====== Calculation methods applied to chemistry ======
===== Synopsis (english) =====
====  Mathematical prerequisites====

==== Programming bases and tools ====
  * Python programming language
    * [[http://www.learnpython.org/|LearnPython.org]] interactive tutorial with code execution
    * [[https://www.datacamp.com/courses/intro-to-python-for-data-science|DataCamp free course "Intro to Python for Data Science"]]
    * [[https://www.sololearn.com/Course/Python/|Python 3 Tutorial]], interactive, with [[https://code.sololearn.com/#py|code use in web browser]]
    * MOOCs (massive open online courses) :
      * [[https://www.coursera.org/course/interactivepython|An Introduction to Interactive Programming in Python (Beginners)]] (Coursera)
      * [[https://www.edx.org/course/introduction-computer-science-mitx-6-00-1x-9|Introduction to Computer Science and Programming Using Python]] (edX)
  * [[wp>Anaconda_(Python_distribution)|Anaconda Python distribution]] (64 bits, with Python 3.x) : [[https://www.anaconda.com/download/]]
    * Includes these tools :
      * Jupyter notebook (interactive web-based environment)
      * qtconsole (high level Python console with graphics & colors)
      * spyder (powerful Python IDE)
    * includes lot of Python libraries : matplotlib, numpy, scipy, pandas,...
    * package management system (conda), virtual environments
    * Anaconda Navigator includes extensive documentation (on anaconda website and dedicated websites)
  * GNU/Linux OS (preferred)

  * Jupyter introductions, tutorials, ...
    * [[https://medium.com/@denkasyanov/jupyter-notebook-tutorial-9c0ffa5ae9a1#.ufaor34wp|Jupyter Notebook Tutorial]], par Den Kasyanov (Medium)
    * [[https://jupyter.readthedocs.io/en/latest/|Jupyter official documentation]]
    * [[https://www.datacamp.com/community/blog/jupyter-notebook-cheat-sheet|Jupyter DataCamp Cheat Sheet]]

<note tip>The Microsoft Azure Notebooks environment can be used to execute sample codes, using a professionnal,  personnal or student login/account (i.e. student login from UMONS). The following public sample notebooks are then available to test Jupyter text and coding features :
  * [[https://notebooks.azure.com/linusable/libraries/samples-public]]
    * //e.g. :// [[https://notebooks.azure.com/linusable/libraries/samples-public/html/notebooks/second_degree_polynomial_and_roots-01.ipynb]] (Second degree polynomial and roots)
</note>

  * Python scientific libraries (official websites including tutorials and documentation)
   * //cf.// [[teaching:progappchim:start#les_librairies_scientifiques_essentielles|this]] (french)
   * [[http://matplotlib.org/|Matplotlib]] (scientific graphs)
  * [[http://www.numpy.org/|NumPy]] (array manipulations, linear algebra, Fourier transforms, random numbers,...
  * [[http://scipy.org/scipylib/index.html|SciPy]] numerical methods (integrations, ODE, PDE,...)
  * [[http://sympy.org/en/index.html|SymPy]] symbolis maths
  * [[http://pandas.pydata.org/|Pandas]], data analysis
  * Pylab, combine Matplotlib, NumPy and SciPy
  * [[http://scikit-learn.org/|Scikit-learn]], machine learning

==== Fundamental numerical methods ====
  * [[system_of_linear_equations|Systems of linear equations]]
    * Diagonalisation and triangularisation
    * LU decomposition : factorization in triangular matrices
  * [[root-finding_algorithm|Root findings :  equations f(x) = 0]]
    * Polynomial equations
    * Dichotomy
    * Secant method, Regula falsi
    * Newton-Raphson method
  * [[numerical_integration|Numerical intégration]] (integrals)
    * Simpson method and gaussian quadratures

<note tip>
Learning outcomes :
  * Systems of linear equations
    * failing of the theoretical way to solve a linear system using determinant and cofactors (np complexity)
    * triangularisation and diagonalisation principles : algorithm and complexity
    * "divide by zero" errors and pivot solutions
    * extension towards the matrix inversion
    * lower-upper LU decomposition and complexity (N³ for the decomposition step and N² for substitution step). How to solve systems with varying independant vectors
    * special matrix require special algorithms : tridiagonal matrix algorithm (Thomas algorithm)
  * Root findings
    * Bisection method (dichotomy) : simple and robust algorithm, invariant loop, slow convergence
    * iterative transformation x = f(x), convergence and divergence situations
    * secant and regula falsi methods, Convergence Criterion of the Fixed Point Method
    * Newton-Raphson method (use of derivatives), quadratic convergence, failure, tolerance and stop condition
    * Van Wijngaardeb-Dekker-Brent method ("black box" in numerical packages)
    * Roots of polynomials and Bairstow's method
  * Numerical intégration
    * Equally Spaced methods (trapezoidal, Simpson), accuracy, error dependance,...
    * Gaussian Quadratures and orthogonal polynomials (special integrals, scale transformations,  error estimates,...)
</note>

==== Classical numerical methods ====
=== Ordinary_differential_equations (ODE) ===

[[numerical_methods_for_ordinary_differential_equations|Numerical solutions of ODE]]
  * principe de discrétisation, méthode d'Euler
  * Améliorations et méthodes de Runge-Kutta
  * Runge-Kutta d'ordre 4
  * Contrôle du pas d'intégration
  * Méthodes predictor-corrector
  * Méthodes d'extrapolation (Richardson, Burlish-Stoer)
  * applications :
    * équations de cinétique chimique
    * Équation logistique
      * [[wp>fr:Modèle_de_Verhulst|Modèle de Verhulst]], [[wp>fr:Dynamique_des_populations|dynamique des populations]] et non-linéarité
      * Bifurcation, doublements de périodes et transition vers le chaos
    * Réactions chimiques oscillantes : Belousov-Zhabotinsky, Brusselator, Oregonator
    * Modèle proie-prédateur
    * Attracteur étrange, modèle atmosphérique de Lorenz

=== Partial_differential_equations (PDE) ===
[[partial_differential_equation|Numerical solutions of PDE]]
  * Domaine d'application des équations : équation de diffusion, équation d'ondes, équations de Navier-Stokes
  * Types de traitements numériques
  * Différences finies et problèmes de diffusion
  * Schémas classiques de différences finies
    * Résolutions stationnaires
    * Résolutions dépendantes du temps
  * Méthodes explicites, critère de (ou d'in)stabilités et méthodes implicites


=== Eigenvalues and eigenvectors ===
[[eigenvalues_and_eigenvectors|Eigenvalues and eigenvectors]]

applications à des problèmes de relaxation et de population, analyse de modes normaux de vibration, PCA (principal component analysis),...

=== Non-linear systems of equations ===
  * Newton-Raphson method


=== Linear and non-linear least squares approximations ===
  * Application to deconvolution (Levenberg–Marquardt algorithm)
  * Reférences :
    * [[https://pubs.acs.org/doi/abs/10.1021/acs.jchemed.8b00649|An Open-Source, Cross-Platform Resource for Nonlinear Least-Squares Curve Fitting]] Andreas Möglich, J. Chem. Educ., 2018, 95 (12), pp 2273–2278 DOI: 10.1021/acs.jchemed.8b00649


=== Chebyshev approximation ===
+ discussion of some approximations like [[wp>Bhaskara_I's_sine_approximation_formula|Bhaskara I's sine approximation formula]]
  * [[https://twitter.com/fermatslibrary/status/1267450081151782913]]
  * [[https://medium.com/@mathcube7/chebyshev-interpolation-with-python-2f2e89bb7c30|Chebyshev Interpolation With Python]] Mathcube, Medium, 03/02/2023

=== Molecules modelisation and visualization ===

=== Minimization ===
Conformational problems

----

===== Additional subjects =====
  * Bioinformatics and related algorithm (biochemistry, mass spectrometry,...)
  * Chemistry
    * quantum calculations, optimization, molecular mechanics
    * visualization, virtual reality
    * chemical informations on structures and reactions
  * Data science, statistics (Python modules : Scipy, Pandas,...)
    * Time series analysis
    * Machine learning (Scikit-learn,...)
  * Data visualization
    * boxplot, 3D, animations, graphs,...
  * Sensors and interfaces, Arduino, Raspberry Pi, IoT
  * Simulations
    * Agent base modelling and complex systems
    * cellular automaton
    * Simpy, active matter simulations...
  * Digital image processing, image recognition
    * particle tracking,...
    * Voronoi diagrams, Delaunay triangulation,... ([[https://towardsdatascience.com/the-fascinating-world-of-voronoi-diagrams-da8fc700fa1b|ref1]])


===== References =====
  * Gradient descent optimization
    * [[https://machinelearningmastery.com/gradient-descent-optimization-from-scratch/|How to Implement Gradient Descent Optimization from Scratch]] By Jason Brownlee on April 16, 2021
    * [[https://towardsdatascience.com/implementing-gradient-descent-in-python-from-scratch-760a8556c31f|Implementing Gradient Descent in Python from Scratch]] Vatsal Sheth, Medium, 18/02/2022
  * Bioinformatics
    * [[http://biopython.org/|Biopython]]
    * [[http://rosalind.info/problems/locations/|Rosalind.info]], learning bioinformatics by problem solutions
  * Machine Learning
    * Scikit-learn
      * [[https://towardsdatascience.com/linear-regression-in-6-lines-of-python-5e1d0cd05b8d|Linear Regression in 6 lines of Python]] (using scikit-learn)
    * [[https://datafloq.com/read/12-algorithms-every-data-scientist-should-know/2024|12 Algorithms Every Data Scientist Should Know]]
  * Deep Learning
    * TensorFlow
    * [[https://www.datacamp.com/community/tutorials/deep-learning-python#gs.PPrcRR4|Keras Tutorial: Deep Learning in Python]]
  * Arduino
    * [[http://pubs.acs.org/doi/abs/10.1021/acs.jchemed.6b00639|Fabrication of an Economical Arduino-Based Uniaxial Tensile Tester]], Julien H. Arrizabalaga, Aaron D. Simmons, and Matthias U. Nollert, J. Chem. Educ., 2017, 94 (4), pp 530–533 DOI: 10.1021/acs.jchemed.6b00639
  * PCA (principal component analysis)
    * [[http://pubs.acs.org/doi/abs/10.1021/acs.jchemed.6b00550|Learning Principal Component Analysis by Using Data from Air Quality Networks]], Luis Vicente Pérez-Arribas, María Eugenia León-González, and Noelia Rosales-Conrado, ,J. Chem. Educ., 2017, 94 (4), pp 458–464 DOI: 10.1021/acs.jchemed.6b00550
    * [[http://benalexkeen.com/principle-component-analysis-in-python/|Principle Component Analysis in Python]]
  * K-Means
    * [[http://benalexkeen.com/k-means-clustering-in-python/|K-means Clustering in Python]]
    * [[https://medium.com/thrive-in-ai/k-means-clustering-in-python-9f00b3d7abc3|K Means Clustering in Python]] Rohit Raj, Medium, 02/03/2022
    * [[https://medium.com/thrive-in-ai/classification-algorithms-in-python-5f58a7a27b88|Classification Algorithms in Python]] Rohit Raj, Medium, 15/03/2022
  * Applications (suggestions, examples,...)
    * [[https://scicomp.stackexchange.com/questions/28195/2d-ising-model-in-python|2D Ising Model in Python]]
  * chemistry
    * misc docs :
      * [[https://pdfs.semanticscholar.org/3161/89e419212794e74bf83442f65de96b5320ba.pdf|Extraction of chemical structures and reactions from the literature]]
    * [[https://medium.com/nerd-for-tech/animating-schrodinger-wave-function-%CF%88-of-a-particle-using-python-with-full-code-5ad9e4852906|Animating Schrodinger Wave Function(ψ) of a Particle Using Python (with full code) - Solving Particle in a Box Using Crank-Nicolson Method]], Kowshik chilamkurthy, Medium, 02/03/2021
  * Blog articles
    * [[https://py.plainenglish.io/linear-regression-in-plain-python-3b20bb56b31d|Simple Linear Regression Explained With Python - Explained in details which are easy to understand]] Arjan de Haan, Medium, Jan 23 (+ [[https://github.com/Vepnar/Script-Collection/blob/master/linear-regression/Linear%20regression.ipynb|jupyter notebook]])
    * simpy : [[https://towardsdatascience.com/simulate-real-life-events-in-python-using-simpy-e6d9152a102f|Simulate Real-life Events in Python Using SimPy]] by Khuyen Tran, May, 2021, Medium, Towards Data Science
    * Active matter simulations ([[https://arxiv.org/pdf/cond-mat/0611743.pdf|Vicsek]], 1995) :
      * [[https://medium.com/swlh/create-your-own-active-matter-simulation-with-python-76fce4a53b6f|Create Your Own Active Matter Simulation (With Python)]] Philip Mocz, The Startup, Medium 2021 + [[https://github.com/pmocz/activematter-python]]
    * Fourier transforms : [[https://towardsdatascience.com/fourier-transforms-an-intuitive-visualisation-ba186c7380ee|Fourier Transforms: An Intuitive Visualisation - An intuitive visualization of discrete Fourier transforms applied to simple time-series data]] Diego Unzueta, Towards Data Science, Medium, 17/09/2021
    * Monte-Carlo simulations : [[Monte Carlo Simulation — a practical guide - A versatile method for parameters estimation. Exemplary implementation in Python programming language]] Robert Kwiatkowski, Medium, 31/01/2022
    * régression logistique : [[https://mlu-explain.github.io/logistic-regression/]]
    * [[https://towardsdatascience.com/cubic-splines-the-ultimate-regression-model-bd51a9cf396d|Cubic Splines: The Ultimate Regression Model - Why cubic splines are the best regression model out there]] Brendan Artley, Medium, 27/07/2022








==== Books ====
  * [[http://www.springer.com/statistics/computational+statistics/book/978-3-642-28069-6|Solving Differential Equations in R]], chez Springer, et  en version électronique sur [[http://www.springerlink.com/content/978-3-642-28070-2#section=1080286&page=1|SpringerLink]]
  * [[http://www.cambridge.org/us/academic/subjects/engineering/engineering-mathematics-and-programming/numerical-methods-engineering-python-3-3rd-edition|Numerical Methods in Engineering with Python 3]] 3rd Edition, Jaan Kiusalaas, 2013, Cambridge University Press, isbn: 9781107033856
  * ...

==== Jupyter notebooks ====
  * [[https://github.com/mandli/intro-numerical-methods|Jupyter notebooks and other materials developed for the Columbia course APMA 4300]]
  * [[http://nbviewer.jupyter.org/url/norvig.com/ipython/Probability.ipynb|A Concrete Introduction to Probability (using Python)]]

==== MOOCs ====
  * [[https://openedx.seas.gwu.edu/courses/course-v1:MAE+MAE6286+2017/about|Practical Numerical Methods with Python]]

==== Miscellaneous ====
  * [[https://github.com/numerical-mooc/numerical-mooc|Practical Numerical Methods with Python]]
  * [[https://github.com/cfgnunes/numerical-methods-python]] (?)
  * [[https://medium.com/@mjspeck/presenting-code-using-jupyter-notebook-slides-a8a3c3b59d67|Presenting Code Using Jupyter Notebook Slides]]
  * [[https://choosealicense.com]]
  * [[https://creativecommons.org/choose/?lang=en]]
  * Published applications
    * [[http://pubs.acs.org/doi/10.1021/acs.jchemed.7b00395|Introduction to Stochastic Simulations for Chemical and Physical Processes: Principles and Applications]] Charles J. Weiss, Journal of Chemical Education 2017 94 (12), 1904-1910 DOI: 10.1021/acs.jchemed.7b00395
  * [[https://medium.freecodecamp.org/an-overview-of-the-gradient-descent-algorithm-8645c9e4de1e|An overview of the Gradient Descent algorithm]]
  * [[https://towardsdatascience.com/building-a-logistic-regression-in-python-step-by-step-becd4d56c9c8|Building A Logistic Regression in Python, Step by Step]]
  * [[https://medium.com/swlh/building-linear-regression-in-python-75a429b0d3ba|Building Linear Regression in Python]]
  * [[https://towardsdatascience.com/17-types-of-similarity-and-dissimilarity-measures-used-in-data-science-3eb914d2681|17 types of similarity and dissimilarity measures used in data science]] : explains various methods for computing distances and showing their instances in our daily lives. Additionally, it will introduce you to the pydist2 package. Mahmoud Harmouch, Medium, 14/03/2021
  * [[http://jakevdp.github.io/|Pythonic Perambulations]] de Jake VanderPlas




----

===== Synopsis (français) =====
(Méthodes de calcul appliqué à la chimie)

  * Learning outcomes teaching unit (UE):
    * Apply standard numerical methods or existing software to solve numerical problems related to scientific research activities
    * Be active in the search for existing numerical methods adapted to problems encountered by chemists
  * Content of the UE:
    * widespread methods : linear systems, numerical integration, root findings
    * Ordinary differential equations (numerical solutions, kinetic applications,...)
    * Partial differential equations (finite differences, diffusion problems)
    * Nonlinear systems of equations (Newton-Raphson method)
    * Eigenvalues eignevectors problems ​​(applications to relaxation and population problems)
    * Approximation by linear and non-linear least squares methods (maximum likelihood, application to deconvolution)
    * Chebyshev approximation
    * Molecular modeling and visualization
    * Minimization and conformational problems
  * Prerequisite skills
    * Basic knowledge of a programming language
    * Basics of Mathematics
  * Exercises and applications: codes written in or mainly written in Python, with the general libraries matplotlib, numpy, scipy, pandas as well as other specialized libraries, especially in chemistry
  * Types of evaluations: Oral examination based on an in-depth study on one of the chapters of the course or an additional theme

  * Acquis d'apprentissage UE :
    * Appliquer des méthodes numériques standards ou des logiciels existant pour résoudre des problèmes fondamentaux ou annexes, liés à des activités de recherche scientifique
    * Être actif dans la recherche de méthodes de résolution numérique existantes et adaptées à des problèmes auxquels les chimistes sont confrontés
  * Contenu de l'UE :
    * Équations différentielles ordinaires (résolutions numériques et applications cinétiques)
    * Équations aux dérivées partielles (différences finies, problèmes de diffusion)
    * Systèmes d’équations non linéaires (méthode de Newton-Raphson)
    * Problèmes aux valeurs propres (applications à des problèmes de relaxation et de population)
    * Approximation par moindre carrés linéaires et non-linéaires (application à la déconvolution)
    * Approximation de Tchébyshev
    * Modélisation et visualisation de molécules
    * Minimisation et problèmes conformationnels
  * Compétences préalables
    * Connaissance de base d'un langage de programmation
    * Bases des mathématiques
  * Exercices et applications : codes écrits ou à écrire principalement en Python, avec les librairies générales matplotlib, numpy et scipy, ainsi que d'autres librairies spécialisées, notamment en chimie
  * Types d'évaluations : Examen oral sur base d'un travail approfondi sur un des chapitres du cours ou un thème additionnel

==== Pré-requis mathématiques ====

==== Base de la programmation ====

==== Méthodes numériques de base ====
  * [[system_of_linear_equations|Systèmes d'équations linéaires]]
    * Diagonalisation et triangularisation
    * Décomposition LU en matrices triangulaires
  * [[numerical_integration|Intégration numérique]]
    * Simpson et quadratures gaussiennes
  * [[root-finding_algorithm|Résolutions d'équations du type f(x) = 0]]
    * Équations polynomiales
    * Recherche dichotomique
    * Méthode de la sécante
    * Méthode de Newton-raphson

==== Méthodes numériques usuelles ====
=== Équations différentielles ordinaires ===

[[numerical_methods_for_ordinary_differential_equations|Résolutions numériques des ODE]]
  * principe de discrétisation, méthode d'Euler
  * Améliorations et méthodes de Runge-Kutta
  * Runge-Kutta d'ordre 4
  * Contrôle du pas d'intégration
  * Méthodes predictor-corrector
  * Méthodes d'extrapolation (Richardson, Burlish-Stoer)
  * applications :
    * équations de cinétique chimique
    * Équation logistique
      * [[wp>fr:Modèle_de_Verhulst|Modèle de Verhulst]], [[wp>fr:Dynamique_des_populations|dynamique des populations]] et non-linéarité
      * Bifurcation, doublements de périodes et transition vers le chaos
    * Réactions chimiques oscillantes : Belousov-Zhabotinsky, Brusselator, Oregonator
    * Modèle proie-prédateur
    * Attracteur étrange, modèle atmosphérique de Lorenz

=== Équations aux dérivées partielles ===
[[partial_differential_equation|Résolutions numériques des équations aux dérivées partielles]]
  * Domaine d'application des équations : équation de diffusion, équation d'ondes, équations de Navier-Stokes
  * Types de traitements numériques
  * Différences finies et problèmes de diffusion
  * Schémas classiques de différences finies
    * Résolutions stationnaires
    * Résolutions dépendantes du temps
  * Méthodes explicites, critère de (ou d'in)stabilités et méthodes implicites


=== Problèmes aux valeurs propres ===
[[eigenvalues_and_eigenvectors|Valeurs propres et vecteurs propres]]

applications à des problèmes de relaxation et de population, analyse de modes normaux de vibration, PCA (principal component analysis),...

=== Systèmes d’équations non linéaires ===
Méthode de Newton-Raphson


=== Approximation par moindre carrés linéaires et non-linéaires ===
application à la déconvolution

=== Approximations de Tchébyshev ===

=== Modélisation et visualisation de molécules ===

=== Minimisation ===
problèmes conformationnels

----

===== Thèmes additionnels =====
  * Bioinformatique et algorithmes spécifiques
  * Chimie
    * calculs quantiques, de minimisation, de mécanique moléculaire
    * représentations
  * Data science, statistiques (librairie Python Pandas,...)
    * Time series analysis
    * Machine learning (Scikit-learn,...)
  * Data visualization
    * boxplot, 3D, animations, graphes,...
  * Senseurs et interfaçage, Arduino, Raspberry Pi, IoT
  * Simulations
    * Agent base modelling et systèmes complexes
    * Automates cellulaires
    * Simpy,...
  * Traitement d'image
    * particle tracking,...