From aa676cf9a994e97032cd77c76e07818b00d6c32d Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Gi=C3=B9=20Marcer?= <giuliamarcer@yahoo.it>
Date: Mon, 1 Jun 2020 16:20:34 +0200
Subject: [PATCH] ex-5: add the plot 5-fit.pdf to the paper

---
 notes/images/5-fit.pdf | Bin 14478 -> 14471 bytes
 notes/sections/5.md    |  45 ++++++++++++++++++++---------------------
 2 files changed, 22 insertions(+), 23 deletions(-)

diff --git a/notes/images/5-fit.pdf b/notes/images/5-fit.pdf
index d8bac2e762004af59e827c17b6b798260183d1db..24b4c6d61320ae168da1ad39db0ecee185078206 100644
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diff --git a/notes/sections/5.md b/notes/sections/5.md
index c70d638..8eb10c7 100644
--- a/notes/sections/5.md
+++ b/notes/sections/5.md
@@ -69,6 +69,11 @@ this can result in an underestimation of the error.
 
 In this case $f(x) = e^{x}$ and $\Omega = [0,1]$, hence $V = 1$.
 
+![Estimated values of $I$ obatined by Plain MC technique with different
+number of function calls; logarithmic scale; errorbars showing their
+estimated uncertainties. As can be seen, the process does a sort o seesaw
+around the correct value.](images/5-MC_MC.pdf){#fig:plain-mc-iter}
+
 Since the distance from $I$ of $I_N$ is related to $N$, the accuracy of the
 method lies in how many points are generated, namely how many function calls
 are executed when the iterative method is implemented. In @fig:plain-mc-iter
@@ -77,11 +82,6 @@ red. In @tbl:plain-mc-res, some of them are listed: the estimated integrals
 $I^{\text{oss}}$ are compared to the expected value $I$ and the differences
 between them are given.
 
-![Estimated values of $I$ obatined by Plain MC technique with different
-number of function calls; logarithmic scale; errorbars showing their
-estimated uncertainties. As can be seen, the process does a sort o seesaw
-around the correct value.](images/5-MC_MC.pdf){#fig:plain-mc-iter}
-
 ---------------------------------------------------------------------------
 calls               $I^{\text{oss}}$        $\sigma$            diff
 ------------------ ------------------ ------------------ ------------------
@@ -99,24 +99,23 @@ Table: Some MC results with three different numbers of function calls.
 As can be seen, $\sigma$ is always of the same order of magnitude of diff,
 except for very low numbers of function calls. Even with \num{5e7} calls,
 $I^{\text{oss}}$ still differs from $I$ at the fifth decimal place, meaning
-that this method shows a really slow convergence. In fact, since the $\sigma$
-dependence on the number $C$ of function calls is confirmed:
+that this method shows a really slow convergence.  
+The $\sigma$ dependence on the number $C$ of function calls was checked with a
+least square minimization by modeling the data with the function:
 $$
-  \begin{cases}
-    \sigma_1 = \num{6.95569e-4} \longleftrightarrow C_1 = \num{5e6} \\
-    \sigma_2 = \num{6.95809e-5} \longleftrightarrow C_1 = \num{5e8}
-  \end{cases}
+  \sigma = \frac{a}{x^b}
 $$
 
-$$
-  \thus
-  \frac{\sigma_1}{\sigma_2} = 9.9965508 \sim 10 = \sqrt{\frac{C_2}{C_1}}
-$$
+As can be seen in @fig:err_fit, the obtained result confirmes the expected value
+of $b^{\text{exp}} = 0.5$, having found $b \sim 0.499$.  
+Given this dependence, for an error of $10^{-n}$, a number $\propto 10^{2n}$ of
+function calls is needed. To compute an integral within double precision, an
+impossibly large number of $\sigma \sim 10^{32}$ calls is needed, which makes
+this method unpractical for high-precision applications.
 
-For an error of $10^{-n}$, a number $\propto 10^{2n}$ of function calls is
-needed. To compute an integral within double precision,
-an impossibly large number of $\sigma \sim 10^{32}$ calls is needed,
-which makes this method unpractical for high-precision applications.
+![Plain MC uncertainties estimations $\sigma$ as a function of the
+  number of function calls $C$. Observed values in red, predicted
+  dependence in gray.](images/5-fit.pdf){#fig:err_fit}
 
 
 ## Stratified sampling
@@ -381,6 +380,10 @@ calls               $I^{\text{oss}}$        $\sigma$            diff
 Table: Some VEGAS results with different numbers of
        function calls. {#tbl:vegas-res}
 
+![Only the most accurate results are shown in order to stress the
+  differences between VEGAS (in gray) and MISER (in black) methods
+  results.](images/5-MC_MI_VE.pdf){#fig:vegas-iter}
+
 As can be appreciated in @fig:vegas-iter, the VEGAS algorithm manages to compute the
 integral value more accurately compared to MISER. The $\chi_r^2$ turns out to
 be close enough to 1 to guarantee a good estimation of $I$, goodness which is
@@ -388,10 +391,6 @@ also confirmed by the very small difference shown in @tbl:vegas-res.
 In fact, with a number of \num{5e7} function calls, the difference is
 smaller than \num{1e-10}.
 
-![Only the most accurate results are shown in order to stress the
-  differences between VEGAS (in gray) and MISER (in black) methods
-  results.](images/5-MC_MI_VE.pdf){#fig:vegas-iter}
-
 In conclusion, between a plain Monte Carlo technique, stratified sampling and
 importance sampling, the last turned out to be the most powerful mean to
 obtain a good estimation of the integrand.