\max}$.

\begin{algorithm}[bht] \footnotesize
\caption{Biggs-Andersen fixed point acceleration of order $o=0,1,2$ }\label{algorithm:Biggs_Andersen}
\caption{Biggs-Andrews vector extrapolation acceleration of order $o=0,1,2$ }\label{algorithm:Biggs_Andrews}
\begin{algorithmic}[1]
\Procedure{update\_vectors}{$\mathbf{g}^{(n-2)}$,$\mathbf{g}^{(n-1)}$,$\mathbf{x}^{(n+1)}$,$\mathbf{x}^{(n)}$,$\mathbf{x}^{(n-1)}$,$\mathbf{x}^{(n-2)}$}
\State $\mathbf{g}^{(n-2)} \gets \mathbf{g}^{(n-1)}$
@@ -405,7 +406,7 @@ \subsection{Acceleration of EM iteration scheme} ~\\
\State $\mathbf{x}^{(n)} \gets \mathbf{x}^{(n+1)}$
\EndProcedure
\State \Comment Acceleration of the fixed point operator $\mathcal{O}[\mathbf{x}]=\mathbf{x}$
\Procedure{Biggs\_Andersen\_acceleration}{$\mathcal{O}[]$, $\mathbf{x}^{(0)}$, o}
\Procedure{Biggs\_Andrews\_acceleration}{$\mathcal{O}[]$, $\mathbf{x}^{(0)}$, o}
\State $\alpha \gets 0$
\State $\mathbf{x}^{(n)} \gets \mathbf{x}^{(0)}$
\State $\mathbf{x}^{(n+1)} \gets \mathcal{O}[\mathbf{x}^{(n)}]$
@@ -437,7 +438,7 @@ \subsection{Acceleration of EM iteration scheme} ~\\

Another more efficient method to accelerate the convergence rate of the EM iteration scheme is the so called Anderson acceleration \cite{Anderson1965} to solve fixed point problems (see also \cite{Walker2011,Toth2015}), which also has been suggested for the EM iteration scheme \cite{Henderson2019}. In \SASfit several fixed point accelerations are supplied by making use of the sundials library \cite{Hindmarsh2005}. However, only the supplies Anderson acceleration speeds up the fixed point iterations whereas all the others like GMRES \cite{Saad1986}, FGMRES \cite{Saad1993}, Bi-CGStab \cite{Vorst1992} or TFQMR \cite{Freund1993} become instable and do not converge.

In \cite{Biggs1997, Biggs1995} an acceleration process has been suggested basing on vector extrapolation which does not require an extra evaluation of the fixed point operator $\mathcal{O}_\mathrm{EM}$ in eq.\ \ref{eq:LucyRichardsonInversionMethod}. The method just needs to remember one or two previous solution vectors for calculating either the first or additionally the second derivative for guessing the next virtual solution vector, which is then used as an input vector for the next step of the fixed point iteration. The algorithm is described in algorithm \ref{algorithm:Biggs_Andersen}. The convergence behaviour of the different acceleration schemes are shown in fig.\ \ref{fig:EMconvergenceSpeed}. Quite good acceleration has been achieved by the Biggs-Andersen method as well as Anderson acceleration, whereas other classical methods like GMRES, Bi-CGSab or TFQMR failed. The Picard iteration scheme and faster overrelaxation method need orders of magnitudes more iterations to reach the true fixed point solution. The preferred acceleration scheme therefore becomes the Biggs-Andersen methods.
In \cite{Biggs1997, Biggs1995, Biggs1998, Wang2014,Jiang2017} an acceleration process has been suggested basing on vector extrapolation which does not require an extra evaluation of the fixed point operator $\mathcal{O}_\mathrm{EM}$ in eq.\ \ref{eq:LucyRichardsonInversionMethod}. The method just needs to remember one or two previous solution vectors for calculating either the first or additionally the second derivative for guessing the next virtual solution vector, which is then used as an input vector for the next step of the fixed point iteration. The algorithm is described in algorithm \ref{algorithm:Biggs_Andrews}. The convergence behaviour of the different acceleration schemes are shown in fig.\ \ref{fig:EMconvergenceSpeed}. Quite good acceleration has been achieved by the Biggs-Andrews method as well as Anderson acceleration, whereas other classical methods like GMRES, Bi-CGSab or TFQMR failed. The Picard iteration scheme and faster overrelaxation method need orders of magnitudes more iterations to reach the true fixed point solution. The preferred acceleration scheme therefore becomes the Biggs-Andrews methods.


\begin{figure}[htb]
@@ -462,10 +463,10 @@ \section{Linear least square regularization (LLS)}
\begin{align}
\hat{\mathbf{w}} &=
\begin{pmatrix}
\frac{1}{\Delta b_0^2} & 0 & \dots & 0 \\
0 & \frac{1}{\Delta b_1^2} & \dots & 0 \\
\frac{1}{\Delta b_0} & 0 & \dots & 0 \\
0 & \frac{1}{\Delta b_1} & \dots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \dots & \frac{1}{\Delta b_{M-1}^2}
0 & 0 & \dots & \frac{1}{\Delta b_{M-1}}
\end{pmatrix}
\end{align}
where $\Delta b_i$ are the experimental errors (standard deviation, Gaussian error or root-mean-square error) of the measured quantities $b_i$.
@@ -118,7 +118,7 @@ \section{Fitting absolute intensities}
$N$ in the size distribution is interpreted in terms of number
density of scatterers.

\section{Contrast - Concentration - Forward Scattering - Particle Volume - Absolute Scale}
\section{Forward Scattering - Volume Fraction - Absolute Scale}
\label{sec:contrast-concentration-I(0)}

A frequently asked question is if the scattering intensity is consistent
@@ -267,15 +267,15 @@ \section{Moments of scattering curves and size distribution}


\begin{figure}[htb]
\begin{center}
\subfigure[GUI for defining fit ranges for Porod and Guinier
approximations]{\includegraphics[width=0.296\textwidth]{QTintegralparameterGUI.png}}
\hfill \subfigure[Tabbed menu displaying the integral structural
\captionsetup[subfigure]{position=b}
\centering
\subcaptionbox{GUI for defining fit ranges for Porod and Guinier
approximations \label{fig:QTintegralstructuralparameters1} }{\includegraphics[width=0.296\textwidth]{QTintegralparameterGUI.png}}
\hfill
\subcaptionbox{Tabbed menu displaying the integral structural
parameters calculated via the different moments of the scattering
curve the Porod and Guinier extrapolations to $Q\rightarrow 0$ and
$Q\rightarrow\infty$.
]{\includegraphics[width=0.604\textwidth]{QTintegralparameterTab.png}}
\end{center}
$Q\rightarrow\infty$. \label{fig:QTintegralstructuralparameters2} }{\includegraphics[width=0.604\textwidth]{QTintegralparameterTab.png}}
\caption{Menu and tabbed window for integral structural parameters.
\SASfit also supports analysis of series of data, whereby the
structural parameters are stored in CSV format readable by many
@@ -214,9 +214,9 @@ \subsection{KholodenkoWorm}
\label{fig:KholodenkoWorm}
\end{figure}

Kholodenko \cite{kholodenko93} presented a new approach using the analogy between Dirac’s fermions
Kholodenko \cite{kholodenko93} presented a new approach using the analogy between Dirac's fermions
and semi-flexible polymers. The form factor $P_0(Q)$ resulting
from Kholodenko’s approach is designed to reproduce
from Kholodenko's approach is designed to reproduce
correctly the rigid-rod limit and the random-coil limit.
Defining $x = 3L/l$ ($L$: contour length, $l$: Kuhn length), it is given by
\begin{align}
@@ -278,14 +278,27 @@ \subsection{Diblock copolymer micelles}
~\\
\begin{figure}[htb]
\centering
\subfigure[spherical
micelle]{\quad\includegraphics[width=0.25\textwidth]{spheregauss.png}\quad}
\quad
\subfigure[ellipsoidal
micelle]{\quad\includegraphics[width=0.25\textwidth]{ellipsoidgauss.png}\quad}
\quad