#+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}
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#+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{\mm}{{\textcolor{blue}{m}} }
#+LATEX: \newcommand{\mk}{{\textcolor{blue}{k}} }
#+LATEX: \newcommand{\mM}{{\textcolor{blue}{M}} }
#+LATEX: \newcommand{\epsik}{{\textcolor{darkgreen}{\epsilon_{I_{\textcolor{blue}{k}}}} }}  

* Efficient direct CI
  
** Sorting

*** Popular misconception

    *Sorting is /not/ $\mathcal{O}(N \log(N))$* :
    #+LATEX: \textcolor{darkgreen}{sorting is $\mathcal{O}(N \log(M))$}
    (linear in $N$, log in $M$)

*** .                                                       :B_ignoreheading:
    :PROPERTIES:                                                                              
    :BEAMER_env: ignoreheading                                                                
    :END: 
    - A is an array of $N$ integer values 
    - The bitmask is an integer with only one bit set to one (00001000)
      #+HEADER: :includes (list "<stddef.h>" "<stdio.h>")
      #+BEGIN_SRC C 
    void radix_sort(int* A, size_t N, int bitmask) {
        if (bitmask == 0) return;
        int left[N], right[N];
        int p=0 ; int q=0 ;
        for (int i=0 ; i<N ; i++) {
            if (A[i] & bitmask) { right[q] = A[i]; q++; }
            else                { left [p] = A[i]; p++; } }
        radix_sort(left , p, bitmask >> 1 ) ;
        radix_sort(right, q, bitmask >> 1 ) ;
        for (int i=0 ; i<p ; i++) { A[  i] = left [i] ; }
        for (int i=0 ; i<q ; i++) { A[p+i] = right[i] ; }
    }
      #+END_SRC

*** No export                                                      :noexport:
       #+BEGIN_SRC C :export none
        int main() {
          int A[14] = {5,2,4,5,7,8,9,65,4,3,3,5,6,72};
          for (int i=0 ; i<14 ; i++)
            printf("%d ", A[i]);
          printf("\n");
          radix_sort(A,14, 1 << 24);
          for (int i=0 ; i<14 ; i++)
            printf("%d ", A[i]);
          printf("\n");
        }
       #+END_SRC

       #+RESULTS:
       | 5 | 2 | 4 | 5 | 7 | 8 | 9 | 65 | 4 | 3 | 3 | 5 |  6 | 72 |
       | 2 | 3 | 3 | 4 | 4 | 5 | 5 |  5 | 6 | 7 | 8 | 9 | 65 | 72 |

** Double-determinant
 A Slater determinant can be written as a Waller-Hartree double determinant

 \[ |I \rangle = \hat{I}\,  |\rangle =
     -1^p \times \hat{I}_\uparrow \, \hat{I}_\downarrow\, |\rangle =
     -1^p \times \hat{I}_\uparrow \, |\rangle \otimes\ \hat{I}_\downarrow \, |\rangle
     \]
 
 \[
 \mPsi = \sum_\mi \mci |\mi\rangle = \sum_{\mk=1}^{\Ndetup}
 \sum_{\mm=1}^{\Ndetdn} C_{\textcolor{darkgreen}{km}} \mDup \mDdn
 \]

 - If $\mDup$ and $\mDdn$ are represented as $\nmo$ -bit strings, this
   transformation can be done in $\mathcal{O}(\Ndet \times \nmo)$.
 

* 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). 
   - Algorithmic improvement
   - 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 
   \[
   \epsi = \sum_{\ma_\mi \in \mA_\mi} \frac{ \left( \langle \mPsi | \mH | \ma_\mi
   \rangle \right)^2}{\Evar - \langle \ma_\mi | \mH | \ma_\mi \rangle}
   \]

** From $\mathcal{O}(N_\text{det}^2)$ to $\mathcal{O}(N_\text{det})$
 \[
 \mPsi = \sum_\mi \mci |\mi\rangle = \sum_{\mk=1}^{\Ndetup} \sum_{\mm=1}^{\Ndetdn} C_{\textcolor{red}{km}} \mDup \mDdn
 \]
  - Sorting is $\mathcal{O}(\Ndet)$
  - $\langle \mi | \mH | \ma \rangle \langle \ma | \mH | \mj \rangle = 0$ when $d(\mi, \mj) > 4$
  - Loop over $\Ndetup$ determinants (rows of the $C$ matrix) \\
    Remove all the rows where $d(\mDup,  \mDupp)>4$ ($\sim \mathcal{O}(\sqrt{\Ndet})$)
  - Loop over $\Ndetdn$ determinants (columns of the $C$ matrix) \\
    Remove all the columns where $d(\mDdn,  \mDdnn)>4)$
  - The remaining number of determinants is bounded by the size of the CISDTQ space

** Finer filtering

   \[
   \epsi  = \sum_{\ma \in \mA_\mi} \frac{ \langle \mPsi | \mH | \ma_\mi \rangle
   \langle \ma_\mi | \mH | \mPsi' \rangle}{\Evar - \langle \ma_\mi | \mH | \ma_\mi
   \rangle}
   \]

   - We know that all the $| \ma_\mi \rangle$ are singles and doubles
     with respect to $| \mi \rangle$
   - $|\mPsi'\rangle$ is the projection of $\mPsi$ on the subspace of
     determinants in $\mD$ which are no more than quadruply excited
     with respect to $| \mi \rangle$
   - For a subset of excitations $\mi \rightarrow \ma$,
     $|\mPsi'\rangle$ is filtered further with possible hole/particle constraints

** Monte Carlo sampling
   \[
   \epsi = \sum_{\ma_\mi \in \mA_\mi} \frac{ \left( \langle \mPsi | \mH | \ma_\mi
   \rangle \right)^2}{\Evar - \langle \ma_\mi | \mH | \ma_\mi \rangle}
   \]
   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]]




* Export                                                           :noexport:
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        ("fontsize" "\\scriptsize")
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(org-beamer-export-to-pdf)
                            
  #+END_SRC

  #+RESULTS:
  : /home/scemama/TEX/Nuclearistes/algorithms.pdf