PT2 stoch OK

algorithms.org

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+#+TITLE: Important algorithms for CIPSI
+#+DATE: 14/12/2020
+#+AUTHOR: Anthony Scemama
+
+#+startup: beamer
+#+LaTeX_HEADER: \institute{Laboratoire de Chimie et Physique Quantiques, IRSAMC, UPS/CNRS, Toulouse}
+#+LATEX_CLASS: beamer
+#+LaTeX_CLASS_OPTIONS:[aspectratio=169]
+#+BEAMER_THEME: pterosor
+#+LaTeX_HEADER: \usepackage{minted}
+#+LaTeX_HEADER: \usepackage[utf8]{inputenc}
+#+LaTeX_HEADER: \usepackage[T1]{fontenc}
+#+LaTeX_HEADER: \usepackage{hyperref}
+#+LaTeX_HEADER: \usepackage{mathtools}
+#+LaTeX_HEADER: \usepackage{physics}
+#+LaTeX_HEADER: \definecolor{darkgreen}{rgb}{0.,0.6,0.}
+#+LaTeX_HEADER: \definecolor{darkblue}{rgb}{0.,0.2,0.7}
+#+LaTeX_HEADER: \definecolor{darkred}{rgb}{0.6,0.1,0.1}
+#+LaTeX_HEADER: \definecolor{darkpink}{rgb}{0.7,0.0,0.7}
+#+EXPORT_EXCLUDE_TAGS: noexport
+#+OPTIONS: H:2 num:t toc:nil \n:nil @:t ::t |:t ^:t -:t f:t *:t <:t
+#+OPTIONS: TeX:t LaTeX:t skip:nil d:nil todo:t pri:nil tags:not-in-toc
+
+#+LATEX: \newcommand{\mcenter}[1]{\multicolumn{1}{c}{#1}}
+#+LATEX: \newcommand{\Ndet}{{\textcolor{darkgreen}{N_\text{det}}}}
+#+LATEX: \newcommand{\Evar}{\textcolor{darkgreen}{E_\text{var}}}
+#+LATEX: \newcommand{\Ept}{\textcolor{red}{E_\text{PT2}}}
+#+LATEX: \newcommand{\de}{\textcolor{red}{\delta E(\alpha)}}
+#+LATEX: \newcommand{\mH}{\mathcal{H}}
+#+LATEX: \newcommand{\mDup}{{\textcolor{orange}{D_k^\uparrow}}}
+#+LATEX: \newcommand{\mDupp}{{\textcolor{orange}{D_{k'}^\uparrow}}}
+#+LATEX: \newcommand{\mDdn}{{\textcolor{orange}{D_m^\downarrow}}}
+#+LATEX: \newcommand{\mDdnn}{{\textcolor{orange}{D_{m'}^\downarrow}}}
+#+LATEX: \newcommand{\Ndetup}{{\textcolor{orange}{N_\text{det}^\uparrow}}}
+#+LATEX: \newcommand{\Ndetdn}{{\textcolor{orange}{N_\text{det}^\downarrow}}}
+#+LATEX: \newcommand{\mD}{{\textcolor{darkgreen}{\mathcal{D}}}}
+#+LATEX: \newcommand{\mPsi}{{\textcolor{darkgreen}{\Psi}}}
+#+LATEX: \newcommand{\mA}{\textcolor{red}{\mathcal{A}}}
+#+LATEX: \newcommand{\ma}{{\textcolor{red}{\alpha}} }
+#+LATEX: \newcommand{\mi}{{\textcolor{darkgreen}{I}} }
+#+LATEX: \newcommand{\mpi}{{\textcolor{darkgreen}{p_I}} }
+#+LATEX: \newcommand{\mci}{{\textcolor{darkgreen}{c_I}} }
+#+LATEX: \newcommand{\mcj}{{\textcolor{darkgreen}{c_J}} }
+#+LATEX: \newcommand{\epsi}{{\textcolor{darkgreen}{\epsilon}_\mi} }
+#+LATEX: \newcommand{\mj}{{\textcolor{darkgreen}{J}} }
+#+LATEX: \newcommand{\nea}{{N_\text{elec}^\uparrow}}
+#+LATEX: \newcommand{\neb}{{N_\text{elec}^\downarrow}}
+#+LATEX: \newcommand{\nmo}{{N_\text{MO}}}
+#+LATEX: \newcommand{\Nsamples}{N_\text{samples}}
+#+LATEX: \newcommand{\uniqueness}{{\textcolor{blue}{uniqueness}} }
+#+LATEX: \newcommand{\mk}{{\textcolor{blue}{k}} }
+#+LATEX: \newcommand{\mM}{{\textcolor{blue}{M}} }
+#+LATEX: \newcommand{\epsik}{{\textcolor{darkgreen}{\epsilon_{I_{\textcolor{blue}{k}}}} }}  
+
+* Stochastic evaluation of the PT2 correction
+  
+** Epstein-Nesbet Second order correction
+
+   Consider a wave function $\mPsi$ expanded on an /arbitrary/ set $\mD$
+   of $\Ndet$ orthonormal Slater determinants,
+   \[
+   \mPsi = \sum_{\mi \in \mD} {\mci} | \mi \rangle,  \;\;\;
+   \Evar = \frac{ \langle \mPsi | \mH | \mPsi \rangle}{\langle \mPsi | \mPsi \rangle}
+   \]
+
+   The Epstein-Nesbet 2nd order correction to the energy is
+   \[
+   \Ept = \sum_{\ma \in \mA} \frac{ \langle \mPsi | \mH | \ma \rangle
+   \langle \ma | \mH | \mPsi \rangle}{\Evar - \langle \ma | \mH | \ma \rangle }
+   \]
+
+   The set $\mA$ contains the Slater determinants
+   - that are not in $\mD$
+   - for which $d(\mi,\ma) = 1$ or $2$ for at least one pair $(\mi,\ma)$
+
+** Formal scaling
+
+   \[
+   \Ept = \sum_{\ma \in \mA} \frac{ \langle \mPsi | \mH | \ma \rangle
+   \langle \ma | \mH | \mPsi \rangle}{\Evar - \langle \ma | \mH | \ma
+   \rangle} = \sum_{\ma \in \mA} \frac{ \left( \sum_{\mi \in \mD} \mci
+   \langle \mi | \mH | \ma \rangle \right)^2 }{\Evar - \langle \ma |
+   \mH | \ma \rangle }
+   \]
+
+   - Size of $\mA$ : size of $(\hat{T}_1 + \hat{T}_2)|\mPsi\rangle$
+   - Number of non-zero terms : $d(\mi,\ma) \le 2  \sim \Ndet \times
+     \left[ \textcolor{darkgreen}{\left( \nea \times (\nmo - \nea) \right)^2 }\right]$ 
+   - *Expensive*
+
+** Solutions to make simulations possible
+*** "Non-general'' but /conventional/ solutions:
+
+   - Partition the MO space into different classes (active, virtual, inactive, \emph{etc})
+   - Use another zeroth-order Hamiltonian (CAS-PT2, NEV-PT2)
+
+*** Solutions applicable to /any/ wave function:
+
+   - Truncation of $\mD$ to consider only contributions due to large
+     ${\mci}$ \\
+     But: Truncation $\longrightarrow$ bias because $\Ept$ is a sum of
+     same-sign values (negative). 
+   - Monte Carlo sampling in $\mA$. \\
+     But: Statistical error decreases as
+     $\mathcal{O}\left(1/\sqrt{\Nsamples}\right) \Longrightarrow$
+     Difficult to get $\textcolor{darkgreen}{10^{-5} a.u}$ precision.
+   - Parallelism
+
+***                                                         :B_ignoreheading:
+    :PROPERTIES:                                                                              
+    :BEAMER_env: ignoreheading                                                                
+    :END: 
+     
+** Central idea
+   #+LATEX: \begin{columns}
+   #+LATEX: \begin{column}{0.5\textwidth}
+   - Choose an arbitrary ordering of $|\mi \rangle$. Natural choice:
+     \[w_\mi = \frac{\mci ^2}{\langle \mPsi | \mPsi \rangle } \]
+   - Make \emph{disjoint} groups $\mA_{\mi}$ of $| \ma \rangle$ originating from the same generator $| \mi \rangle$
+   - Each $\mA_\mi$ has its own contribution $\epsi$ to $\Ept$
+   #+LATEX: \end{column}
+   #+LATEX: \begin{column}{0.5\textwidth}
+   #+ATTR_LATEX: :width 0.9\columnwidth
+   [[./fig1.pdf]] 
+   #+LATEX: \end{column}
+   #+LATEX: \end{columns}
+
+** Central idea
+   \begin{eqnarray*}
+   \Ept &=& \sum_{\ma \in \mA} \frac{ \left( \langle \mPsi | \mH | \ma \rangle
+   \right)^2 }{\Evar - \langle \ma | \mH | \ma \rangle } \\ &=& \sum_{\mi \in \mD}
+   \sum_{\ma_\mi \in \mA_\mi} \frac{ \left( \langle \mPsi | \mH | \ma_\mi \rangle
+   \right)^2}{\Evar - \langle \ma_\mi | \mH | \ma_\mi \rangle} \\
+   &=& \sum_{\mi \in \mD} \epsi
+   \end{eqnarray*}
+
+*** Contribution per /internal/ determinant 
+   \[
+   \epsi = \sum_{\ma_\mi \in \mA_\mi} \frac{ \left( \langle \mPsi | \mH | \ma_\mi
+   \rangle \right)^2}{\Evar - \langle \ma_\mi | \mH | \ma_\mi \rangle}
+   \]
+
+** Consequences
+   \[
+   \epsi = \sum_{\ma_\mi \in \mA_\mi} \frac{ \left( \langle \mPsi | \mH | \ma_\mi
+   \rangle \right)^2}{\Evar - \langle \ma_\mi | \mH | \ma_\mi \rangle}
+   \]
+   1. $\langle \mPsi | \mH | \ma_\mi \rangle = \sum_{\mj \ge \mi} \mcj\, \langle\, \mj | \mH | \ma_\mi \rangle$
+   2. $\langle \ma_\mi | \mH | \ma_\mi \rangle$ is always large
+      (otherwise $|\ma_\mi \rangle$ would be better in the
+      \textcolor{red}{variational space}, and PT is questionable)
+***                                                         :B_ignoreheading:
+    :PROPERTIES:
+    :BEAMER_env: ignoreheading
+    :END: 
+   - $\forall \mi \in \mD : \epsi \le 0$ 
+   - $|\epsi|$ is expected to decrease as $\mci^2$
+   - The computational cost decreases with $\mi$
+
+*** Monte Carlo formulation
+    \[
+    \Ept = \sum_{\mi \in \mD} \epsi = \sum_{\mi \in \mD} \mpi \frac{\epsi}{\mpi} =
+    \left\langle \frac{\epsi}{\mpi} \right\rangle_{\mpi}
+    \]
+** Naive sampling
+   #+LATEX: \begin{columns}
+   #+LATEX: \begin{column}{0.2\textwidth}
+   Uniform sampling:
+   \mpi = \frac{1}{\Ndet}$
+   #+LATEX: \end{column}
+   #+LATEX: \begin{column}{0.8\textwidth}
+   #+CAPTION: F$_2$, cc-pVDZ, \textcolor{red}{$10^6$} determinants in the variational space
+   #+ATTR_LATEX: :height 0.75\textheight
+   [[./dist2_noise.png]]
+   #+LATEX: \end{column}
+   #+LATEX: \end{columns}
+
+** Improved sampling
+   #+LATEX: \begin{columns}
+   #+LATEX: \begin{column}{0.2\textwidth}
+   Sampling : $\mpi = \mci^2$
+   #+LATEX: \end{column}
+   #+LATEX: \begin{column}{0.8\textwidth}
+   #+CAPTION: F$_2$, cc-pVDZ, \textcolor{red}{$10^6$} determinants in the variational space
+   #+ATTR_LATEX: :height 0.75\textheight
+   [[./eici2.png]]
+   #+LATEX: \end{column}
+   #+LATEX: \end{columns}
+
+** Lazy evaluation
+   Only $\Ndet$ contributions $\epsi \longrightarrow$ all $\epsi$ can be stored in memory.
+
+*** Lazy Evaluation (Wikipedia)
+    In programming language theory, /lazy evaluation/, or /call-by-need/ is
+    an evaluation strategy which delays the evaluation of an expression until its
+    value is needed (non-strict evaluation) and which also avoids repeated
+    evaluations (sharing).
+    
+***                                                         :B_ignoreheading:
+    :PROPERTIES:
+    :BEAMER_env: ignoreheading
+    :END: 
+  #+BEGIN_SRC python
+def lazy_e(i):
+    if not e_is_computed[i]:
+        e[i] = compute_e(i)
+        e_is_computed[i] = true
+    return e[i]
+  #+END_SRC
+
+** Monte Carlo with Lazy Evaluation
+   \[
+   \Ept = \sum_{\mi \in \mD} \epsi = \sum_{\mi \in \mD} \mpi \frac{\epsi}{\mpi} =
+   \left\langle \frac{\epsi}{\mpi} \right\rangle_{\mpi}
+   \]
+
+   - Draw a generator determinant $|\mi\,\rangle$ with probability $\mpi$
+   - Increment $n_\mi$, the number of evaluations of $\epsi$
+   - If $\epsi$ is not already computed, compute it and store its value
+   - $\Ept \sim \sum_{\mi \in \mD} \frac{n_\mi}{\Nsamples} \frac{\epsi}{\mpi}$
+   - Statistical error : $\mathcal{O}\left(1/\sqrt{\Nsamples}\right)$
+   - Lazy evaluation : Exponential acceleration (time to solution)
+
+** Monte Carlo with Lazy Evaluation
+   #+CAPTION: F$_2$, cc-pVDZ, \textcolor{red}{$10^6$} determinants in the variational space
+   #+ATTR_LATEX: :height 0.75\textheight
+   [[./samples.pdf]]
+   
+** Monte Carlo with Lazy Evaluation
+   #+CAPTION: F$_2$, cc-pVDZ, \textcolor{red}{$10^6$} determinants in the variational space
+   #+ATTR_LATEX: :height 0.75\textheight
+   [[./lazy_e.pdf]]
+   
+** Monte Carlo with Lazy Evaluation
+   #+CAPTION: F$_2$, cc-pVDZ, \textcolor{red}{$10^6$} determinants in the variational space
+   #+ATTR_LATEX: :height 0.75\textheight
+   [[./lazy_err.pdf]]
+   
+** Monte Carlo with Variance reduction
+
+   - Noise can be smoothed out by averaging 
+   - Split $\mD$ into $\mM$ \emph{equiprobable} sets : "Comb"
+
+   \[
+   \Ept = \sum_{\mi \in \mD} \epsi = \sum_{\mk=1}^{\mM} \sum_{\mi_\mk \in \mD_\mk} \epsik
+   \]
+
+*** New Monte Carlo estimator
+    \[
+    \Ept = \left \langle
+    \frac{1}{\mM} \sum_{\mk=1}^{\mM} \frac{\epsik}{{\textcolor{red}{p_{I_k}}}}
+    \right \rangle_{\textcolor{red}{(p_{I_1}, \dots, p_{I_M})}}
+    \]
+
+** Monte Carlo with Variance reduction
+   #+CAPTION: F$_2$, cc-pVDZ, \textcolor{red}{$10^6$} determinants in the variational space
+   #+ATTR_LATEX: :height 0.75\textheight
+   [[./Comb1.pdf]]
+
+** Monte Carlo with Variance reduction
+   #+CAPTION: F$_2$, cc-pVDZ, \textcolor{red}{$10^6$} determinants in the variational space
+   #+ATTR_LATEX: :height 0.75\textheight
+   [[./Comb2.pdf]]
+
+** Monte Carlo with Variance reduction
+   #+CAPTION: F$_2$, cc-pVDZ, \textcolor{red}{$10^6$} determinants in the variational space
+   #+ATTR_LATEX: :height 0.75\textheight
+   [[./Comb3.pdf]]
+
+** Monte Carlo with Variance reduction
+   #+CAPTION: F$_2$, cc-pVDZ, \textcolor{red}{$10^6$} determinants in the variational space
+   #+ATTR_LATEX: :height 0.75\textheight
+   [[./dist2.png]]
+
+** Monte Carlo with Variance reduction
+   #+CAPTION: F$_2$, cc-pVDZ, \textcolor{red}{$10^6$} determinants in the variational space
+   #+ATTR_LATEX: :height 0.75\textheight
+   [[./lazy_e.pdf]]
+
+** Monte Carlo with Variance reduction
+   #+CAPTION: F$_2$, cc-pVDZ, \textcolor{red}{$10^6$} determinants in the variational space
+   #+ATTR_LATEX: :height 0.75\textheight
+   [[./comb_e.pdf]]
+
+** Monte Carlo with Variance reduction
+   #+CAPTION: F$_2$, cc-pVDZ, \textcolor{red}{$10^6$} determinants in the variational space
+   #+ATTR_LATEX: :height 0.75\textheight
+   [[./comb_err.pdf]]
+
+** Hybrid deterministic/stochastic scheme
+   - When all the determinants have been drawn, the \emph{exact} $\Ept$ can be computed
+   - $\Longrightarrow$ The result with zero statistical error can be reached
+     in a finite time
+   - In typical wave functions, $90\%$ of the norm is on a few determinants
+   - Compute the few first contributions $\epsi$, and perform the MC in the rest
+   \[
+   \Ept = \sum_{\mi \in \mD_D} \epsi + \left \langle
+   \frac{1}{\mM} \sum_{\mk=1}^{\mM} \frac{\epsik}{\textcolor{red}{p_{I_k}}}
+   \right \rangle_{(\textcolor{red}{p_{I \in \mD_S})}}
+   \]
+
+** Hybrid deterministic/stochastic scheme
+   Make the deterministic part grow during the calculation.
+
+*** At each MC step
+   - Draw a random number
+   - Find the determinants selected by the comb (increment $n_\mi$'s)
+   - Compute the $\epsi$ which have not been yet computed
+   - Compute deterministically the first non-computed determinant
+   - If a tooth of the comb is completely filled $\Longrightarrow$ Deterministic
+
+*** At any time
+   \[
+   \Ept(t) = \sum_{\mi \in \mD_D(t)} \epsi + \sum_{\mi \in \mD_S(t)} \frac{1}{\mM(t)}
+   \frac{n_\mi(t)}{\Nsamples(t)} \frac{\epsi}{\mpi}
+   \]
+
+** Hybrid deterministic/stochastic scheme
+   #+CAPTION: F$_2$, cc-pVDZ, \textcolor{red}{$10^6$} determinants in the variational space
+   #+ATTR_LATEX: :height 0.75\textheight
+   [[./lazy_e.pdf]]
+
+** Hybrid deterministic/stochastic scheme
+   #+CAPTION: F$_2$, cc-pVDZ, \textcolor{red}{$10^6$} determinants in the variational space
+   #+ATTR_LATEX: :height 0.75\textheight
+   [[./comb_e.pdf]]
+
+** Hybrid deterministic/stochastic scheme
+   #+CAPTION: F$_2$, cc-pVDZ, \textcolor{red}{$10^6$} determinants in the variational space
+   #+ATTR_LATEX: :height 0.75\textheight
+   [[./hybrid_e.pdf]]
+
+** Hybrid deterministic/stochastic scheme
+   #+CAPTION: F$_2$, cc-pVDZ, \textcolor{red}{$10^6$} determinants in the variational space
+   #+ATTR_LATEX: :height 0.75\textheight
+   [[./hybrid_err.pdf]]
+
+
+** Some timings: Cr$_2$, $2\,10^7$ determinants, 800 cores
+
+   |---------+-------------------------------------+--------------------------|
+   | Basis   | $\Ept$                              | Wall-clock time          |
+   |---------+-------------------------------------+--------------------------|
+   | cc-pVDZ | \textcolor{red}{$-0.068\,3(1)$}     | 14 min                   |
+   |         | \textcolor{red}{$-0.068\,36(1)$}    | 55 min                   |
+   |         | \textcolor{red}{$-0.068\,361(1)$}   | 2.4 hr                   |
+   |         | \textcolor{red}{$-0.068\,360\,604$} | 3 hr                     |
+   |---------+-------------------------------------+--------------------------|
+   | cc-pVTZ | \textcolor{red}{$-0.124\,4(5)$}     | 19 min                   |
+   |         | \textcolor{red}{$-0.124\,7(1)$}     | 58 min                   |
+   |         | \textcolor{red}{$-0.124\,63(1)$}    | 3.5 hr                   |
+   |         | \textcolor{red}{$-0.124\,642(1)$}   | 8.7 hr                   |
+   |         | ---                                 | $\sim$ 15 hr (estimated) |
+   |---------+-------------------------------------+--------------------------|
+   | cc-pVQZ | \textcolor{red}{$-0.155\,8(5)$}     | 56 min                   |
+   |         | \textcolor{red}{$-0.155\,9(1)$}     | 2.5 hr                   |
+   |         | \textcolor{red}{$-0.155\,95(1)$}    | 9.0 hr                   |
+   |         | \textcolor{red}{$-0.155\,952(1)$}   | 18.5 hr                  |
+   |         | ---                                 | $\sim$ 29 hr (estimated) |
+   |---------+-------------------------------------+--------------------------|
+
+** Parallel efficiency
+   #+ATTR_LATEX: :height 0.8\textheight
+   [[./speedup_pt2.png]]
+
+
+* Efficient direct CI
+
+
+* Export                                                           :noexport:
+  #+BEGIN_SRC elisp :output none
+(setq org-latex-listings 'minted
+      org-latex-packages-alist '(("" "minted"))
+      org-latex-pdf-process
+      '("pdflatex -shell-escape -interaction nonstopmode -output-directory %o %f"
+        "pdflatex -shell-escape -interaction nonstopmode -output-directory %o %f"
+        "pdflatex -shell-escape -interaction nonstopmode -output-directory %o %f"))
+(setq org-latex-minted-options '(("breaklines" "true")
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+(setq org-latex-minted-options
+      '(("frame" "lines")
+        ("fontsize" "\\scriptsize")
+        ("linenos" "")))
+(org-beamer-export-to-pdf)
+                            
+  #+END_SRC
+
+  #+RESULTS:
+  : /home/scemama/TEX/Nuclearistes/algorithms.pdf

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+}{%
+  \PackageWarning{beamerthemepterosor}{%
+    You need to compile with XeLaTeX or LuaLaTeX to use the Fira fonts%
+  }
+}
+\setbeamerfont{title}{size=\Large,%
+                      series=\bfseries}
+\setbeamerfont{author}{size=\small}
+\setbeamerfont{date}{size=\small}
+\setbeamerfont{section title}{size=\Large,%
+                              series=\bfseries}
+\setbeamerfont{block title}{size=\normalsize,%
+                            series=\bfseries}
+\setbeamerfont{block title alerted}{size=\normalsize,%
+                                    series=\bfseries}
+\setbeamerfont*{subtitle}{size=\large}
+\setbeamerfont{frametitle}{size=\large,%
+                           series=\bfseries}
+\setbeamerfont{caption}{size=\small}
+\setbeamerfont{caption name}{series=\bfseries}
+\setbeamerfont{description item}{series=\bfseries}
+\setbeamerfont{page number in head/foot}{size=\scriptsize}
+\setbeamerfont{bibliography entry author}{size=\normalsize,%
+                                          series=\normalfont}
+\setbeamerfont{bibliography entry title}{size=\normalsize,%
+                                         series=\bfseries}
+\setbeamerfont{bibliography entry location}{size=\normalsize,%
+                                            series=\normalfont}
+\setbeamerfont{bibliography entry note}{size=\small,%
+                                        series=\normalfont}
+\setbeamerfont{standout}{size=\Large,%
+                         series=\bfseries}
+\pgfkeys{
+  /pterosor/font/titleformat title/.cd,
+    .is choice,
+    regular/.code={%
+      \let\pterosor@titleformat\@empty%
+      \setbeamerfont{title}{shape=\normalfont}%
+    },
+    smallcaps/.code={%
+      \let\pterosor@titleformat\@empty%
+      \setbeamerfont{title}{shape=\scshape}%
+    },
+    allsmallcaps/.code={%
+      \let\pterosor@titleformat\lowercase%
+      \setbeamerfont{title}{shape=\scshape}%
+      \PackageWarning{beamerthemepterosor}{%
+        Be aware that titleformat title=allsmallcaps can lead to problems%
+      }
+    },
+    allcaps/.code={%
+      \let\pterosor@titleformat\uppercase%
+      \setbeamerfont{title}{shape=\normalfont}
+      \PackageWarning{beamerthemepterosor}{%
+        Be aware that titleformat title=allcaps can lead to problems%
+      }
+    },
+}
+\pgfkeys{
+  /pterosor/font/titleformat subtitle/.cd,
+    .is choice,
+    regular/.code={%
+      \let\pterosor@subtitleformat\@empty%
+      \setbeamerfont{subtitle}{shape=\normalfont}%
+    },
+    smallcaps/.code={%
+      \let\pterosor@subtitleformat\@empty%
+      \setbeamerfont{subtitle}{shape=\scshape}%
+    },
+    allsmallcaps/.code={%
+      \let\pterosor@subtitleformat\lowercase%
+      \setbeamerfont{subtitle}{shape=\scshape}%
+      \PackageWarning{beamerthemepterosor}{%
+        Be aware that titleformat subtitle=allsmallcaps can lead to problems%
+      }
+    },
+    allcaps/.code={%
+      \let\pterosor@subtitleformat\uppercase%
+      \setbeamerfont{subtitle}{shape=\normalfont}%
+      \PackageWarning{beamerthemepterosor}{%
+        Be aware that titleformat subtitle=allcaps can lead to problems%
+      }
+    },
+}
+\pgfkeys{
+  /pterosor/font/titleformat section/.cd,
+    .is choice,
+    regular/.code={%
+      \let\pterosor@sectiontitleformat\@empty%
+      \setbeamerfont{section title}{shape=\normalfont}%
+    },
+    smallcaps/.code={%
+      \let\pterosor@sectiontitleformat\@empty%
+      \setbeamerfont{section title}{shape=\scshape}%
+    },
+    allsmallcaps/.code={%
+      \let\pterosor@sectiontitleformat\MakeLowercase%
+      \setbeamerfont{section title}{shape=\scshape}%
+      \PackageWarning{beamerthemepterosor}{%
+        Be aware that titleformat section=allsmallcaps can lead to problems%
+      }
+    },
+    allcaps/.code={%
+      \let\pterosor@sectiontitleformat\MakeUppercase%
+      \setbeamerfont{section title}{shape=\normalfont}%
+      \PackageWarning{beamerthemepterosor}{%
+        Be aware that titleformat section=allcaps can lead to problems%
+      }
+    },
+}
+\pgfkeys{
+  /pterosor/font/titleformat frame/.cd,
+    .is choice,
+    regular/.code={%
+      \let\pterosor@frametitleformat\@empty%
+      \setbeamerfont{frametitle}{shape=\normalfont}%
+    },
+    smallcaps/.code={%
+      \let\pterosor@frametitleformat\@empty%
+      \setbeamerfont{frametitle}{shape=\scshape}%
+    },
+    allsmallcaps/.code={%
+      \let\pterosor@frametitleformat\MakeLowercase%
+      \setbeamerfont{frametitle}{shape=\scshape}%
+      \PackageWarning{beamerthemepterosor}{%
+        Be aware that titleformat frame=allsmallcaps can lead to problems%
+      }
+    },
+    allcaps/.code={%
+      \let\pterosor@frametitleformat\MakeUppercase%
+      \setbeamerfont{frametitle}{shape=\normalfont}
+      \PackageWarning{beamerthemepterosor}{%
+        Be aware that titleformat frame=allcaps can lead to problems%
+      }
+    },
+}
+\pgfkeys{
+  /pterosor/font/.cd,
+  titleformattitle/.code=\pgfkeysalso{titleformat title=#1},
+  titleformatsubtitle/.code=\pgfkeysalso{titleformat subtitle=#1},
+  titleformatsection/.code=\pgfkeysalso{titleformat section=#1},
+  titleformatframe/.code=\pgfkeysalso{titleformat frame=#1},
+}
+\newcommand{\pterosor@font@setdefaults}{
+  \pgfkeys{/pterosor/font/.cd,
+    titleformat title=regular,
+    titleformat subtitle=regular,
+    titleformat section=regular,
+    titleformat frame=regular,
+  }
+}
+\def\pterosor@titleformat#1{#1}
+\def\pterosor@subtitleformat#1{#1}
+\def\pterosor@sectiontitleformat#1{#1}
+\def\pterosor@frametitleformat#1{#1}
+\patchcmd{\beamer@title}%
+  {\def\inserttitle{#2}}%
+  {\def\inserttitle{\pterosor@titleformat{#2}}}%
+  {}%
+  {\PackageError{beamerfontthemepterosor}{Patching title failed}\@ehc}
+\patchcmd{\beamer@subtitle}%
+  {\def\insertsubtitle{#2}}%
+  {\def\insertsubtitle{\pterosor@subtitleformat{#2}}}%
+  {}%
+  {\PackageError{beamerfontthemepterosor}{Patching subtitle failed}\@ehc}
+\patchcmd{\sectionentry}
+  {\def\insertsectionhead{#2}}
+  {\def\insertsectionhead{\pterosor@sectiontitleformat{#2}}}
+  {}
+  {\PackageError{beamerfontthemepterosor}{Patching section title failed}\@ehc}
+\@tempswafalse
+\patchcmd{\beamer@section}
+  {\def\insertsectionhead{\hyperlink{Navigation\the\c@page}{#1}}}
+  {\def\insertsectionhead{\hyperlink{Navigation\the\c@page}{%
+    \pterosor@sectiontitleformat{#1}}}}
+  {\@tempswatrue}
+  {}
+\patchcmd{\beamer@section}
+  {\protected@edef\insertsectionhead{\noexpand\hyperlink{Navigation\the\c@page}{#1}}}
+  {\protected@edef\insertsectionhead{\noexpand\hyperlink{Navigation\the\c@page}{%
+    \noexpand\pterosor@sectiontitleformat{#1}}}}
+  {\@tempswatrue}
+  {}
+\if@tempswa\else
+  \PackageError{beamerfontthemepterosor}{Patching section title failed}\@ehc
+\fi
+\@tempswafalse
+\patchcmd{\beamer@subsection}
+  {\def\insertsubsectionhead{\hyperlink{Navigation\the\c@page}{#1}}}
+  {\def\insertsubsectionhead{\hyperlink{Navigation\the\c@page}{%
+    \pterosor@sectiontitleformat{#1}}}}
+  {\@tempswatrue}
+  {}
+\patchcmd{\beamer@subsection}
+  {\protected@edef\insertsubsectionhead{\noexpand\hyperlink{Navigation\the\c@page}{#1}}}
+  {\protected@edef\insertsubsectionhead{\noexpand\hyperlink{Navigation\the\c@page}{%
+    \noexpand\pterosor@sectiontitleformat{#1}}}}
+  {\@tempswatrue}
+  {}
+\if@tempswa\else
+  \PackageError{beamerfontthemepterosor}{Patching section title failed}\@ehc
+\fi
+\patchcmd{\beamer@@frametitle}
+  {{%
+      \gdef\insertframetitle{{#2\ifnum\beamer@autobreakcount>0\relax{}\space%
+      \usebeamertemplate*{frametitle continuation}\fi}}%
+    \gdef\beamer@frametitle{#2}%
+    \gdef\beamer@shortframetitle{#1}%
+    }}
+  {{%
+      \gdef\insertframetitle{{\pterosor@frametitleformat{#2}\ifnum%
+      \beamer@autobreakcount>0\relax{}\space%
+      \usebeamertemplate*{frametitle continuation}\fi}}%
+    \gdef\beamer@frametitle{#2}%
+    \gdef\beamer@shortframetitle{#1}%
+    }}
+  {}
+  {\PackageError{beamerfontthemepterosor}{Patching frame title failed}\@ehc}
+\pterosor@font@setdefaults
+\ProcessPgfPackageOptions{/pterosor/font}
+\endinput
+%%
+%% End of file `beamerfontthemepterosor.sty'.

beamerinnerthemepterosor.sty

@@ -0,0 +1,53 @@
+\mode<presentation>
+
+\setbeamertemplate{background}{
+\ifnum\thepage>1
+\includegraphics[height=\paperheight]{beamerbackgroundpterosor.png}
+\else
+\includegraphics[height=\paperheight]{beamertitlepterosor.png}
+\fi
+}
+
+% Title page
+\defbeamertemplate*{title page}{pterosor}[1][]
+{
+   \vskip3cm%
+    \begin{beamercolorbox}[wd=6.cm,leftskip=-0.5cm,#1]{title page header}
+      \usebeamerfont{title}\inserttitle\par%
+    \end{beamercolorbox}%
+    \vskip0.2cm%
+    \begin{beamercolorbox}[wd=6.cm,leftskip=-0.5cm,#1]{author}
+      \usebeamerfont{author}\insertauthor%
+    \end{beamercolorbox}
+     \vskip0.3cm%
+    \begin{beamercolorbox}[wd=6.cm,leftskip=-0.5cm,#1]{date}
+      \usebeamerfont{author}\insertdate%
+    \end{beamercolorbox}
+     \vskip0.3cm%
+    \begin{beamercolorbox}[wd=7.cm,leftskip=-0.5cm,#1]{institute}
+      \usebeamerfont{author}{\tiny \insertinstitute}%
+    \end{beamercolorbox}
+  \vfill
+}
+
+% Section page
+\defbeamertemplate*{section page}{pterosor}{
+  \centering
+  \begin{minipage}{22em}
+    \raggedright
+    \usebeamercolor[fg]{section title}
+    \usebeamerfont{section title}
+    \insertsectionhead\\[-1ex]
+    \par
+    \ifx\insertsubsectionhead\@empty\else%
+      \usebeamercolor[fg]{subsection title}%
+      \usebeamerfont{subsection title}%
+      \insertsubsectionhead
+    \fi
+  \end{minipage}
+  \par
+  \vspace{\baselineskip}
+}
+
+
+\mode<all>

beamerouterthemepterosor.sty

@@ -0,0 +1,20 @@
+\mode<presentation>
+
+% Frame title
+\defbeamertemplate*{frametitle}{pterosor}[1][]
+{
+ \begin{beamercolorbox}[wd=\paperwidth,ht=0.95cm]{frametitle} 
+ \begin{tikzpicture}
+% \useasboundingbox[fill=white](0,0) rectangle(\the\paperwidth,1.);
+%\fill[red!80!black] (0,0) rectangle(2.95,1.);
+%\fill[blue!50!cyan!80!black] (3.05,0) rectangle(\the\paperwidth,1.);
+%\fill[white!50] (3.05,0) rectangle(\the\paperwidth,1.);
+  \useasboundingbox[](0,0) rectangle(\the\paperwidth,1.);
+  \node[anchor=east, white,font=\large] at
+  (\paperwidth,0.25){\insertframetitle \phantom{m}};
+ \end{tikzpicture}
+ \end{beamercolorbox}
+}
+
+\mode<all>
+

beamerthemepterosor.sty

@@ -0,0 +1,29 @@
+\mode<presentation>
+
+
+% Requirement
+\RequirePackage{tikz}
+
+% Settings
+\useinnertheme{pterosor}
+\useoutertheme{pterosor}
+\usecolortheme{pterosor}
+\usefonttheme{pterosor}
+
+
+% we remove the navigation symbols and define blocks aspect.
+\setbeamertemplate{navigation symbols}{}
+\setbeamertemplate{blocks}[rounded][shadow=true]
+
+\AtBeginSection[]{%
+    \begin{frame}[plain]
+        \usebeamertemplate{section page}
+    \end{frame}
+}
+
+\setbeamertemplate{footline}{
+    \vspace{0.6cm}
+}
+
+
+\mode<all>