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b/Lectures_my/NumMet/2016/Lecture12/images/fig-gauss-newton.pdf new file mode 100644 index 0000000..7e1c79e --- /dev/null +++ b/Lectures_my/NumMet/2016/Lecture12/images/fig-gauss-newton.pdf Binary files differ diff --git a/Lectures_my/NumMet/2016/Lecture12/images/fig-levenberg-marquardt.pdf b/Lectures_my/NumMet/2016/Lecture12/images/fig-levenberg-marquardt.pdf new file mode 100644 index 0000000..eb79365 --- /dev/null +++ b/Lectures_my/NumMet/2016/Lecture12/images/fig-levenberg-marquardt.pdf Binary files differ diff --git a/Lectures_my/NumMet/2016/Lecture12/lecture12.tex b/Lectures_my/NumMet/2016/Lecture12/lecture12.tex index 80a3ff3..2fce60f 100644 --- a/Lectures_my/NumMet/2016/Lecture12/lecture12.tex +++ b/Lectures_my/NumMet/2016/Lecture12/lecture12.tex @@ -70,7 +70,7 @@ \vspace{1em} \vspace{0.5em} - \textcolor{normal text.fg!50!Comment}{Numerical Methods, \\ 07. November, 2016} + \textcolor{normal text.fg!50!Comment}{Numerical Methods, \\ 21. November, 2016} \end{center} \end{frame} } @@ -79,9 +79,8 @@ \begin{itemize} \item \alert{What types of function do we usually need to minimize?}\\ \begin{itemize} - \item least-squares - \item likelihood/posterior \item general non-linear problem + \item least-squares \end{itemize} \vfill \item \alert{What information can we use?}\\ @@ -111,10 +110,27 @@ \begin{equation*} A = \lbrace (a_1, \dots, a_N)\,|\,g(a_1, \dots, a_N) > 0\rbrace\,. \end{equation*} + In order to find maxima, flip the function around: $f \to -f$. \end{frame} \begin{frame}{Example} - Posterior with non-gaussian likelihood. For example, exponential likelihood. + Posterior probability density function with non-gaussian likelihood. + + \begin{columns} + \begin{column}{.6\textwidth} + \includegraphics[height=.8\textheight]{fig-bclvub-comparison-0-1} + \hfill {\small taken from 1409.7186} + \end{column} + \begin{column}{.4\textwidth} + \begin{itemize} + \item shallow gradients in one direction + \item step gradients in the other + \item non-symmetric shape + \item slight ``banana'' qualities\\ + (slightly bent in along one axis) + \end{itemize} + \end{column} + \end{columns} \end{frame} \begin{frame}{Limitations and Monte-Carlo Methods} @@ -124,10 +140,12 @@ A very popular way to explore $f$ is by using Monte Carlo methods, e.g. plain random walks, Markov chain methods, or genetic algorithms. Most of these methds are very good at delineating \emph{local environments} - around some/most/all(?) of the modes. As always, analytic knowledge + around some/most/all(?) of the modes.\\ + \vfill + As always, analytic knowledge of the problem will help. For example, symmetry relations among the parameters - or (a)periodic boundary conditions should be exploited if possible. - + or (a)periodic boundary conditions should be exploited if possible.\\ + \vfill The specific Monte-Carlo methods are beyond the scope of these lectures. \end{frame} @@ -138,24 +156,24 @@ \vfill The basic idea is as follows: \begin{itemize} - \item[0] A set of parameter points must be provided as an initial + \item<1->[0] A set of parameter points must be provided as an initial value. If all else fails, random points can be used. - \item[1] Order the points by their respetive size of $f$ - \item[2] Compute the midpoint $\vec{a}_\text{mid}$ of all points except the worst, + \item<2->[1] Order the points by their respetive size of $f$ + \item<3->[2] Compute the midpoint $\vec{a}_\text{mid}$ of all points except the worst, and reflect the worst point $\vec{a}_{N+1}$ at the midpoint yielding $\vec{a}_\text{ref}$. \begin{itemize} - \item[a] If $f(\vec{a}_\text{ref})$ fulfills some minimization criteria, replace + \item<4->[a] If $f(\vec{a}_\text{ref})$ fulfills some minimization criteria, replace one of the existing simplex points with $\vec{a}_\text{ref}$. Continue at step 1. - \item[b] Otherwise compute $\vec{a}_\text{contr}$ as a linear combination of the worst + \item<5->[b] Otherwise compute $\vec{a}_\text{contr}$ as a linear combination of the worst point of the simplex $\vec{a}_N$, and $\vec{a}_\text{ref}$. If $\vec{a}_\text{contr}$ is better than the worst point, replace the worst point. Continue at step 1. - \item[c] Otherwise compress the the simplex by moving the points + \item<6->[c] Otherwise compress the the simplex by moving the points $\vec{x}_1$ to $\vec{x}_N$ closer to $\vec{x}_0$ on their respective connecting lines. Continue at step 1. \end{itemize} - \item[3] If at any point the volume of the simplex falls below a given + \item<7->[3] If at any point the volume of the simplex falls below a given treshold, then stop. \end{itemize} \end{frame} @@ -170,7 +188,7 @@ \begin{equation*} \vec{a}_\text{mid} = \frac{1}{N} \sum_{n=0}^{N - 1} \vec{a}_n \end{equation*} - The reflection is computed as\hfill~[default: $\alpha = 1$] + The reflection is computed as\hfill~\alert{[default: $\alpha = 1$]} \begin{equation*} \vec{a}_\text{ref} = (1 + \alpha) \vec{a}_\text{mid} - \alpha \vec{a}_N \end{equation*} @@ -201,7 +219,7 @@ \only<5-6>{ \item[2c] Compress the entire simplex\hfill~\alert{[default: $\kappa = 1/2$]} \begin{equation*} - \vec{a}_n = \sigma \vec{a}_0 + (1 - \sigma) \vec{a}_n\qquad \forall n=1,\dots,N + \vec{a}_n = \kappa \vec{a}_0 + (1 - \kappa) \vec{a}_n\qquad \forall n=1,\dots,N \end{equation*} } \only<6>{ @@ -226,18 +244,129 @@ \end{frame} \begin{frame}{Example} + \begin{equation*} + f(x, y) = x^2 + y^2 + \end{equation*} + \begin{columns}[T] + \begin{column}{.55\textwidth} + \resizebox{1.10\textwidth}{!}{ + \begin{tikzpicture} + \begin{axis}[ + xmin=-2,xmax=+2, + ymin=-2,ymax=+2, + grid=both, + axis equal + ] + \addplot [domain=0:360,samples=100] + ({cos(x)}, {sin(x)}); + \addplot [domain=0:360,samples=100] + ({sqrt(2) * cos(x)}, {sqrt(2) * sin(x)}); + \addplot [domain=0:360,samples=100] + ({sqrt(3) * cos(x)}, {sqrt(3) * sin(x)}); + \addplot [domain=0:360,samples=100] + ({sqrt(4) * cos(x)}, {sqrt(4) * sin(x)}); + \addplot [domain=0:360,samples=100] + ({sqrt(5) * cos(x)}, {sqrt(5) * sin(x)}); + + \only<1-2>{ + \draw[red,thick,mark=none,fill=red,fill opacity=0.5] + (axis cs: +1,-1) -- (axis cs: 1,+1) -- (axis cs: +2,+1) -- cycle; + } + \only<2>{ + \addplot[black,mark=x,mark size=3pt,thick] coordinates { + (+1,+0) + (+0,-1) + }; + \addplot[black,mark=none,thick,opacity=0.5] coordinates { + (+2,+1) + (+1,+0) + (+0,-1) + }; + } + \only<3-5>{ + \draw[red,thick,mark=none,fill=red,fill opacity=0.5] + (axis cs: 0, -1) -- (axis cs: +1,-1) -- (axis cs: 1,+1) -- cycle; + } + \only<4>{ + \addplot[black,mark=x,mark size=3pt,thick] coordinates { + (+0.5,-1) + (+0.0,-3) + }; + \addplot[black,mark=none,thick,opacity=0.5] coordinates { + (+1.0,+1) + (+0.5,-1) + (+0.0,-3) + }; + } + \only<5>{ + \addplot[black,mark=x,mark size=3pt,thick] coordinates { + (+0.50,-1) + (+0.75,-0) + }; + \addplot[black,mark=none,thick,opacity=0.5] coordinates { + (+1.00,+1) + (+0.75,-0) + (+0.50,-1) + }; + } + \only<6>{ + \draw[red,thick,mark=none,fill=red,fill opacity=0.5] + (axis cs: 0.75, 0) -- (axis cs: 0, -1) -- (axis cs: +1,-1) -- cycle; + } + \only<7>{ + \draw[red,thick,mark=none,fill=red,fill opacity=0.5] + (axis cs: -0.3125, 0.25) -- (axis cs: 0.75, 0) -- (axis cs: 0, -1) -- cycle; + } + \end{axis} + \end{tikzpicture} + } + \end{column} + \begin{column}{.45\textwidth} + \begin{overlayarea}{\textwidth}{.3\textheight} + \only<1-2>{ + \begin{align*} + a_0 & = (+1, -1) & f(a_0) & = 2\\ + a_1 & = (+1, +1) & f(a_1) & = 2\\ + \only<2>{\color{red}} + a_2 & = (+2, +1) & f(a_2) & = 5 + \end{align*} + } + \only<3-4>{ + \begin{align*} + a_0 & = (+0, -1) & f(a_0) & = 1\\ + a_1 & = (+1, -1) & f(a_1) & = 2\\ + \only<4>{\color{red}} + a_2 & = (+1, +1) & f(a_2) & = 2 + \end{align*} + } + \end{overlayarea} + \begin{overlayarea}{\textwidth}{.3\textheight} + \only<2>{ + \begin{align*} + a_\text{mid} + & = (+1, 0) & \\ + a_\text{ref} + & = (0, -1) & f(a_\text{ref}) & = 1 + \end{align*} + } + \only<4>{ + \begin{align*} + a_\text{mid} + & = (+1/2,-1) & \\ + a_\text{ref} + & = (0, -3) & f(a_\text{ref}) & = 9 + \end{align*} + } + \end{overlayarea} + \end{column} + \end{columns} \end{frame} \begin{frame}{Least-squares Problems} - The least-squares problem arises from the case of a Gaussian - likelihood function if all uncertainties are equal. It is a - very good example for understanding a non-linear problem - through linearization.\\ - \vfill The target function is called the residue $r(a_1, \dots, a_N)$ with $N$ parameters. \begin{equation*} - r_k \equiv f(\vec{a}, \vec{x}_k) - y_k + r_k \equiv y(\vec{a}, \vec{x}_k) - y_k \end{equation*} \vfill Here $k$ denotes one of the $K$ possible coordinate on a curve, @@ -245,8 +374,13 @@ \vfill The problem now aims to minimize \begin{equation*} - \sum_k^K |r_k|^2 + f(\vec{a}) \equiv \sum_k^K |r_k|^2 \end{equation*} + \vfill + The least-squares problem arises from the case of a Gaussian + likelihood function if all uncertainties are equal. It is a + very good example for understanding a non-linear problem + through linearization. \end{frame} \begin{frame}{Gauss-Newton Method} @@ -258,7 +392,7 @@ \vfill The algorithm now involves the following quantities: \begin{align*} - D & = \left(\begin{matrix} + J & = \left(\begin{matrix} r'_{1,1} & r'_{1,2} & \dots & r'_{1,n}\\ r'_{2,1} & r'_{2,2} & \dots & r'_{2,n}\\ \vdots & \vdots & \ddots & \vdots \\ @@ -267,6 +401,7 @@ \vec{r} & = (r_1, \dots, r_k)^T \end{align*} + Here $J$ is the Jacobi matrix of the residue. \end{frame} \begin{frame}{Gauss-Newton Method (cont'd)} @@ -274,7 +409,7 @@ \item[0] As always, we will be requiring a starting point $\vec{a}_0$ in parameter space.\\ \item[1] Update the current point: \begin{equation*} - \vec{a}_{i + 1} = \vec{a}_i - (D^T\cdot D)^{-1} \cdot D \cdot \vec{r} + \vec{a}_{i + 1} = \vec{a}_i - (J^T\cdot J)^{-1} \cdot J \cdot \vec{r} \end{equation*} \item[2] If $||\vec{a}_{i + 1} - \vec{a}_i|| < T$, where $T$ is some a-prior threshold, we stop. Otherwise, continue with step 1. @@ -283,32 +418,52 @@ \begin{itemize} \item The literature usually recommends to compute the auxiliary variable $\vec{s}$ via: \begin{equation*} - (D^T \cdot D) \cdot \vec{s} = D^T \cdot \vec{r} + (J^T \cdot J) \cdot \vec{s} = J^T \cdot \vec{r} \end{equation*} The above linear system of equations can be solved with known methods. - \item Since $(D^T \cdot D)$ is symmetric, it is a good idea to use Cholesky decomposition. + \item Since $(J^T \cdot J)$ is symmetric, it is a good idea to use Cholesky decomposition. \end{itemize} \end{frame} \begin{frame}{Derivation} - \url{https://en.wikipedia.org/wiki/Gauss-Newton_algorithm} + Why does this work? Necessary and sufficient conditions for an optimum are: + \begin{equation*} + \vec{g} \equiv \frac{\partial f(\vec{a})}{\partial a_n} \overset{!}{=} 0 \qquad\text{and}\qquad\det{h} \overset{!}{\neq} 0\qquad\text{with} \quad h \equiv\frac{\partial^2 f}{\partial a_{n}\,\partial a_{n'}} + \end{equation*} + One could therefore use Newton's method to find the zeros of the gradient: + \begin{equation*} + \vec{a}_{i + 1} = \vec{a}_i - (h)^{-1} \vec{g} + \end{equation*} + Express $\vec{g}$ and $h$ in terms of the $r'_{k,n}$: + \begin{align*} + g_n & = \sum_k^{K} 2 \frac{\partial r_k}{\partial a_n} r_k & + H_{n,n'} & = \sum_k^{K} 2 \frac{\partial^2 r_k}{\partial a_n\,\partial a_{n'}} r_k + + 2 \frac{\partial r_k}{\partial a_n} \frac{\partial r_k}{\partial a_{n'}}\\ + & = 2 J \cdot \vec{r} & + & = {\color{red}\left[\sum_{k}^K 2 \frac{\partial^2 r_k}{\partial a_n\,\partial a_{n'}}\right]} + 2 J^T \cdot J + \end{align*} + Assuming the {\color{red}second derivatives} are small compared to the square first-deriv.~term, than + we can neglect hem. \end{frame} \begin{frame}{Example} - \url{https://en.wikipedia.org/wiki/Gauss-Newton_algorithm} + \begin{equation*} + y(x, a_1, a_2) = \frac{a_1 x}{a_2 + x}\qquad\text{7 data points} + \end{equation*} + \includegraphics[height=.78\textheight]{fig-gauss-newton.pdf} \end{frame} \begin{frame}{Levenberg-Marquardt Method} The Levenberg-Marquardt method arises from a modification to the Gauss-Newton method. - - Trust region. - - Dampen: + \vfill + The adjustment length is attenuated through a dampening parameter {\color{red}$\lambda$}. + Steers adjustment away from Gauss-Newton direction to the gradient's direction. + \vfill \begin{itemize} \item[0] As always, we will be requiring a starting point $\vec{a}_0$ in parameter space.\\ \item[1] Update the current point: \begin{equation*} - \vec{a}_{i + 1} = \vec{a}_i - (D^T\cdot D {\color{red}\,+ \lambda I})^{-1} \cdot D \cdot \vec{r} + \vec{a}_{i + 1} = \vec{a}_i - (J^T\cdot J {\color{red}\,+ \lambda I})^{-1} \cdot J \cdot \vec{r} \end{equation*} where $I$ is the unit matrix in $N \times N$ {\color{red} and $\lambda$ a real-valued parameter}. \item[2] If $||\vec{a}_{i + 1} - \vec{a}_i|| < T$, where $T$ is some a-prior threshold, @@ -318,7 +473,11 @@ The optimal choice of {\color{red}$\lambda$} is specific to the problem. \end{frame} -\begin{frame} +\begin{frame}{Example} + \begin{equation*} + y(x, a_1, a_2) = \frac{a_1 x}{a_2 + x}\qquad\text{7 data points} + \end{equation*} + \includegraphics[height=.78\textheight]{fig-levenberg-marquardt.pdf} \end{frame} \backupbegin diff --git a/Lectures_my/NumMet/2016/Lecture12/levenberg-marquardt.nb b/Lectures_my/NumMet/2016/Lecture12/levenberg-marquardt.nb new file mode 100644 index 0000000..3da4c9c --- /dev/null +++ b/Lectures_my/NumMet/2016/Lecture12/levenberg-marquardt.nb @@ -0,0 +1,1155 @@ +(* Content-type: application/vnd.wolfram.mathematica *) + +(*** Wolfram Notebook File ***) +(* http://www.wolfram.com/nb *) + +(* 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