Introduction to Numerical Simulation Methods

0. Introduction in Introduction

What is the common base of knowledge for our graduate school?

That's the problem.

この授業を組み立てるに当たっての悩みであり解決はしていない。

共通科目の難しさ

• 大学院修士課程の改組(2016) ⇒ 医工農総合教育部・工学専攻
  • 教育部・専攻の共通科目をつくるように（と文科省から暗に指導）
  • （日本人は？）総合・融合が好き？
  • 年配者の使命と思い、専攻共通科目を引き受けた
• 学部改組(2012)の前に社会科学系を含む総合的な学科「循環システム工学科」にいたので、いわゆる文系向けのIT実習を引き受けたこともあった。その経験は活かせると思う。
• 基本的には、「融合」はその必然性があるテーマ（研究対象）において起こる。
• 「融合」それ自体に重要な価値があるわけではない。
• 異分野人との「対話」、「協力」は常に重要である。相手の用語や考え方を理解し、それぞれの専門の立場から協力してことに当たる能力は重要
  • 必要になったとき専門外だったことを学ぶ力も

Numerical experiment (computer simulation) methods or data sciences (data analysis methods)

現時点ならば、後者を選択したいが、4年前はそうではなかった。前者を選択した。

豊木は今年度末で定年（つまり、今年度が最後の授業）なので、data scienceの教材をつくる気力がちょっと前まで湧いていなかったが、実行することとした。

事前アンケート結果

https://moodle.yamanashi.ac.jp/2020/mod/feedback/analysis.php?id=26477

受講生数の推移

この授業の方法

多様なスキル、履修履歴の学生がいるので、

• 話は短く
• できるだけ多くのドキュメントとプログラム例を用意
• 提示したプログラムを実行しながら理解を深める
  • 各自の経験・スキルに応じて、プログラムの改変、独自のプログラム作成、別のデータへの適用などを行ってみる
• 個々の質問に答える
• 最終的に、試したこと(Jupyter Notebook形式あるいはPDF形式）で提出
• 個々の質問は、当面、zoomのチャットで。(moodleのチャットの方が便利ならそちらに移行するが。）
  • 小さな質問でも遠慮なく。
  • 質問への回答ができる人がいたら、回答も遠慮なく。

Numerical Experiments

In natural sciences and engineering as the use of scientific principles,

• Modeling on the basis of known simple principles or theoretical hypothesis (conjecture)
• Comparison with experiments
• Result → New principles, formula, laws and extensible prediction methods

Data Analysis

• Objects not necessarily governed by simple principles (Causality logic is not always clear but often left as a black box.)

  例: 様々な指標　→ スポーツ選手の評価(活躍予測)

• Statistics and Machine Learning as theoretical scheme and methods
  • (needless to say,) those mathematics is excellent and continues to develop
• Results → prediction scheme and software 　

1. The importance of computer simulation (from ¹) (in physics and related areas of natural science)

自然科学、工学におけるコンピュータ利用は多岐にわたるが、まずは、物理系の自然科学とそれに近い工学分野での「コンピュータシミュレーション」の位置づけを概観する。

"physics and related sciences" :

• Various phenomena can be understood by a small number of principles.
• Aim of research is to seek those principles by use of experiments and mathematical formulations (analytical procedure).

However,

• Problems difficult to be understood analytically　（理論的には(つまり、手計算では)解が求められない問題がたくさんある。）
  • Most of dynamics (spatio-tempral behavior) of physical objects are described by differential equations. (Newton's eq., Maxwell's eq., Schroedinger eq., Navier-Stokes eq., etc.)
  • Because many natural systems are nonlinear and have many degrees of freedom, we cannot solve them analytically. （非線形、多自由度）
• Computer simulation as an alternative experimental method
  • Computer simulations are sometimes referred as computer experiments because they share much in common with laboratory experiments.
  • The results of a computer simulation can be served as a bridge between laboratory experiments and theoretical calculations.　（実験と理論を結ぶ架け橋）

    Laboratory Experiment	Computer Simulation
    sample	model
    physical apparatus	computer program
    calibration	testing of program
    measurement	computation
    data analysis	data analysis

Computer simulations, like laboratory experiments, are not substitutes for thinking, but are tools that we can use to understand natural phenomena (including manufactured instruments).

コンピュータシミュレーションは、実験室での実験と同様、思考の代替物ではなく、現象を理解するための道具である。

2. Numerical modeling methods

(i) First-principle calculation

For systems described by a few fundamental principle such as Newton's law and quantum mechanics but including too many variables for our computers , we develop effective approximation schemes.

In the class, we will try to enrich the above table by discussions.

(ii) Phenomenological models

There have been investigated some simulation models whose scheme itself is created as a model (equations). Some examples are shown in the following.

(1) Traffic flow model

See this page of "Introductory E. E. experiments" of dept. EE.

cf.

An example and a trial for data analysis in My Lab.

(2) Chemical Reaction-Diffusion phenomena

Growing spatio-temporal macroscopic pattern is a remarkable phenomenon in non-equilibrium processes. The Belousov-Zhabotinsky reaction is a well-known example which yields temporal rhythms and dynamical spiral patterns.

• Oregonator: a welknown numerical equation for BZ reaction( brief review based on Ohta's text² is here(in Japanese).)

We will try to make a simulation program of the two-dimensional Oregonator. The goal is to visualize the spatiotemporal evolution.

movie

(3) Numerical simulations for social phenomena

Mathematical model for social network such as SNS, ecomomical phenomena have been extensively researched.

(iii) Data analysis (anothor crutial role of computer in science)

In phenomena we want to analyse, there are a lot of things which is not the result of a simple principle but one caused by many factors. While the important thing in the above computer simulation is to find relatively simple principles in objective phenomena, the data analysis is used for such as stock price prediction which is intrinsically affected by many factors.

Deep lerarning is a most remarkable developing method.

e.g.

• AlphaGo
• IBM Watson (e.g. see http://healthpress.jp/2016/08/ai-10ibm-watson_2.html)

(iv) Simulation for artificial (engineering) systems

Other usage of computers

Extension of human abilities: Senses, Thinking and Working

• Controlling engineering systems
• Electrical sensing
• Telecommunication

3. python and development tools

Python

In 2018 and later, the lecture is given based on other materials.

A goal of the data analysis in this lecture

We will learn how to analyze some data such as the following. 次のようなデータがあったとします。xとyの関係をどのように推定すればよいでしょうか。

• What methods should we choose?
• What assumptions should we set? Fitting function?, initial parameters?
• How should we estimate precision or likelihood?

source ipynb for the above figure

それよりも、各自の経験にこれまでに身に着けたスキルに応じて、python及びそのモジュールの利用やデータの扱いの方法について何らかの進歩があれば良いと思っている。 I recommend that you set your own goal on the usage of python and modules of python, and data analysis methods based on your own previous experiences and skills.

Return the homepage and see the next items.

(The following part is obsolete.)

We use a interpreter-type language Python which have many libraries for scientific calculation and data analysis. The libraries are rapidly expanded and improved. Those are mostly witten by C and Fortran language, so Python can be regarded as the front-end for users.

(At the first course, we learn python by the following tutorial page in the classroom.)

python and its libraries for scientific use

A useful tutorial for scientific use is available at http://www.scipy-lectures.org/,

and the Japanese translation is http://www.turbare.net/transl/scipy-lecture-notes/index.html

official documents of python

• English: https://docs.python.org/3.8.3/
• Japanese: http://docs.python.jp/3/

Development tools

Anaconda, one of major packages including python development tools and libraries, is installed in the classroom computers.

There are three styles of developement environment:

• ipython console + multi-purpose editor, such as notepad++, emacs, vim and so on.
• integrated developement enviroment: such as Spyder in Anaconda.
• jupyter: ipython notebook (You can make a composite of textbook and codes, so I will mainly use this in this course.)

There are many IDEs. Please try to search them on Web and use some IDEs. (e.g. see http://noeticforce.com/best-python-ide-for-programmers-windows-and-mac )

console: ipython console

At the compter room, we can run "ipython console" from the windows menu "Anaconda3" or "Anaconda2".

jupyter (ipython notebook)

The certain part of the text of this lecture is created by "jupyter notebook" formally called "Ipython notebook" and some instructions and examples are provided as "ipython notebook" file whose name has the extension ".ipynb".

"jupyter" also is in the Anaconda menu.

You can find many tutorial pages, e.g. https://ai-inter1.com/jupyter-notebook/