Introduction to Computational Methods in Science and Engineering

https://webclass.cdel.uec.ac.jp

Computational methods have replaced analytical methods already
in many fields of science and engineering, and their importance
is still increasing. The aim of the lecture is to provide
fundamental criteria for the choice of numerical methods, give an
overview about some available methods in some fields,
and give ideas about performance-oriented implementation
for such methods. Depending on the background and interest
of the auditory, some topics may be subject to changes.

First year Analysis and Linear Algebra, one procedural Programming Language

無し

Script can be downloaded from http://webclass.cdel.uec.ac.jp/,
further reading:
A. L. Garcia, Numerical Methods for Physics, Benjamin-Cummings Pub Co,1999
G.J. Borse: Numerical Methods with Matlab, International Thomson Publishing, 1997

In the first half of the semester, the lectures on general topics will be cover the lesson time (90 minutes).
In the second half, every student will get a project (topic will be discussed between students and lecturer), and the first half of the lesson will be
lecture on more specific and specialized topics, and the second half will be dedicated
to the completion of the project.

1. Introduction
Interpreters and Compilers, basic MATLAB syntax, interacting with the operating system
2. More advanced Syntax
Implicit loops, vector- and matrix commands
3. Stochastic Methods I a) Random numbers and direct Monte Carlo
Averages and Variance; Computing Pi with random numbers and the power of Monte Carlo Methods for problems of arbitrary dimension
4. Stochastic Methods I b) Modeling
Producing test data, Modeling
5. Numerical analysis I
Why bother about errors; integer vs. floating point numbers, precision and rounding errors; Truncation error and strategies to reduce it
6. Graphics I 2D- and 3D-plots
Basic plotting functions and not so basic methods of manipulating the graphs
7. Graphics II More complex
Surfaces, overlaying graphics and textures, transparency alpha; From animated graphics to making movies
-> End of first part
8. Linear Algebra I: From implicit loops to vectors and matrices
How many matrix products are there, Performance and loop ordering; Norms, Matrix inversion and other matrix commands for linear algebra
9. Linear Algebra II
Eigenvalue decomposition, Determinants, Landau-Order symbol for computational effort / complexity
10. Linear Algebra III: Non-square matrices
Least squares fitting, singular value decomposition, condition number; Overfitting and Underfitting; Difference between fitting and interpolation
11. Stochastic Methods II: Spin Systems
From Magnets to Spin systems: Frustration and physics problem with no good solution: Spin glasses, ground states, thermodynamics weights: Form importance Sampling Monte Carlo to Simulated Annealing at zero and finite Temperature
12. Stochastic Methods III: Neural networks as a foot note to spin glasses
From infinite range spin glasses to nerve systems; Pattern recognition with Neural Networks; fast Fourier Transform and convoluting the input; the incremental advances from Neural Networks to Deep learning
13. Numerical Analysis II a) Types of numerical ordinary differential Equations
Symplectic, non-stiff and stiff ODEs; standard methods with constant step size
14. Numerical Analysis II b) Types of numerical ordinary differential Equations
From constant step size to variable step size

There will always be short programming examples during the lecture, so students should always have their MATLAB-environment ready for use.

Exercises will be given as homework.

20% Participation (including punctuality) and activity (asking meaningful questions and giving meaningful answers when asked) in the Lecture
80% Homework exercises

Contact me by E-Mail and we organise date and time at the earliest possible moment.

A craving to work with "Object orientation" has derailed several students in 2022 in this course: This course is about algorithms (mostly floating point arrays), and the structuring and packaging of the data is irrelevant. You have to work out and implement algorithms, not play around with data structures in this course.

The lecture starts after the the introduction to the computer system in the JUSST-Program has been held.